---
abstract: |
    Linear polymers adsorbing on a wall can be modelled by half-space self-avoiding walks adsorbing on a line in the square lattice, or on a surface in the cubic lattice. In this paper a numerical approach based on the GAS algorithm is used to approximately enumerate states in the partition function of this model. The data are used to approximate the free energy in the model, from which estimates of the location of the critical point and crossover exponents are made. The critical point is found to be located at $$a_c^+ =
    \cases{
    1.779 \pm 0.003, & \hbox{in the square lattice}; \\
    1.306 \pm 0.007, & \hbox{in the cubic lattice}.
    }$$ These results are then used to estimate the crossover exponent $\phi$ associated with the adsorption transition, giving $$\phi =
    \cases{
    0.496 \pm 0.009, & \hbox{in two dimensions}; \\
    0.505 \pm 0.006, & \hbox{in three dimensions}.
    }$$ In addition, the scaling of these thermodynamic quantities is examined using the numerical data, including the scaling of metric quantities, and the partition and generating functions. In all cases results and numerical values of exponents were obtained which are consistent with estimates in the literature.
address: '$^1$Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada\'
author:
- 'E.J. Janse van Rensburg$^1$'
bibliography:
- 'References.bib'
title: 'Microcanonical Simulations of Adsorbing Self-Avoiding Walks'
---

Introduction {#section1}
============

The adsorption of a linear polymer on an attractive surface is a conformational rearrangement of the polymer to a state where it explores conformations which remain near or on the surface. This is the so-called polymer adsorption transition, and the many models of this phenomenon (see, for example, reference [@JvR15]) remain a rich source of mathematical and numerical studies. Polymer adsorption is a phase transition [@BDG83], and the properties of the adsorbed polymer have been examined both experimentally (see, for example, references [@DR71; @BDD95; @MBM14]) and theoretically [@deG79]. These models include directed path models of adsorbing polymers [@PFF88; @W98], as well as self-avoiding walk models [@HTW82], and these have received considerable attention in the literature [@HG94], using both rigorous methods [@RW11] and numerical methods (for example, the Monte Carlo simulation of adsorbing self-avoiding walks [@JvRR04]).

In this study, a new Monte Carlo method for sampling adsorbing self-avoiding walks is proposed and implemented. In particular, the GAS algorithm [@JvRR09] is generalised to sample adsorbing walks in the microcanonical ensemble, and the data obtained are analysed to estimate the locations of critical points, and the values of critical exponents and scaling of adsorbing walks, in the square and cubic lattices. The algorithm is related to the Rosenbluth algorithm [@RR55] and to the GARM algoritm [@RJvR08]. The GARM algorithm is related to the PERM algorithm [@G97; @HG11], and flat histogram implementations of PERM [@PK04] have been used to sample states from a flat histogram over state space in a variety of different models of interacting walks, including collapsing walks [@PO00] and adsorbing walks [@KPOL04].

figure1-.tex

The GAS algorithm was introduced in reference [@JvRR09] and was used in several studies as an algorithm to approximately enumerate walks or polygons [@JvRR11; @JvRR11A]. However, it was not clear how to generalise the algorithm to sample states in models of interacting walks. In this paper our purpose is (1) to generalise the algorithm to a model of adsorbing walks, (2) to examine the behaviour of the algorithm by computing critical points and exponents of the walk, and to compare this to results found elsewhere, and (3) to use our data to examine scaling in adsorbing walks by computing critical exponents and examining the scaling of thermodynamic functions.

Adsorbing walks {#subsection1.2}
---------------

An adsorbing self-avoiding walk in the square lattice is illustrated in figure \[figure0\]. Let $\mathL^d$ denote the $d$-dimensional hypercubic lattice and denote a unit length edge with endpoints $\vec{x}$ and $\vec{y}$ in $\mathL^d$ by $\edge{\vec{x}}{\vec{y}}$. The hypercubic half-lattice $\mathL^d_+$ is defined by $$\mathL^d_+ = 
\{ \edge{\vec{x}}{\vec{y}} \in \mathL^d \vv 
\hbox{$\vec{x}(d) \geq 0$ and $\vec{y}(d) \geq 0$} \},$$ where $\vec{x}(d)$ is the $d$-th Cartesian component of $\vec{x}$, and $\vec{y}(d)$ is the $d$-th Cartesian component of $\vec{y}$. The boundary of $\mathL^d_+$ is given by $$\partial\mathL^d_+ = 
\{ \edge{\vec{x}}{\vec{y}} \in \mathL^d_+ \vv 
\hbox{$\vec{x}(d)=0$ and $\vec{y}(d)=0$} \},$$ and it is isomorphic to $\mathL^{d-1}$ if $d\geq 2$. Please note that these definitions, and the definitions of additional functions and quantities, are listed in table \[Defs\].

The number of self-avoiding walks of length $n$ from the origin in $\mathL^d$ is denoted by $c_n$. The *growth constant* of the self-avoiding walk [@H57; @H60] is defined by the limit $$\lim_{n\to\infty} c_n^{1/n} = \mu_d,$$ and $\kappa_d = \log \mu_d$ is the connective constant of the self-avoiding walk. This shows that $c_n = \mu_d^{n+o(n)}$. Self-avoiding walks from the origin in $\mathL^d_+$ are *positive walks*. The number of positive walks of length $n$ is denoted by $c_n^+$, and it is known that $c_n^+ = \mu_d^{n+o(n)}$ [@HTW82].

If positive walks are counted with respect to the number of vertices in $\partial\mathL^d_+$ (these are *visits*), then the walks are *adsorbing walks*. For example, the walk in figure \[figure0\] is an adsorbing walk with $6$ visits.

Let $c_n^+(v)$ be the number of adsorbing walks of length $n$ from the origin in $\mathL^d_+$, with $v$ visits to $\partial \mathL^d_+$. The canonical partition function of adsorbing walks is obtained by introducing an activity $a$ conjugate to the number of visits: $$Z_n(a) = \sum_{v=0}^n c_n^+(v)\,a^v  .
\label{eqnZ}   $$ When $a$ is large, then $Z_n(a)$ is dominated by walks with a large number of visits, and if $a$ is small (but positive), then $Z_n(a)$ is dominated by walks with a small number of visits.

The *finite size free energy* of these models is computed from the partition function $Z_n(a)$ (see equation ), and is given by $$\C{F}_n(a) = \Sfrac{1}{n} \log \sum_v c_n^+(v)\, a^v .
\label{eqn23}   $$ The *limiting free energy* of the model is given by the thermodynamic limit in the model: $$\C{F}(a) = \lim_{n\to\infty} \Sfrac{1}{n} \log \sum_v c_n^+(v)\, a^v .
\label{eqn25F}  $$ This limit exists (see reference [@HTW82], and also, for example, reference [[@JvR15]]{}), and it is a convex function of $\log a$ with a singular point at $a=a_c^+$ (which is the *adsorption critical point* in the model). For $a<a_c^+$ the model is in a *desorbed state*, and for $a>a_c^+$ the model is in an *adsorbed state*. It is known that $a_c^+>1$ [@JvR98], and $a_c^+ < \sfrac{\mu_d}{\mu_{d-1}}$ [@HTW82], and $$\C{F}(a) \cases{
= \log \mu_d, & if $a\leq a_c^+$; \\
> \log \mu_d, & if $a>a_c^+$.
}$$ Since $\C{F}(a)$ is a convex function of $\log a$, it is differentiable for almost all $a>0$, and it follows that the density of visits to the adsorbing plane is $a\sfrac{d}{da}\C{F}(a) = 0$ if $a<a_c^+$ (this is the *desorbed phase*), and $a\sfrac{d}{da} \C{F}(a) >0$ if $a>a_c^+$ $\alsu$ (whenever $\C{F}(a)$ is differentiable). This is the *adsorbed phase*. In the desorbed phase the walk tends to make few returns to the adsorbing plane (walks of length $n$ will return, on average, $o(n)$ times to the adsorbing plane in the desorbed phase). Thus, a desorbed walk will tend to move away from the boundary into the bulk of $\mathL^d_+$. An adsorbed walk, on the other hand, is expected to have a positive density of returns to the adsorbing plane. This implies that the walk will remain near the adsorbing plane, and so have the properties of a walk which is stretched out in $(d\minus 1)$ dimensions near $\partial\mathL^d_+$ (and compressed in the $d$-th dimension). Separating these two regimes is the adsorption critical point $a_c^+$.

The finite size scaling of the free energy $\C{F}_n(a)$ is given by $$\C{F}_n(a) \simeq \log \mu_d + (a\minus a_c^+)^{2-\alpha}\,f(n^\phi(a\minus a_c^+)),
\label{eqn24}  $$ where $f$ is a scaling function, $\alpha$ is the specific heat exponent, and $\phi$ is the finite size crossover exponent. The exponents $\alpha$ and $\phi$ are related by the hyperscaling relation $$2-\alpha=\Sfrac{1}{\phi} .
\label{eqn25}  $$

The bulk entropy contribution to $\C{F}_n(a)$ is $\log\mu_d$ in $\mathL^d_+$ (where $\mu_d$ is the growth constant of self-avoiding walks in the square lattice). Slightly redefining the scaling function, it is found that $$\C{F}_n(a) \simeq \log \mu_d + \Sfrac{1}{n}\, g(n^\phi (a\minus a_c^+)).
\label{eqn26}  $$ By plotting $n(\C{F}_n(a) \minus \log\mu_d)$ against $n^\phi(a\minus a_c^+)$, the function $g$ can be uncovered (for $n$ large and $|a\minus a_c^+|$ small).

Taking derivatives of $\C{F}_n(a)$ to $\log a$ gives the energy (density) $\C{E}_n(a)$ and specific heat $\C{C}_n(a)$ of the model. The scaling of these quantities follows directly from equation : $$\C{E}_n(a) \simeq n^{\phi-1}\,h_e(n^\phi(a\minus a_c^+)),
\q \hbox{and}\q
\C{C}_n(a) \simeq n^{\alpha\phi}\,h_c(n^\phi(a\minus a_c^+)),
\label{eqn24EC}  $$ for some scaling functions $h_e$ and $h_c$. In the limit as $n\to\infty$, $\C{E}_n(a) \to \C{E}(a)$ (the limiting energy density) and $\C{C}_n(a)\to\C{C}(a)$ (the limiting specific heat). Existence of these limits (almost everywhere) is a consequence of the convexity properties of the limiting free energy (see for example reference [@JvR15]). Physically, $\C{E}(a)$ is the density of visits per unit length, and $\C{C}(a)$ is the rate of change in $\C{E}(a)$ as a function of changes in $\log a$ (it has a maximum at $a_c^+$).

For adsorbing walks it is thought that $\phi=\sfrac{1}{2}$ in all dimensions $d\geq 2$ [@BEG89; @BY95], and numerical evidence supporting this in dimensions lower than $d=4$ (the upper critical dimension) are available in references [@LM88A; @ML88A; @BWO99; @JvRR04; @KPHMBS13]. If $\phi = \sfrac{1}{2}$, then $\alpha = 0$, so, for example, the specific heat has scaling $\C{C}_n(a) = h_c(n^\phi(a\minus a_c^+))$, and plotting measurements of $\C{C}_n(a)$ against the rescaled variable $\tau=n^\phi(a\minus a_c^+)$ for small values of $\tau$ should collapse the curves to a limiting curve (with some finite size corrections to scaling), exposing the scaling function $h_c$.

The partition function has a more complicated scaling law (see, for example, equation (23) in reference [@JvRR04], or section 4.2.2 in reference [@JvR15]). In the high temperature (or small $a$) regime, the partition function scales as $$Z_n(a) \simeq B_\lambda n^{\gamma_t -1} h_\lambda(n^\phi |t|)\,
\kappa_-^{n\,|t|^{1/\phi}},\q
\hbox{if $a< a_c^+$},
\label{eqn3}   $$ where $t = (a\minus a_c^+)$ and $\lambda$ denotes the high temperature regime, and where $h_\lambda(x) \simeq |x|^{(\gamma_1-\gamma_t)/\phi}$ and $\log \kappa_- \simeq |a\minus a_c^+|^{-1/\phi}\log \mu_d$. Putting $a=1$, for example, and adsorbing constants and functions of $t$ into $B_\lambda$, give $$Z_n(1) \simeq B_\lambda n^{\gamma_1-1} \mu_d^n, 
\q\hbox{since $\C{F}(a) = \log \mu_d$ if $a<a_c^+$},$$ and the exponent $\gamma_1$ is the entropic exponent of half-space walks, namely $c_n^+ \sim n^{\gamma_1-1} \mu_d^n$.

At the critical adsorption point $a_c^+$, the above scaling is modified to $$Z_n(a_c^+) \simeq B_c n^{\gamma_t -1}\,\mu_d^n ,\q
\hbox{if $a= a_c^+$},
\label{eqn15}  $$ where $\gamma_t$ is the entropic exponent associated with adsorbing walks at the critical adsorption point. The ensemble of half-space walks at the critical point has associated *surface entropic exponent* $\gamma_s$, and this is related to $\gamma_t$ by $\gamma_t
= \gamma_s$ (see for example section 9.1.3 in reference [@JvR15]).

The scaling in the adsorbed phase is similar to the above, but now with different exponents $$Z_n(a) \simeq B_{\tau_0} n^{\gamma_t -1} h_{\taus0} (n^\phi |t|)\,
\kappa_+^{n\,|t|^{1/\phi}},\q
\hbox{if $a> a_c^+$},$$ where $h_{\taus0} (x) \simeq |x|^{(\gamma_+ -\gamma_t)/\phi}$, and where the subscript $\taus0$ denotes the low temperature (and large $a$) scaling. Since $\sfrac{1}{n} \log Z_n(a) = \C{F}(a)\,(1\plus o(1))$, it follows that $\log \kappa_+  \simeq |a\minus a_c^+|^{-1/\phi} \C{F}(a)$. This, in particular, gives the scaling $$\C{F}_s(a) \sim |a \minus a_c^+|^{1/\phi}$$ for the singular part of the free energy in the adsorbed phase (consistent with the hyperscaling relation ). The scaling of $Z_n(a)$ simplifies here to $$Z_n(a) \simeq B_{\tau_0} n^{\gamma_+-1} \; e^{n\thin\C{F}(a)} ,\q
\hbox{if $a>a_c^+$}.$$ The exponent $\gamma_+$ should be that of adsorbed walks, and so given by $\gamma_+ = \gamma^{(d-1)}$, the entropic exponent of walks in one dimension lower.

  Function                   Definition
  -------------------------- ------------------------------------------------------------------------------------------
  $\mathL^d$                 The hypercubic lattice
  $\mathL^d_+$               The half-hypercubic lattice
  $\partial\mathL^d_+$       The boundary of $\mathL^d_+$ (it is isomorphic is $\mathL^{d-1}$)
  $\mu_d$                    The growth constant of self-avoiding walks in $\mathL^d$
  $c_n^+(v)$                 The number of positive walks of length $n$ and $v$ visits from $\vec{0}$ in $\mathL^d_+$
  $Z_n(a)$                   Partition function of positive adsorbing walks of length $n$ and activity $a$
  $\C{F}_n(a)$               Finite size free energy: $\displaystyle F_n(a) = \sfrac{1}{n} \log Z_n(a)$
  $\C{F}(a)$                 Limiting free energy: $\displaystyle \C{F}(a) = \lim_{n\to\infty} \C{F}_n(a)$
  $\C{E}_n(a)$, $\C{E}(a)$   Energy density and limiting energy density
  $\C{C}_n(a)$, $\C{C}(a)$   Specific heat and limiting specific heat
  $H_n$                      The mean height of the endpoint of the walk (a function of $a$)
  $R_n^2$                    The mean square radius of gyration of the walk (a function of $a$)
  $P^+(\eps)$                The microcanonical density function (see equation )
  $a_c^+$                    The adsorption critical point
  $\phi$                     The crossover exponent
  $\alpha$                   The specific heat exponent
  $\gamma_1$                 The half-space entropic exponent, see also $\gamma_s$ (the surface exponent)
  $\nu$                      The metric exponent
  $G(a,t)$                   Generating function of $Z_n(a)$ (see equation )
  $G_N(a,t)$                 Truncated generating function (see equation )
  $t_c^+(a)$                 The radius of convergence of $G(a,t)$

  : Short list of definitions[]{data-label="Defs"}

Organisation of the manuscript
------------------------------

This paper is a report on two aspects of the Monte Carlo simulation of adsorbing walks. The first is the generalisation of the GAS algorithm to a model of interacting walks, and in particular, an implementation of this algorithm to achieve flat histogram sampling over state space of adsorbing square and cubic lattice walks. The second aspect of the paper is a report on the properties of adsorbing self-avoiding walks in the square and cubic lattices. The aim here is to verify the results obtained in the Multiple Markov Chain Monte Carlo study in reference [@JvRR04], and also to use the data generated here to test the scaling of the thermodynamic and metric quantities of adsorbing walks.

The model of adsorbing walks is defined in section \[subsection1.2\], and its partition function and free energy are discussed. The limiting free energy of this model exists, and its properties have been examined elsewhere [@HTW82]; see, for example, reference [@JvR15]. The basic scaling relations for the free energy, energy, and specific heat were introduced above, and the scaling of the partition function was briefly reviewed.

In section \[section2\] the GAS algorithm [@JvRR09] is reviewed and then generalised to interacting models. The algorithm normally has only one set of parameters (associated with the size of the walks), but it is shown here that introducing a second set of parameters (associated with the energy of the walks) can give an algorithm which samples effectively in both length and energy. It is shown that the algorithm can be tuned to give flat histogram sampling in the spirit of the PERM algorithm [@G97; @HG11] (but without using enrichment or pruning of states).

Numerical results for adsorbing walks are analysed in section \[section3\]. The location of the adsorption critical point is determined from the mean energy of adsorbing walks, giving $$a_c^+ =
\cases{
1.779 \pm 0.003, & \hbox{in the square lattice}; \\
1.306 \pm 0.007, & \hbox{in the cubic lattice}.
}$$ These results are then used to estimate the crossover exponent $\phi$ associated with the adsorption transition, giving $$\phi =
\cases{
0.496 \pm 0.009, & \hbox{in two dimensions}; \\
0.505 \pm 0.006, & \hbox{in three dimensions}.
}$$ The microcanonical density function of adsorbing walks is determined as well, and shown to have properties consistent with the location of the critical points above. In addition, the specific heat of the model is determined, and it is found that it has scaling behaviour consistent with the estimates of the critical points above.

It is also shown that the adsorption transition is seen in a change in the metric scaling of walks at the critical point. The desorbed phase is a phase of positive walks with the scaling properties of self-avoiding walks in a good solvent, while the adsorbed walk has the metric properties of a walk in one dimension less; that is, of walks adhering to the adsorbing surface.

Scaling of the generating and partition functions are found to be consistent with the exact values of critical exponents determined elsewhere in two dimensions [@BEG89; @BY95], and with the value $\phi=\shalf$ in three dimensions [@HG94; @JvRR04; @ML88A]. Similary, the data are also consistent with estimates for the surface exponent $\gamma_s$; this is seen particularly in the scaling of the partition function (see equation ).

The paper is concluded with a few brief remarks and a summary in section \[section4\].

figure2-.tex

GAS Sampling of self-avoiding walks {#section2}
===================================

The GAS algorithm is a generalisation of kinetic growth algorithms. It is designed to sample along weighted sequences in state space in such a way that the ratios of average weights of sequences ending in walks of length $n$ and $m$ are estimates of the ratio of the numbers of walks of lengths $n$ and $m$. In this section I show how to generalise this algorithm so that it can be used to estimate the number of walks of length $n$ and *energy* $m$. That is, the algorithm will be used to sample walks in the microcanonical ensemble.

Let $w=\LA \omega_0,\omega_1,\ldots,\omega_n\RA$ be a self-avoiding walk of length $n$ from its source vertex $\omega_0 =\vec{0}$ at the origin, to its terminal vertex $\omega_n$, giving $n$ steps $\LA \omega_{j-1},\omega_j\RA$ for $j=1,2,\ldots,n$. The walk $w$ may be made longer by adding a step $\LA \omega_n,\omega_{n+1}\RA$ to $\omega_n$, or it may be made shorter by removing its last step. These two operations compose an end-point elementary move for sampling self-avoiding walks, as illustrated in figure \[figure1\]: A *positive elementary move* is the addition of an edge to the endpoint of a growing walk. The reverse of a positive elementary move is a *negative elementary move*, namely the deletion of the least edge in a walk. Notice that every positive move is immediately reversible by a negative elementary move.

Endpoint elementary moves have been used widely in the simulation of self-avoiding walks (for example the Rosenbluth algorithm [@RR55], and the Beretti-Sokal algorithm [@BS85]). In what follows the discussion will be restricted to endpoint elementary moves; however, the algorithms generalise directly if other elementary moves, such as BFACF elementary moves [@BF81; @AA83], or generalised atmospheric moves [@RJvR08], are used instead.

figure3-.tex

As an example, consider the elementary move in figure \[figure1\], which is an implementation of an endpoint elementary move on a self-avoiding walk in the positive half-lattice $\mathL^+_2$ (where $\vec{w}(2)$ is the $y$-coordinate of $\vec{w}$). The set of lattice edges in $\mathL^+_2$ which may be appended to the walk $w$ to extend it by one step is the *positive atmosphere* of $w$, and the number of edges in the positive atmosphere is denoted by $a^+(w)$. For example, for the walk on the left in figure \[figure1\], $a^+(w)=2$.

Similarly, the set of edges which may be removed from the endpoint of a walk to decrease its length by one, is the *negative atmosphere* of the walk. For endpoint elementary moves, the last step is always the sole negative atmospheric edge, so that the size of the negative atmosphere for endpoint elementary moves is always $a^-(w)=1$, if $w$ is not the trivial walk of length $0$.

An elementary move may change the energy of a walk. For example, in a model of adsorbing walks in $\mathL^+_2$, the energy is the number of *visits* of the walk to the adsorbing line $\partial\mathL^+_2$ (the boundary of the half-lattice $\mathL^+_2$). The positive move in figure \[figure1\] does not change the number of visits, but the move in figure \[figure2\] increases the number of visits (and so the energy) by $1$. The negative move in figure \[figure2\] similarly reduces the number of visits by $1$.

figure4-.tex

A similar situation arises if a model of collapsing walks with energy given by nearest neighbour *contacts* between vertices in the walk which are adjacent in $\mathL$, but not in the walk. This is illustrated in figure \[figure3\]; the positive elementary move creates $2$ new contacts in the walk, and so it changes the energy of the walk by $2$.

More general elementary moves (for example, the BFACF elementary moves) may contribute to the atmospheric statistics $a^+$ and $a^-$ in various ways, and may even give rise to *neutral atmospheres* which do not change the length of the walk (but which may change the energy of the walk).

Thus, in what follows, let $a^+_v(w)$ be the size of the positive atmosphere of a walk $w$ of length $\ell_n=|w|$, and which changes the energy of $w$ by $v$ units. For example, the walk on the left in figure \[figure1\] has $a^+_{0}(w)=1$ and $a^+_{1}(w)=1$, and if this was a walk in the three dimensional half-lattice $\mathL_3^+$, then $a^+_{0}(w)=3$ and $a^+_{1}(w)=1$. The walk on the left in figure \[figure3\] (with energy given by the number of contacts), has, in a similar way, $a^+_{0} (w)=2$, $a^+_{1}(w)=0$ and $a^+_{2} (w)=1$.

In exactly the same way one may define the size of the negative atmosphere of a walk $w$ of length $\ell$ which changes the energy by $v$, denoted by $a^-_{v} (w)$.

The neutral atmosphere (an elementary move which does not change the length of the walk) of a walk $w$ is similarly given by $a^0_{v} (w)$, if it changes the energy of a walk by $v$. If the endpoint elementary move in figure \[figure1\] is used, then $a^0_{v} (w)=0$ by default (since there are no neutral elementary moves implemented), but in general assume that a more general set of elementary moves (for example BFACF elementary moves) is used to sample walks, and in that case the neutral atmosphere may have positive size.

Implementation of GAS-sampling
------------------------------

Suppose that an elementary move is implementated on the state space of self-avoiding walks from the origin, and assume the implementation is irreducible (that is, the elementary move gives a connected graph on the state space of walks).

Suppose that the sequence $$\phi_N = \LA w_0,w_1,w_2,\ldots,w_n,\ldots,w_N\RA
\label{eqn1}     $$ is realised after $N$ steps and that the atmospheres of the states $w_n$ have sizes (or *statistics*) $a_{v}^+(w_n)$, $a_{v}^0(w_n)$ and $a_{v}^-(w_n)$. These elementary moves may be classified as follows, depending on whether they increase or decrease the lengths of the walks (or are neutral), or whether they increase or decrease, or leave unchanged, the energy of the walk. This is done by defining atmospheric statistics as follows: Let the states $w_n$ be walks of length $\ell_n = |w_n|$. Define $$\begin{aligned}
\alpha_n^{--} = \sum_{v< 0} a_{v}^- (w_n),\qq & 
\alpha_n^{-0} = \sum_{v=0} a_{v}^-(w_n), \qq 
\alpha_n^{-+} = \sum_{v>0} a_{v}^-(w_n); \\
\alpha_n^{0-} = \sum_{v< 0} a_{v}^0 (w_n),\qq & 
\alpha_n^{00} = \sum_{v=0} a_{v}^0(w_n), \qq 
\alpha_n^{0+} = \sum_{v>0} a_{v}^0(w_n); \\
\alpha_n^{+-} = \sum_{v< 0} a_{v}^+ (w_n),\qq & 
\alpha_n^{+0} = \sum_{v=0} a_{v}^+(w_n), \qq 
\alpha_n^{++} = \sum_{v>0} a_{v}^+(w_n).  \end{aligned}$$ For example, $\alpha_n^{--}$ is the number of negative elementary moves which also *decreases* the energy of the walk, and $\alpha_n^{-+}$ is the number of negative elementary moves which also *increases* the energy of the walk. The rest of the $\alpha$’s are similarly defined.

With these atmospheric statistics defined, a rule needs to be constructed in order to realise the sequence $\phi_N$ in equation .

The endpoint elementary moves in figures \[figure2\] and \[figure3\] have the property that no positive elementary move can decrease the energy, and no negative elementary move can increase the energy. Moreover, there are no neutral moves amongst the elementary moves. Thus, assume that $$\alpha_n^{-+} = 0,\q\alpha_n^{+-}=0,\q\alpha_n^{0-}=0,\q\alpha_n^{00}=0 
\q\hbox{and}\; \alpha_n^{0+} = 0.$$ The algorithm can be modified appropriately to account for such transitions in models where this is not the case. This assumption leaves the following atmospheric statistics: $\{\alpha_n^{--},\alpha_n^{-0},\alpha_n^{+0},\alpha_n^{++}\}$. That is, distinguish between negative moves which decrease the energy, or negative moves which leave the energy unchanged, or positive moves which leave the energy unchanged, and positive moves which increase the energy.

Introduce parameters $\{\beta_{\ell,u}\}$ to control positive elementary moves on walks of length $\ell$ and energy $u$, and which leave the energy unchanged (that is, the elementary moves contributing to $\alpha_n^{+0}$). Similarly, introduce the parameters $\{\gamma_{\ell,u}\}$ to control positive elementary moves which increase the energy on walks of length $\ell$ and energy $u$.

The transition probabilities of positive elementary moves which leave the energy unchanged will be proportional to $\beta_{\ell,u}$; if the state has length $\ell$ and energy $u$. For example, since $\ell=16$ and $u=2$ in the walk in figure \[figure1\], the transition probability of the positive move in that figure is proporsional to $\beta_{16,2}$. Similarly, the positive elementary move in figure \[figure2\] increases the energy; so here the transition probability is proportional to $\gamma_{16,2}$, instead. In figure \[figure3\] the positive elementary move also increases the energy, and so its transition probability is proportional to $\gamma_{16,4}$.

Thus, if $w_n$ is the current state (of length $\ell_n$ and energy $u_n$), and $w_{n+1}$ is the next state (of length $\ell_{n+1}$ and energy $u_{n+1}$), then define the change in length by $\Delta_n = \ell_{n+1}\minus \ell_n$, and the change in energy by $\delta_n = u_{n+1} \minus u_n$. Notice that $\Delta_n=\pm 1$ for endpoint elementary moves, and that $\delta_n = \pm 1$, or $\delta_n = 0$, for the model of adsorbing walks in figures \[figure1\] and \[figure2\] (but these quantities may take on other values in the model in figure \[figure3\]). The transition probabilities are chosen such that $$\Pr(w_n\to w_{n+1}) \propto
\cases{
\beta_{\ell,u}, & \hbox{if $\Delta_n = +1$ and $\delta_n=0$;} \\
1 & \hbox{if $\Delta_n = -1$ and $\delta_n=0$},
}$$ where $\ell=\ell_n$ and $u=u_n$ are functions of $n$. In the case that the energy is changed, then the transition probability is constructed such that $$\Pr(w_n\to w_{n+1}) \propto
\cases{
\gamma_{\ell,u}, & \hbox{if $\Delta_n=+1$ and $\delta_n>0$;} \\
1 & \hbox{if $\Delta_n=-1$ and $\delta_n < 0$}.
}$$ Normalising the transition probabilities gives $$\fl
\Pr(w_n \to w_{n+1})
= \cases{
\frac{\beta_{\ell,u}}{\alpha_n^{--} + \alpha_n^{-0} + \alpha_n^{+0}\thin\beta_{\ell,u} 
+ \alpha_n^{++}\thin\gamma_{\ell,u}}, & \hbox{if $\Delta_n=1$ and $\delta_n=0$}; \\
\frac{\gamma_{\ell,u}}{\alpha_n^{--} +\alpha_n^{-0} + \alpha_n^{+0}\thin\beta_{\ell,u} 
+ \alpha_n^{++}\thin\gamma_{\ell,u}}, & \hbox{if $\Delta_n=1$ and $\delta_n>0$}; \\
\frac{1}{\alpha_n^{--} +\alpha_n^{-0} + \alpha_n^{+0}\thin\beta_{\ell,u} 
+ \alpha_n^{++}\thin\gamma_{\ell,u}}, & \hbox{if $\Delta_n=-1$ and $\delta_n\leq 0$}; \\
}
\label{eqn4}   $$ The probability for any type of move can be explicitly computed for any given state $w_n$ (of length $\ell$ and energy $u$) by computing the $\alpha$’s. For example, the probability for a positive elementary move which leaves the energy unchanged is $\alpha_n^{+0}\thin\beta_{\ell,u}/(\alpha_n^{--} \plus \alpha_n^{+0}\thin\beta_{\ell,u} \plus  \alpha_n^{++}\thin\gamma_{\ell,u})$. For the state on the left in figure \[figure1\] this becomes $\beta_{16,2}/(1+\beta_{16,2}+\gamma_{16,2})$ since $\alpha_{16}^{--}=1$, $\alpha_{16}^{+0} = 1$ and $\alpha_{16}^{++}=1$.

GAS-weights
-----------

Let $w_0$ be a starting state (possibly the walk consisting of a single vertex at the origin, of length $\ell_0=0$ and energy $u_0=0$). Implement the endpoint elementary moves on $w_0$ by computing its atmosphere recursively and updating it using the transition probabilities in equation , starting at $n=0$. This generates a Markov Chain of states in a sequence $\phi_N$ (see equation ).

Assume that the parameters $\{\beta_{n,u},\gamma_{n,u}\}$ are known and fixed, so that the sampling can be implemented by simply computing the transtition probabilities and selecting positive and negative elementary moves with appropriate probabilities.

The probability of the sequence $\phi_N$ is given by $$\fl
\Pr (\phi_N) = \prod_{n=0}^{N-1}
\frac{1}{\alpha_n^{--} + \alpha_n^{-0} 
+ \alpha_n^{+0}\thin\beta_{\ell,u} 
+ \alpha_n^{++}\thin \gamma_{\ell,u}}
\; {\prod}_m^\prime \beta_{\ell,u}
\; {\prod}_k^{\prime\prime} \gamma_{\ell,u}
\label{eqn6}            $$ where the primed product ${\prod}^\prime$ is over all the $\beta_{\ell,u}$ for transitions through positive elementary moves leaving the energy unchanged, and the double primed product ${\prod}^{\prime\prime}$ is over all the $\gamma_{\ell,u}$ where the transition is a positive elementary move increasing the energy. In this expression the $\ell$ and $u$ are functions of $n$, $m$, and $k$ in each of the products.

A weight $W(\phi_N)$ will be assigned to the sequence $\phi_N$. In order to compute the weight, define $$\fl
\sigma(j,j\plus 1) = \sigma(w_j\to w_{j+1}) = 
\cases{
-1, &\hbox{if $\Delta_j=+1$ and $\delta_j=0$}; \\
+1, &\hbox{if $\Delta_j=-1$ and $\delta_j=0$}; \\
\hspace{0.8em} 0, &\hbox{otherwise}. \\
}$$ Similarly, define $$\fl
\rho (j,j\plus 1) = \rho (w_j\to w_{j+1}) = 
\cases{
-1 , &\hbox{if $\Delta_j=+1$ and $\delta_j>0$}; \\
+1, &\hbox{if $\Delta_j=-1$ and $\delta_j<0$}; \\
\hspace{0.8em} 0, &\hbox{otherwise}. \\
}$$ That is, the function $\sigma(j,j\plus 1)$ tracks the negative and positive moves along $\phi_N$ where the energy is not changed, and the function $\rho(j,j\plus 1)$ tracks the negative and positive transitions along $\phi_N$ where the energy is also changed.

Assign the weight $$\fl
W(\phi_N) = \LB \Sfrac{\alpha^{--}_0 + \alpha^{-0}_0 + \alpha^{+0}_0 \beta_0 
+ \alpha^{++}_0 \gamma_0}{
\alpha^{--}_N +\alpha^{-0}_N +\alpha^{+0}_N \beta_N+ \alpha^{++}_N \gamma_N} \RB
\prod_{j=0}^{N-1} \beta_{\ell,u}^{\sigma(j,j{+}1)}
\prod_{j=0}^{N-1} \gamma_{\ell,u}^{\rho(j,j{+}1)}
\label{eqn9}            $$ to the sequence $\phi_N$.

The expected value of the weight over sequences of length $N$ from state $w_0$ to state $w_N$ is $$\LA W(w_0\to w_N) \RA_N = \sum_{\phi:w_0 \to w_N}  \Pr(\phi) W(\phi) .
\label{eqn10}    $$ Inserting equations and in this, and simplifying, gives $$\fl
\LA W(w_0\to w_N) \RA_N = \sum_{\phi:w_0\to w_N}
\prod_{j=1}^N 
\frac{1}{\alpha_j^{--} \plus \alpha_j^{-0} \plus \alpha_j^{+0}\thin\beta_{\ell,u} 
\plus  \alpha_j^{++}\thin \gamma_{\ell,u}}
\; {\prod}^{n} \beta_{\ell,u}
\; {\prod}^{nn} \gamma_{\ell,u}
\label{eqn11}    $$ where the product ${\prod}^{n}$ is over all the $\beta_{\ell,u}$ for transitions through *negative* elementary moves leaving the energy unchanged, and the product ${\prod}^{nn}$ is over all $\gamma_{\ell,u}$ where the transition is a *negative* elementary move decreasing the energy. As before, the $\ell
\equiv \ell(j)$ and $u\equiv u(j)$ are functions of $j$ (in other words, functions of the states $w_j$ in the sequence $\phi$).

Reverse all the sequences in equation so that the starting state is $w_N$ and the final state is $w_0$. Under this reversal all negative elementary moves become positive elementary moves and vice versa. That is, equation becomes $$\fl
\LA W(w_0\to w_N) \RA_N = \sum_{\psi:w_N\to w_0}
\prod_{j=1}^N 
\frac{1}{\alpha_j^{--} \plus \alpha_j^{-0} \plus \alpha_j^{+0}\thin\beta_{\ell,u} 
\plus \alpha_j^{++}\thin \gamma_{\ell,u}}
\; {\prod}^{n} \beta_{\ell,u}
\; {\prod}^{nn} \gamma_{\ell,u} ,
\label{eqn12}    $$ where, as before, the product ${\prod}^{n}$ is over all the $\beta_{\ell,u}$ for transitions through *negative* elementary moves along the *reverse sequence* $\psi$ leaving the energy unchanged, and the product ${\prod}^{nn}$ is over all $\gamma_{\ell,u}$ where the transition is a *negative* elementary move along the *reverse sequence* $\psi$ decreasing the energy.

For example, consider the model of collapsing walks in figure \[figure3\] and suppose the sequence $\phi$ is realised, where

at -25 5 at 0 0 at 20 0 40 0 50 0 / at 40 0 50 0 / at 70 0 90 0 100 0 100 10 / at 90 0 100 0 100 10 / at 120 0 140 0 150 0 150 10 140 10 / at 140 0 150 0 150 10 140 10 / at 170 0 190 0 200 0 200 10 / at 190 0 200 0 200 10 / at 220 0 240 0 250 0 250 10 240 10 / at 240 0 250 0 250 10 240 10 / at 270 0 300 0 310 0 310 10 300 10 290 10 / at 300 0 310 0 310 10 300 10 290 10 / at 320 5 at 0 -10 at 45 -10 at 95 -10 at 145 -10 at 195 -10 at 245 -10 at 300 -10 at 0 -20

The probability of this sequence is $$\fl
\Pr(\phi) = \Sfrac{\beta_{0,0}}{4\beta_{0,0}}
\Sfrac{\beta_{1,0}}{(1+3\thin\beta_{1,0})}
\Sfrac{\gamma_{2,0}}{(1+2\thin\beta_{2,0}+\gamma_{2,0})}
\Sfrac{1}{(1+2\thin\beta_{3,1})}
\Sfrac{\gamma_{2,0}}{(1+2\thin\beta_{2,0}+\gamma_{2,0})}
\Sfrac{\beta_{3,1}}{(1+2\thin\beta_{3,1})}$$

The weight of $\phi$ can similarly be computed from equation . This gives $$W(\phi) = \Sfrac{4\thin\beta_{0,0}}{1+3\thin\beta_{4,1}}
\times \LB  \beta_{0,0}^{-1}\,\beta_{1,0}^{-1}\,\gamma_{2,0}^{-1}
\,\gamma_{3,1}^{+1}\,\gamma_{2,0}^{-1}\,\beta_{3,1}^{-1}\RB .$$ The consequence is that $$\fl
\Pr(\phi)W(\phi) = 
\Sfrac{1}{(1+3\thin\beta_{4,1})}
\Sfrac{1}{(1+2\thin\beta_{3,1})}
\Sfrac{1}{(1+2\thin\beta_{2,0}+\gamma_{2,0})}
\Sfrac{\gamma_{3,1}}{(1+2\thin\beta_{3,1})}
\Sfrac{1}{(1+2\thin\beta_{2,0}+\gamma_{2,0})}
\Sfrac{1}{(1+3\thin\beta_{1,0})} ,$$ and this is the probability that a sequence $\psi$, starting in the state $w_6$ and terminating in the state $w_0$, is realised by the algorithm.

The same observation is true generally for equation : The summand is the probability that a particular sequence $\psi$ from state $w_N$ to $w_0$ is realised by the algorithm, and the summation is over all such sequences $\psi$. That is, $\LA W(w_0\to w_N) \RA_N = \Pr(w_N\to w_0)$ is the probability that the algorithm realises a sequence $\psi$ of length $N$ from state $w_N$ to state $w_0$.

The sequence $\psi$ is a Markov Chain, and if it is aperiodic and irreducible, and if $w_0$ is a recurrent state, then asymptotically (for large $N$) the probability that the sequence hits the state $w_0$ is positive and independent of the starting state $w_N$. That is, $\Pr(w_N\to w_0) \to C(w_0)>0$ as $N\to\infty$ where $C(w_0)$ is dependent on the parameters of the algorithm, and independent of $w_N$. Thus, the average weight $\LA W(w_0\to w_N) \RA_N \to C(w_0)$ as $N\to\infty$. Summing $\LA W(w_0\to w_N) \RA_N$ over all the states $w_N$ of length $n = \ell(w_N)$ and energy $u$, shows that the average weights of sequences of length $N$ ending in walks of length $n$ and energy $u$ is $$\LA W_{n,u} \RA_N = \sum_{w_N:\,n,u}  \LA W(w_0\to w_N) \RA_N \to c_n(u)\, C(w_0) ,$$ where the summation is over all walks of length $n$ and energy $u$ (and $c_n(u)$ is the number of walks of length $n$ and energy $u$).

Taking ratios of average weights give $$\frac{\LA W_{n,u} \RA_N}{\LA W_{m,v} \RA_N} \to \frac{c_n(u)}{c_m(v)} .$$ That is, if $c_n(u)$ is known for some values of $n$ and $u$, then the ratios of average weights can be used to estimate $c_m(v)$.

  $n\backslash u$   0       1       2       3       4       5       6       7       8        9       10
  ----------------- ------- ------- ------- ------- ------- ------- ------- ------- -------- ------- -------
  0                 1.000                                                                            
  1                 0.333   1.000                                                                    
  2                 0.428   0.250   1.001                                                            
  3                 0.368   0.499   0.200   1.000                                                    
  4                 0.387   0.380   0.416   0.166   1.000                                            
  5                 0.373   0.396   0.427   0.429   0.142   1.000                                    
  6                 0.386   0.375   0.377   0.359   0.438   0.124   1.000                            
  7                 0.377   0.392   0.384   0.401   0.348   0.445   0.111   1.000                    
  8                 0.383   0.378   0.388   0.356   0.368   0.327   0.449   0.100   0.999            
  9                 0.378   0.386   0.385   0.395   0.370   0.368   0.308   0.454   0.0908   0.999   
  10                0.381   0.379   0.383   0.373   0.368   0.352   0.364   0.289   0.458    0.083   0.999

  : Numerical estimates of $\beta_{n,u}$ for adsorbing walks in $\mathL^2_+$[]{data-label="betanu"}

Sampling with GAS
-----------------

The algorithm is implemented by choosing a starting state $w_0$ and then sampling along a sequence $\phi$ of length $N$. The weight is updated along $\phi$, and collected into bins for walks of length $n$ and energy u. Once the sequence is completed, then the average of each bin is computed, giving the average weight $W_{n,u}$ of walks of length $n$ and energy $u$ seen along $\phi$.

This process is repeated $M$ times, so that $M$ sequences of length $N$, denoted by $\LA \phi_1, \phi_2,\ldots,\phi_M\RA$ are realised, and for each sequence $\phi_j$ the average weight $W_{n,u}^{(j)}$ is calculated. The estimated average weight is computed over the $M$ sequences: $$[W_{n,u}]^{est}_{N,M} = \Sfrac{1}{M} \sum_{j=1}^M W_{n,u}^{(j)} .$$ The estimated weight $[W_{n,u}]_{N,M}^{est}$ is an estimator of $\LA W_{n,u}\RA_N$, and so as $M\to\infty$ and $N\to\infty$, it is expected that $[W_{n,u}]_{N,M}^{est}
\to c_n(u)\,C(w_0)$. Taking ratios give $$\frac{[W_{n,u}]_{N,M}^{est}}{[W_{m,v}]_{N,M}^{est}} \to \frac{c_n(u)}{c_m(v)} .$$

The sampling requires that both $N$ and $M$ are sufficiently large, and there is a trade-off between these quantities. $M$ should be large enough to have sufficient independent estimates of the weights to compute sound averages, and $N$ should be large enough to have sufficiently long sequences to have sampled large enough regions of state space.

  $n\backslash u$   0         1         2         3         4         5         6         7         8         9         10
  ----------------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- ---------
  0                 0\. 499                                                                                             
  1                 0\. 499   1\. 000                                                                                   
  2                 0\. 374   1\. 001   0\. 999                                                                         
  3                 0\. 436   0\. 801   0\. 998   0\. 999                                                               
  4                 0\. 451   0\. 668   0\. 832   0\. 998   1\. 000                                                     
  5                 0\. 461   0\. 751   0\. 857   0\. 855   1\. 000   0\. 999                                           
  6                 0\. 463   0\. 716   0\. 721   0\. 873   0\. 874   0\. 999   0\. 999                                 
  7                 0\. 470   0\. 734   0\. 767   0\. 845   0\. 887   0\. 889   0\. 998   1\. 000                       
  8                 0\. 470   0\. 728   0\. 709   0\. 775   0\. 834   0\. 898   0\. 899   0\. 999   1\. 00              
  9                 0\. 473   0\. 741   0\. 721   0\. 807   0\. 835   0\. 845   0\. 908   0\. 907   1\. 000   0\. 999   
  10                0\. 474   0\. 736   0\. 699   0\. 752   0\. 795   0\. 836   0\. 853   0\. 915   0\. 915   1\. 000   0\. 999

  : Numerical estimates of $\gamma_{n,u}$ for adsorbing walks in $\mathL^2_+$[]{data-label="gammanu"}

There remains the additional issue of the GAS parameters $\beta_{n,u}$ and $\gamma_{n,u}$. These can be estimated using training runs prior to the simulation. Best results are obtained when the sampling is *flat*, so that states of size and energy $\{n,u\}$ are sampled uniformly in $\{n,u\}$. This is best achieved when the GAS sequences are random walks on $n$ and $u$. Thus, the probability of a positive move should be, on average, equal to the probability of a negative move. A good choice for $\beta_{n,u}$ is $$\beta_{n,u} = \frac{\LA \alpha_{n,u}^{-0}\RA}{\LA \alpha_{n,u}^{+0}\RA }
\label{eqn40C}  $$ where $\LA\alpha_{n,u}^{-0}\RA$ is the average negative atmosphere which does not decrease the energy, and $\LA\alpha_{n,u}^{+0}\RA$ is the average positive atmosphere which does not increase the energy, of walks of length $n$ and energy $u$.

A similar argument shows that a good choice for $\gamma_{n,u}$ is $$\gamma_{n,u} = \frac{\LA \alpha_{n,u}^{--}\RA}{\LA \alpha_{n,u}^{++}\RA }
\label{eqn41C}  $$ where $\LA\alpha_{n,u}^{--}\RA$ is the average negative atmosphere which decreases the energy, and $\LA\alpha_{n,u}^{++}\RA$ is the average positive atmosphere which increases the energy, of walks of length $n$ and energy $u$.

  $n\backslash u$   0      1       2       3       4       5       6       7       8       9       10
  ----------------- ------ ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
  0                 8658                                                                           
  1                 8616   13208                                                                   
  2                 9642   7711    13261                                                           
  3                 8971   9622    7144    13012                                                   
  4                 8919   9576    9261    6528    12883                                           
  5                 8957   8947    9501    9388    5998    13072                                   
  6                 8982   8736    9220    9019    9281    6055    13263                           
  7                 9059   8562    9026    9095    8711    9733    5812    13278                   
  8                 8996   8398    9030    8844    8568    8914    9493    5728    13362           
  9                 9169   8162    8788    8772    8579    8843    8417    9689    5678    13204   
  10                9134   8182    8580    8434    8559    8898    8390    8221    10101   5477    12883

  : The frequency of states sampled to $n=10$ and $u=10$ for adsorbing walks in $\mathL^2_+$[]{data-label="statessampled"}

Computed values for $\beta_{n,u}$ and $\gamma_{n,u}$ for a square lattice adsorbing walk model in figure \[figure1\] are given in tables \[betanu\] and \[gammanu\] for $0\leq n\leq 10$ and $0 \leq u \leq 10$. These data are chopped at an accuracy of three decimal places.

Observe that $\beta_{n,n} = 1$ (when rounded) and $\gamma_{n,n}=1$. This is expected, since there are exactly two states if $n>0$ and $n=u$ (a completely adsorbed walk which never leaves the adsorbing line).

With these values of $\beta_{n,u}$ and $\gamma_{n,u}$ the sampling is reasonably flat, as shown in table \[statessampled\]. The data in table \[statessampled\] is the number of times a sequence of length $10^6$ visited states of length $n$ and energy $u$. The maximum length was set at $n=500$, so that the number of pairs $(n,u)$ is $124750$; thus, the expected number of visits to states of length $n$ and energy $u$ is roughly $8000$. The data in table \[statessampled\] are spread around this number, and the distribution is a reasonably flat histogram. There are exceptions for data along the diagonal where a larger number is seen. The explanation for this is that the sequence can only visit states along the diagonal if it starts in $(n,u)=(0,0)$ and then stay on the diagonal (doing a random walk on states of length and energy both equal to $n$) – the elementary moves chosen in the simulation do not include neutral moves, and so the sequence cannot enter the diagonal except at $n=0$.

Observe that since $c_n^+(n)=2$ for $n>0$ in this model, that these states are rare, but are sampled frequently by the algorithm. This is an example of *rare event* sampling, where an algorithm spends significant time sampling rare states in the tails of a distribution in order to get good estimates of microcanonical quantities.

Simulations were performed by collecting data over $500$ realised sequences, each of length $10^9$. Numerical estimates of $c_{n,u}$ are shown in table \[countsads2d\]. The data in table \[countsads2d\] are rounded to the nearest integer. These data can be compared to exact counts to verify the algorithm and its implementation.

  $n\backslash u$   0      1      2      3      4     5     6     7    8    9   10
  ----------------- ------ ------ ------ ------ ----- ----- ----- ---- ---- --- ----
  0                 1                                                           
  1                 1      2                                                    
  2                 3      2      2                                             
  3                 7      8      2      2                                      
  4                 19     16     10     2      2                               
  5                 49     42     24     12     2     2                         
  6                 131    106    56     28     14    2     2                   
  7                 339    283    148    76     32    16    2     2             
  8                 897    720    385    193    92    36    18    2    2        
  9                 2338   1905   990    543    249   110   40    20   2    2   
  10                6178   4932   2571   1372   672   298   130   44   22   2   2

  : Estimates of $c_n^+(u)$ to $n=10$ and $u=10$ for adsorbing walks in $\mathL^2_+$[]{data-label="countsads2d"}

Partition functions (see equation ) can be directly estimated from the microcanonical data in table \[countsads2d\], for example, for $n=6$ the partition function is approximated by $$Z_6(a) = 131 + 106\thin a+ 56\thin a^2 + 28\thin a^3 + 14\thin a^4 + 2\thin a^5 + 2\thin a^6$$ where $a=e^{\upsilon/kT}$ is a Boltzmann factor ($\upsilon$ is the interaction energy associated with a single visit).

Additional data were collected in the square and cubic lattices. For example, the average mean square radius of gyration of walks of length $n$ and energy $u$ were also determined, as was the average height of the endpoint of the walks. In addition, data were also collected on the average positive and negative atmospheric statistics from which estimates of the $\beta_{n,u}$ and $\gamma_{n,u}$ were made (see equations and ).

Numerical Results {#section3}
=================

Adsorbing walks in the square lattice
-------------------------------------

The finite size free energy $\C{F}_n(a)$ was determined from the data and is plotted in figure \[figure4\] for walks of lengths $n=50\thin N$ with $=1,2,3,\ldots,10$. $\C{F}_n(a)$ is a function of the combined variable $\tau=n^{1/2}(a\minus a_c^+)$ (see equations and , and note that the crossover exponent is $\phi=\sfrac{1}{2}$). Plotting the free energy $\C{F}_n(a)$ against $\tau$ should collapse the data to a single underlying curve (near the critical point $a_c^+$; that is, for small values of $\tau$) which exposes the scaling function $g$ in equation . This is displayed in figure \[figure4B\].

figure5-.tex

figure6-.tex

The energy $\C{E}_n(a)$ and specific heat $\C{C}_n(a)$ (see equation ) can be determined by differentiation of $\C{F}_n(a)$ and is plotted in figure \[figure5\] against $\log a$ and in figure \[figure6\] against $\tau$. These curves clearly show the adsorption transition in the model at a critical value of $a$.

figure7-.tex

figure8-.tex

Closer inspection of the specific heat curves in figure \[figure5\] shows that they intersect each other close to a fixed point. To the left of this point the curves decrease with increasing $n$ to zero, and to the right of this point the curves increase with increasing $n$. The common point of intersection is located approximatedly at the critical adsorption point $a_c^+$ in the model. In figure \[figure5B\] the specific heat curves are magnified in a region close to the point where they intersect.

figure9-.tex

### Location of the critical point $a_c^+$: {#section311}

In order to determine $a_c^+$, consider the finite size energy density $\C{E}_n(a) = a\sfrac{d}{da} \C{F}(a)$ per edge in the model. It is known that (see, for example, equation and reference [@JvRR04]) $$\C{E}_n(a) = n^{\phi-1}\,h_n(\tau)$$ where $h_n$ is a scaling function. From this equation, construct the ratio $$\frac{\log (n\, \C{E}_n(a))}{\log (m\,\C{E}_m(a))}
= \frac{\phi\log n + \log h_n(\tau)}{\phi\log m + \log h_m(\tau)}$$ If $a=a_c^+$, then $\tau = 0$, and the above simplifies to $$\frac{\log (n\, \C{E}_n(a_c^+))}{\log (m\,\C{E}_m(a_c^+))}
= \frac{\phi\log n + \log h_n(0)}{\phi\log m + \log h_m(0)} .$$ For large values of $n$ and $m$, $\log h_n(0)$ and $\log h_m(0)$ approaches the same constant $C$, so that the above becomes $$\frac{\log (n\,\C{E}_n(a_c^+))}{\log (m\, \C{E}_m(a_c^+))}
= \frac{\phi\log n + C}{\phi\log m + C} + \hbox{small correction}.$$ When $C$ is small compared to $\log m$, the right hand side may be expanded to obtain $$\hspace{-1cm}
\frac{\log (n\,\C{E}_n(a_c^+))}{\log (m\, \C{E}_m(a_c^+))}
= \Sfrac{\log n}{\log m} + \LH \Sfrac{C}{\phi \log m}
- \Sfrac{C \log n}{\log^2 m} \RH - 
\Sfrac{C^2}{\phi^2\log^2 m} + \hbox{small correction}.$$ The signs of the two terms in square brackets are opposite, and for $n$ and $m$ not too far apart, these terms grow about at the same rate as the last term $\sfrac{C^2}{\phi^2\log^2 m}$. That is, the approximation $$\hspace{-1cm}
\frac{\log (n\,\C{E}_n(a_c^+))}{\log (m\, \C{E}_m(a_c^+))}
= \Sfrac{\log n}{\log m}  - 
\Sfrac{C_0}{\phi^2\log^2 m} + \hbox{small correction},$$ where $C_0$ is a constant, should be accurate at the critical point. Divide both sides by $\sfrac{\log n}{\log m}$ to obtain $$\hspace{-1cm}
P_{n,m}(a_c^+) 
= \frac{\log (n\,\C{E}_n(a_c^+))}{\log (m\, \C{E}_m(a_c^+))}\frac{\log m}{\log n}
= 1 - \Sfrac{C^2}{\phi^2\log n \log m} + \hbox{small correction},
\label{eqn44C}   $$ Solving for the critical point by inverting $P_{n,m}$ gives the solution $a_{n,m}^+$ as an approximation of the critical point, namely $a_{n,m}^+ 
= P_{n,m}^{-1} (1 \minus \sfrac{C^2}{\phi^2\log n \log m} \plus \hbox{small correction})$ which may be expanded to $$a_{n,m}^+ = P_{n,m}^{-1}(1) - \Sfrac{C_1}{\log n \log m} +
\hbox{small correction},
\label{eqn44D}  $$ where $C_1$ is a constant. That is, for given values of $n$ and $m$, an estimate of $a_c^+$ can be obtained by determining the solution of $P_{n,m}(a) = 1$, or $a_{n,m}^+$ is estimated obtained by solving for $a$ in $$\hspace{-1cm}
\frac{\log (n\,\C{E}_n(a))}{\log (m\, \C{E}_m(a))}\frac{\log m}{\log n}
= 1.$$ In the above, $m$ was put equal to $n\minus 100$, and $n$ was assigned values starting at $n=200$ to $n=500$ in steps of $1$. The estimates $a_n^+ \equiv a_{n,n-100}^+$ showed no systematic dependence on $n$, and a simple average over all $n\in[200,500]$ gives the best value $$a_c^+ = 1.7788 \pm 0.0029 .
\label{abest}   $$ The confidence interval is obtained by doubling the square root of the variance of the estimates $a_n^+$. This result compares well with the result in reference [@BGJ12], namely $a_c^+ = 1.77564$ (obtained by the exact enumeration of adsorbing walks), and also with $a_c^+ = 1.759\pm 0.018$ in reference [@JvRR04] (obtained by using a Multiple Markov Chain implementation of the Berretti-Sokal algorithm [@BS85]).

The estimate can be used to determine the crossover exponent $\phi$. By equation , the specific heat scales as $\C{C}_n(a_c^+) \sim n^{\alpha\phi} h_c(0)$ when $a=a_c^+$. That is, an estimate of $\alpha\phi$ is obtained by computing $\sfrac{\log \C{C}_n(a_c^+)}{\log n}$. Computing this for $100 \leq n \leq 500$, and taking the mean as the best estimate (and estimating a confidence interval by doubling the square root of the variance of the estimates), gives $\alpha\phi = -0.0091 \pm 0.0162$. Determining $\phi$ by using equation then gives the best estimate for $\phi$: $$\phi = 0.4955 \pm 0.0081 .
\label{phibest}   $$ This result is consistent with $\phi=\shalf$ [@BY95; @BWO99; @BEG89], and compares well with other estimates in the literature (for example, $\phi=0.501\pm 0.014$ in reference [@JvRR04]).

figure10-.tex

### The critical point $a_c^+$ and the specific heat $\C{C}_n(a)$:

The best estimates above may be compard to estimates obtained from the specific heat curves in figure \[figure5\]. These curves intersect each other near $a_c^+$, and the region containing the intersections (in figure \[figure5B\]) is magnified in figure \[figure5C\].

In general the location of the intersection between $\C{C}_n(a)$ and $\C{C}_m(a)$ is a function of $n$ and $m$. The location of the critical point $a_c^+$ can be estimated by extrapolating this dependence. Consider for example the intersections between the curves $\C{C}_n(a)$ and $\C{C}_{n+100}(a)$. The location of these intersections are plotted against $\sfrac{1}{\sqrt{n}}$ in figure \[figure5C\], where $n=2\, N$ and $N\in [23,200]$. The data lie along a straight line, except for a few points at the largest values of $n$ (where the data is more uncertain). The best line through the data can be extrapolated to its intersection with the vertical axis (where $\sfrac{1}{\sqrt{n}} = 0$). This gives an estimate of $a_c^+$ as being located in the interval $[1.77,1.79]$. Using a linear least squares model for $n\geq 10$ gives the (extrapolated) estimate $a_c^+ \approx 1.7839$. By examining the spread of the data in figure \[figure5C\], a confidence interval can be estimated. The result is $$a_c^+ = 1.784\pm 0.010 .
\label{eqn29}  $$ This estimate is consistent with the best estimate in equation .

An alternative approach to determining the critical point $a_c^+$, is to consider the scaling of the specific heat in equation . Taking ratios for $n$ and $m$, and then logarithms, give $$\log \LB \Sfrac{\C{C}_n(a)}{\C{C}_m(a)} \RB
 = \alpha \phi \log \LB \Sfrac{n}{m} \RB
+ \log \LB \Sfrac{h_c(n^\phi(a\minus a_c^+))}{h_c(m^\phi(a\minus a_c^+))} \RB .
\label{eqn27}  $$ Observe that the last term is zero when $a=a_c^+$. Since, in addition, $\alpha = 2\minus \sfrac{1}{\phi}=0$ in this model, this shows that an estimate of the critical point is given by the solution of $$\log \LB \Sfrac{\C{C}_n(a)}{\C{C}_m(a)} \RB = 0 .$$ Solving this for $150\leq n \leq 500$ (and $n$ a multiple of $10$), and for $m=n\minus k$ where $k\in\{10,20,\ldots,100\}$ gives a large collection of estimates of $a_c^+$, with mean $$a_c^+ = 1.762 \pm 0.016 .
\label{eqn27AA}  $$ The confidence interval is one-half of difference between the maximum and minimum estimates of $a_c^+$. This estimate is slightly less than,but still consistent with, the results in equation and equation . Recall that it is also predicated on the assumption that $\alpha=0$ (or $\phi = \sfrac{1}{2}$).

figure11-.tex

### The microcanonical density function:

The microcanonical density function of visits in adsorbing positive walks is determined from the microcanonical data in the model, and is given by $$P^+(\eps) =\lim_{n\to\infty} (c_n^+(\lfl \eps n \rfl))^{1/n} = \lim_{n\to\infty} P_n^+(\eps),
\label{eqn59X}  $$ where $P_n^+(\eps) = (c_n^+(\lfl \eps n\rfl))^{1/n}$ is a finite size approximation to $P^+(\eps)$. Existence of the limit is known (see for example reference [@JvR15]), and $\log P^+(\eps)$ is a concave function of $\eps$.

$P^+(\eps)$ can be estimated by interpolating the finite size approximations $P_n^+(\eps)$ and then extrapolating to $n=\infty$ by fitting a least squares model to the data. In figure \[figure6-B\] the data for the extrapolated function $P^+(\eps)$ is plotted together with $P_n^+(\eps)$ for $n=100$ and $n=500$.

A least squares fit of a quadratic to $\log P^+(\epsilon)$ for $\eps\in[0,0.1]$ gives the $\log P^+(\eps) \approx 0.97007\minus 0.58190\eps\minus 0.13030\eps^2$, and by taking the right derivative and then taking $\eps\to 0^+$, an estimate for the critical point is obtained: $$a_c^+ \approx 1.789 .
\label{eqn27BB}  $$ This is close to the estimates obtained in equations and , showing consistency in the data and the analysis above.

The free energy $\C{F}(a)$ is the Legendre transform of $\log P^+ (\epsilon)$. This may be estimated by fitting a polynomial to $\log P^+(\epsilon)$. If a cubic polynomial in $\eps$ is fitted to $\log P^+(\eps)$ for $0 \leq \eps \leq 0.5$, then the estimated free energy for $a>a_c^+$ is approximately $$\fl
F (a) \approx  1.0376 -0.1153\log a  -  (0.3306 \minus 0.5693\log a)
\sqrt{-1.3603 \plus 2.3422 \log a}.$$ The critical point can be estimated as that location where the square root in the above is zero. This gives $a_c^+ \approx 1.787$. Similarly, the factor $(0.3306-0.5693\log a)$ vanishes when $a_c^+ \approx 1.787$.

### Metric data:

The mean square radius of gyration $R_n^2$, and the mean height $H_n$ of the endpoint of the walk, are functions of the adsorption activity $a$. In the desorbed phase (for $a<a_c^+$) it is expected that $R_n^2 \sim n^{2\nu}$, and $H_n \sim n^\nu$, where $\nu = \sfrac{3}{4}$ is the metric exponent [@D89C]. This scaling changes in the adsorbed phase (when $a>a_c^+$) to $R_n^2 \sim n^2$ and $H_n \sim \hbox{constant}$. These expectations are confirmed by the data, as seen, for example, in figure \[figure8\], where data for the mean square radius of gyration are normalised and then plotted as a function of $a$. These graphs clearly show two scaling regimes, namely a high temperature phase (when $a< a_c^+$) where the walk has bulk critical exponents and is desorbed, and a low temperature phase where the walk stays near the adsorbing boundary and has critical exponents of a linear object.

figure12-.tex

In general, the metric exponent associated with $R^2_n$ is a function of $a$, and it will be denoted by $\nu_a$, where $\nu_a=\sfrac{3}{4}$ in the desorbed phase, and $\nu_a=1$ in the adsorbed phase. This exponent may be estimated from the mean square radius of gyration $R_n^2$ data by examining the ratio $$2\,\nu_{n,m}(a) = \frac{\log ( R_n^2/R_m^2 )}{\log (n/m)}.
\label{eqn41}   $$ Here, $\nu_{n,m}(a)$ is a function of $n$ and $m$. By averaging over $m$, the estimate $\nu_n(a) = \LA \nu_{n,m}(a)\RA_m$ may be determined. In particular, fixing $n$ and taking the average over $m$ for $100 \leq m \leq 500$ in multiples of $5$ (and for $m\not=n$) gives estimates of $\nu_n(a)$. These results are plotted in figure \[figure8-nu\] for $n\in\{50,100,150,\ldots,500\}$. The data for $a\leq 1.5$ gives $\nu \approx 0.747$, and for $a \geq 1.95$, $\nu \approx 1.01$. These results are evidence for the exact value $\nu_a = \sfrac{3}{4}$ in the desorbed phase, and $\nu_a=1$ in the adsorbed phase.

figure13-.tex

The function $\nu_n(a)$ should scale with the combined variable $\tau = n^\phi(a\minus a_c^+)$. That is, one may expect that $\nu_n (a) = \hbox{\Large$\nu$}(\tau)$, where is a scaling function. In figure \[figure6-D\] the data in figure \[figure8-nu\] are rescaled by plotting against $\tau$ to uncover the scaling function .

figure14-.tex

The (normalised) average height of the endpoint of the walk is plotted as a function of $a$ in figure \[figure8HH\]. These data show a clear transition where the scaling of $H_n$ changes. That is, the metric exponent associated with $H_n$ is a function of $a$, and is denoted by $\nu_a^\perp$, where $\nu_a^\perp = \nu= \sfrac{3}{4}$ if $a<a_c^+$, and $\nu_a^\perp = 0$ if $a>a_c^+$, so that $H_n \sim n^{\nu_a^\perp}$. The graph of $n^{-3/4}H_n$ contains a set of curves which decreases with increasing $a$. These curves intersect each other close to $a_c^+$.

figure15-.tex

figure16-.tex

An approach similar to the ratio method in equation may be used to estimate $\nu_a^\perp$ (which will be referred to as the *vertical metric exponent*): $$\nu^{\perp}_{n,m} (a) = \frac{\log (H_n/H_m)}{\log (n/m)}.
\label{eqn42}   $$ This estimate of $\nu_a^\perp$ is a function of $n$ and $m$, and may be averaged over $m$ to obtain $\nu^{\perp}_n(a) = \LA \nu_{n,m}(a)\RA_n$. Taking the average for $100\leq m \leq 500$ in multiples of $5$ to estimate $\nu^\perp_n(a)$ gives the curves in figure \[figure6-nuH\] when plotted against the combined variable $\tau=n^\phi(a\minus a_c^+)$. It is seen in the graph that if $a<a_c^+$, then $\nu_n (a) \approx \sfrac{3}{4}$, but for $a>a_c^+$, $H_n \simeq \hbox{const}$ so that $\nu_n^{\perp}(a) \approx 0$.

figure17-.tex

Since $R^2_n \sim n^{2\nu_a}$ where $\nu_a = \nu = \sfrac{3}{4}$ if $a<a_c^+$, and $\nu_a = 1$ if $a>a_c^+$, ratios of $R^2_n$ may be defined by $$\frac{R^2_{n}}{R^2_{2n}} \approx 2^{-2\nu_a} ,\q\hbox{or}\q
\frac{2^{2\nu}R^2_{n}}{R^2_{2n}} \approx 
\cases{
1, & if $a< a_c^+$; \\
2^{2\nu - 2\nu_a} = 2^{-1/2} , & if $a>a_c^+$.
}
\label{eqn58} $$ In figure \[figure8RR\] the quantity $\sfrac{2^{2\nu}R_{n}^2}{R_{2n}^2}$ is plotted as a function of the rescaled variable $\tau = n^{1/2}(a\minus a_c^+)$ for $n$ from $25$ to $250$ in steps of $25$. In the desorbed phase this ratio should be equal to $1$, but, in the adsorbed phase, it should be equal to $2^{1.5-2.0} \approx 0.71$. This is clearly seen in the graph. The curves coincide well over the entire range of $n$ and $\tau$, and decreases sharply close to the critical adsorption point at $\tau=0$. A similar approach using the heights of the endpoint involves the ratios $$\frac{H_{n}}{H_{2n}} \approx 2^{-\nu_a^\perp} ,\q\hbox{or}\q
\frac{2^\nu H_{n}}{H_{2n}} \approx 
\cases{
1, & if $a< a_c^+$; \\
2^{\nu-\nu^\perp}  = 2^{3/4}, & if $a>a_c^+$.
}
\label{eqn59} $$ The quantity $\sfrac{2^\nu H_{n}}{H_{2n}}$ is plotted against $\tau$ on the right panel in figure \[figure8RR\]. In the desorbed phase this ratio should be equal to $1$, as seen in the graph. In adsorbed phase the scaling of $H_n$ changes, and the ratio should be equal to $2^{3/4} \approx 1.68$, as seen in the graph.

figure18-.tex

The generating function
-----------------------

The generating function is given by the series $$G (a,t) = \sum_{n=0}^\infty \sum_{v=0}^n c_n^+(v)\,a^vt^n .
\label{eqn45}  $$ Approximations of $G(a,t)$ is given by the truncated sum $$G_N(a,t) = \sum_{n=0}^N \sum_{v=0}^n c_n^+(v)\,a^vt^n ,
\label{eqn46a}  $$ where $c_n^+(v)$ is approximately enumerated by the GAS algorithm. In this study, $G_{500}(a,t)$ was estimated using the approximate values of $c_n^+(v)$ obtained in the simulations.

![A plot of $\log G_{500}(a,t)$ as a function of $(a,t)$ for adsorbing walks in $\mathL_2^+$. The critical curve is the black curve along the surface, and the critical point separating the $\tau_0$ and $\lambda$ curves (see figure \[figure4-6\]) is denoted. For $t<t_c^+(a)$ the generating $G(a,t)$ is finite, and for $t>t_c^+(a)$ it is divergent. These two regimes are clearly visible.[]{data-label="figureAdsrb-2"}](figure19-.jpg)

The *critical curve* $t_c^+(a)$ of $G(a,t)$ is its radius of convergence as a function of $a$. By equation and by equation , $$\log t_c^+(a) = - \C{F}(a) = - \lim_{n\to\infty} \sfrac{1}{n} \log  Z_n(a) .
\label{eqn46A}  $$ The critical curve is shown schematically in figure \[figure4-6\]. $G(a,t)$ is singular when $t=t_c^+(a)$, and if $t>t_c^+(a)$, then $G(a,t)$ is divergent.

The critical curve is parametrized by the scaling fields $(\sigma,g)$ as shown in figure \[figure4-6\]. The critical point when $a=a_c^+$ is located at $(a_c^+,t_c^+)$, where $t_c^+\equiv t_c^+(a_c^+)$, and it divides the critical curve $t_c^+(a)$ into two parts. The part marked by $\tau_0$ corresponds to a transition to desorbed walks, so that $t_c^+(a)=1/\mu_2$ in this regime (which is a transition to a high temperature phase). For $a>a_c^+$ the approach to $t_c^+(a)$ is to adsorbed walks, along the critical curve marked by $\lambda$ (which is a transition to a low temperature phase).

The phase diagram in figure \[figure4-6\] may be described in terms of the coordinates $g=(\sfrac{1}{\mu_2} \minus t)$ and $\sigma = (a\minus a_c^+)$. The behaviour of $G(a,t)$ along its singular points along the critical curve is described by $$G(a,t) \sim \cases{
g^{-\gamma_1}, & \hbox{along $\tau_0$;} \\
g^{-\gamma_s}, & \hbox{at the critical point $a=a_c^+$;}\\
g^{-\gamma_+}, & \hbox{along $\lambda$.}
}
\label{eqn46}  $$ In two dimensions exact values are known for the exponents: The exponent $\gamma_1 = \sfrac{61}{64}$ [@C87] is the entropic exponent of half-space walks, and $\gamma_s = \sfrac{93}{64}$ [@BEG89] is the surface exponent of adsorbing half-space walks at the critical point $a_c^+$. The exponent $\gamma_+$ is the entropic exponent of adsorbed walks, and is given by the entropic exponent of self-avoiding walks in one dimension lower. In one dimension, this is $\gamma_+=1$.

A plot of $G_N(a,t)$ is shown in figure \[figureAdsrb-2\]. The horizontal plane is the $(a,t)$-plane, and the critical curve in figure \[figure4-6\] is shown as a black curve with the critical point shown. Below the critical curve $G(a,t)$ is finite, and approximated well by $G_N(a,t)$ (for large values of $N$ not too close to the critical curve). Above the critical curve $G(a,t)$ is divergent, while $G_N(a,t)$, which is a polynomial, is finite.

figure20-.tex

The exponent $\gamma_1$ can be estimated by putting $a=1$ so that $G(1,t) \sim g^{-\gamma_1}$ where $g=(t_c^+(1)\minus t)$ and $t_c^+(1) = \sfrac{1}{\mu_2}$ (and where $\mu_2$ is the growth constant of the walks in two dimensions). Thus, estimate $\gamma_1$ by noting that $$\frac{\log G(1,t)}{\log g} = - \gamma_1 + \Sfrac{C_1}{\log g} + \Sfrac{C_2}{\log^2 g}
+ \ldots .
\label{eqn47}  $$ Proceed by approximating $G(1,t)$ by $G_N(1,t)$ (with $t < t_c^+(1)$ so that $G_{500}(1,t)$ is a good approximation of $G(1,t)$). A least squares fit of the ratio on the left to a quadratic in $\sfrac{1}{\log g}$ gives the estimate $$\gamma_1=0.952\ldots .$$ This result is very close to the exact value $\sfrac{61}{64} = 0.953\ldots$. Similar analysis for $a>1$ and $a<a_c^+$ gives results slightly larger, since the critical point at $a_c^+$ influences the data in its vicinity for finite values of $N$.

figure21-.tex

A similar analysis with $a=a_c^+$ gives the estimate $$\gamma_s=1.429\ldots,$$ for the surface exponent at the critical adsorption point. This is close in value to the exact result $\sfrac{93}{64} = 1.453\ldots$.

The case that $a>a_c^+$ may also be analysed. The adsorbed walk should have the statistics of the self-avoiding walk in one dimension, so that the entropic exponent is $\gamma_+ = 1$ (this is the value of the exponent $\gamma$ in one dimension). Putting $a=3.5$ and plotting $Z_n^{1/n}(3.5)$ against $n$ gives an estimate for the critical value $t_c^+(3.5)$. In this case the data quickly converges to $t_c^+(3.5)=0.260\ldots$ (to three decimal places). Assuming that $t_c^+(3.5)=0.260$ and choosing the scaling field $g=(0.260 \minus t)$ gives the model $$\Sfrac{\log G(3.5,t)}{\log g} = - \gamma_+ + \Sfrac{C_1}{\log g} + \Sfrac{C_2}{\log^2 g}
+ \ldots ,
\label{eqn48}  $$ similar to the above, but now with the exponent $\gamma_+$. Plotting the left hand side as a function of $\kappa=\sfrac{1}{\log g}$, and fitting the data to a quadratic in $\kappa$, give the estimate $\gamma_+ = 1.00\ldots$. There remains, however, some curvature in the model for small values of $g$ (and of $\sigma$), so that there may remain strong systematic corrections to this result (however, $\gamma_+ = 1$ is consistent with this result).

In the vicinity of the critical point $(a_c^+,t_c^+)$ the generating function should exhibit scaling given by $$G(a,t) \sim g^{-\gamma_s} \,f(g^{-\phi} \sigma),
\label{eqnGscale}   $$ where $\phi=\sfrac{1}{2}$ is the crossover exponent and $f$ is a scaling function. That is, plotting $g^{\gamma_s}\,G(a,t)$ against the combined variable $g^{-\phi} \sigma$ should expose the scaling function $f$. In figure \[figureGscale\] this is done by plotting $\log (g^{\gamma_s}\,G(a,t) )$ against $g^{-1/2}\sigma$ for $g=\sfrac{1}{\mu_2}\minus t \in [0.01,0.05]$ and $\sigma=a\minus a_c^+ 
\in [-\sfrac{1}{2}g^{1/2},g^{1/2}]$.

The partition function (see equation ) should also exhibit scaling for large $n$, given by $$Z_n(a) \sim n^{\gamma_t-1}\,h(n^\phi (a_c^+ \minus a))\,\mu_a^n,
\label{eqn53}   $$ where $\log \mu_a = \C{F}(a)$ and $\log \mu_a = - \log t_c^+(a)$. The exponent $\gamma_t$ can be related to the $\gamma$-exponents as $a\to a_c^+$ (that is, as $\sigma\to 0$), and namely to the surface exponent $\gamma_s$, where the walk is critical with respect to the adsorption transition. By noting that $G(a,t) = \sum_n Z_n(a)\,t^n$, and approximating the summation by an integral, it follows that $G(a,t) \sim
g^{-\gamma_t} h(0)$ if $a=a_c^+$ and $g=t_c^+\minus t$. This shows that $$\gamma_t-1 = \gamma_s - 1 = \Sfrac{93}{64} - 1 = \Sfrac{29}{64}.
\label{eqn71g}   $$ That is, when $a$ is close to $a_c^+$ (and $a<a_c^+$), the partition function has asymptotic behaviour $$Z_n(a) \sim n^{29/64} h(n^\phi(a_c^+\minus a))\, \mu_a^n .
\label{eqnZscaling}   $$ This result may be tested by plotting $n^{-29/64} Z_n(a) \, (t_c^+(a))^n$ against $|\tau | = n^{1/2} |a\minus a_c^+|$. All the data should collapse to the same universal curve exposing the scaling function $h$. This is done with $a<a_c^+$ (and $t_c^+(a) = \sfrac{1}{\mu_2}$) in figure \[figureZscale\] for $n\in\{50,100,150,\ldots,500\}$ and $0 \leq a < a_c^+$.

figure22-.tex

Adsorbing walks in the cubic lattice
------------------------------------

The (finite size) free energy $\C{F}_n(a)$ is a function of the combined variable $\tau=n^{1/2}(a\minus a_c^+)$ (see equations and ; note that $\phi=\sfrac{1}{2}$ for adsorbing walks in three dimensions [@HG94]). Plotting $\C{F}_n(a)$ against $\tau$ for data in the cubic lattice gives a graph similar to figure \[figure4B\]. In figure \[figure4B-3\] the scaled free energy $n(\C{F}_n(a)\minus\log\mu_3)$ is plotted against $\tau$. The data shows a clear transition in the model from a desorbed to an adsorbed phase.

Derivatives of the free energy to $\log a$ gives the (finite size) energy density $\C{E}_n(a)$ and (finite size) specific heat $\C{C}_n(a)$. These are plotted in figure \[figure1-3a\] against $\log a$ and in figure \[figure4-3a\] against $\tau$. In these plots, as in figure \[figure4B-3\], the critical point was approximated by $a_c^+ = 1.31$. This is a close approximation of the best estimate for the critical point from our data (see equation ).

figure23-.tex

figure24-.tex

### Location of the critical point $a_c^+$: {#location-of-the-critical-point-a_c}

The location of the critical adsorption point can be determined using the same analysis as in section \[section311\], and in particular using equation as a starting point. That, is for given values of $n$ and $m$, an estimate $a_{n,m}^+$ of $a_c^+$ can be obtained by solving for $a$ in $$\hspace{-1cm}
\frac{\log (n\,\C{E}_n(a))}{\log (m\, \C{E}_m(a))}\frac{\log m}{\log n}
= 1.$$ Here, the choice $m=n\minus 100$ worked well, and $n$ was assigned values starting at $n=200$ to $n=500$ in steps of $1$. The estimates $a_n^+ \equiv a_{n,n-100}^+$ showed a dependence on $n$, systematically decreasing with increasing $n$. The best estimate is obtained by using the model $a_n^+ = a_c^+ - \sfrac{c_1}{\log^2 n}$ suggested by equation . A least squares fit for all $n\in [200,500]$ gives the best estimate $$a_c^+ = 1.3055 \pm 0.0061 .
\label{abest3}   $$ The confidence interval is obtained by doubling the square root of the variance of the estimates $a_n^+$. This result is slightly smaller than the result in reference [@JvRR04], namely $a_c^+ = 1.334\pm 0.027$ (obtained by using a Multiple Markov Chain implementation of the Berretti-Sokal algorithm [@BS85]).

The estimate can be used to determine the crossover exponent $\phi$. This is again done by considering the scaling of the specific heat (equation ). It is expected that $\C{C}_n(a_c^+) 
\sim n^{\alpha\phi} h_c(0)$ when $a=a_c^+$. An estimate of $\alpha\phi$ is obtained by computing $\sfrac{\log \C{C}_n(a_c^+)}{\log n}$ for a range of values of $n$ (in this case $100 \leq n \leq 500$). The average is taken as the best estimate and a confidence interval is estimated by doubling the square root of the variance of the estimates. This gives $\alpha\phi = 0.0106 \pm 0.0116$. Determine the best estimate for $\phi$ by using equation : $$\phi = 0.5053 \pm 0.0053 .
\label{phibest3}  $$ This result compares well with the estimate $\phi=0.5005\pm 0.0036$ for adsorbing walks in reference [@JvRR04].

### The critical point $a_c^+$ and the specific heat $\C{C}_n(a)$:

The best estimate for $a_c^+$ above (see equation ) should be examined by comparing it to estimates obtained from the specific heat curves in figure \[figure1-3a\]. These curves intersect each other near $a_c^+$, and the region containing the intersections is magnified in figure \[figure2-3a\].

figure25-.tex

The locations of the intersections between the specific heat curves in figure \[figure2-3a\] is a function of $n$ and are estimates of the critical adsorption point $a_c^+$. By plotting the intersections between $\C{C}_n(a)$ and $\C{C}_{n+100}(a)$ against $\sfrac{1}{\sqrt{n}}$ (see figure \[figure3-3a\], where $n=2\, N$ and $N\in [23,200]$); it is seen that the intersections fall approximately along a curve, which may be extrapolated to its intersection with the vertical axis. This gives a rough estimate of the critical point $\log a_c^+$ as being located in the the interval $[1.33,1.35]$. A more accurate extrapolation is done by using a linear least squares model to extrapolate to $n=\infty$. Fitting to the model $a_c^+\plus \sfrac{a_0}{\sqrt{n}}
\plus \sfrac{a_1}{n}$, for all $n \geq 50$, gives the estimate $\log a_c^+ \approx 1.337$. By examining the spread of the data in figure \[figure3-3a\], a confidence interval can be estimated: $$\log a_c^+ = 1.337\pm 0.020 .
\label{eqn29-3}  $$ This estimate is slightly larger than the estimate in equation , but is consistent with the estimate $a_c^+ = 1.334$ in reference [@JvRR04]. However, the noise in the data in figure \[figure3-3a\] makes this a less reliable estimate.

figure26-.tex

Equation is equally valid for adsorbing walks in the cubic lattice. The last term on the right hand side is equal to zero when $a=a_c^+$. Thus, by plotting $\log (\C{C}_n(a)/\C{C}_m(a))$ against $\log (n/m)$, a set of curves should be seen which intersect when $a=a_c^+$. At this point the coefficient of $\log \sfrac{n}{m}$ is an estimate of $\alpha\phi$. Since $\alpha=0$ in this model (and $\phi=\sfrac{1}{2}$), the critical point can also be determined by solving for $a$ in $$\log \LB \Sfrac{\C{C}_n(a)}{\C{C}_m(a)} \RB = 0 .$$ Solving this for $n\in[150,500]$ and $m\in[n\minus 100,n\minus 10]$ gives a large collection of estimates. The average is $$a_c^+ = 1.324 \pm 0.012,
\label{eqn27CC}  $$ where the confidence interval is one-half of the largest difference between two estimates in the collection. This result is smaller than the estimate in equation , and larger than the best estimate in equation .

These results indicate that there may remain sources of systematic errors in the data and in the analysis, and that the estimates for $a_c^+$ should be considered in this context.

figure27-.tex

### The microcanonical density function:

The microcanonical density function of visits in adsorbing positive walks is determined from the microcanonical data in the model, and is given by $$P^+(\eps) =\lim_{n\to\infty} (c_n^+(\lfl \eps n \rfl))^{1/n} = \lim_{n\to\infty} P_n^+(\eps),$$ where $P_n^+(\eps) = (c_n^+(\lfl \eps n\rfl))^{1/n}$ is a finite size approximation to $P^+(a)$. Existence of $P^+(\eps)$ can be shown (see for example reference [@JvR15]), and $\log P^+(\eps)$ is a concave function of $\eps$.

$P^+(\eps)$ can be determined by interpolating the finite size approximations $P_n^+(\eps)$ and then extrapolating to $n=\infty$ by fitting a least squares model to the data. In figure \[figure9-3\] the data for the extrapolated function $P^+(\eps)$ is plotted together with $P_n^+(\eps)$ for $n=100$ and $n=500$.

A least squares fit of a quadratic to $\log P^+(\epsilon)$ for $\eps\in[0,0.1]$ gives $\log P^+(\eps) \approx 1.54378\minus 0.28704\eps\minus 0.08784\eps^2$. By taking the right derivative and then taking $\eps\to 0^+$, an estimate for the critical point is obtained: $$a_c^+ \approx 1.332 .
\label{eqn27EE}  $$ The free energy is the Legendre transform of $\log P^+ (\epsilon)$. This may be estimated by fitting a polynomial to $\log P^+(\epsilon)$. If a cubic polynomial in $\eps$ is fitted to $\log P^+(\eps)$ for $0 \leq \eps \leq 0.5$, then the estimated free energy for $a>a_c^+$ is approximately $$\fl
F (a) \approx  1.6526 -0.4240\log a  -  (0.3184 \minus 1.319\log a)
\sqrt{-0.2438 \plus 1.0105 \log a}.$$ The critical point can be estimated as that location where the square root in the above is zero. This gives $a_c^+ \approx 1.273$. Similarly, the factor $(0.3184-1.319\log a)$ vanishes when $a_c^+ \approx 1.273$. These estimates are far less secure than the estimates above, and are also smaller.

figure28-.tex

figure29-.tex

### Metric data:

The mean square radius of gyration $R_n^2$ and mean height $H_n$ of the endpoint of the walk can be calculated as a function of $a$. In the desorbed phase (for $a<a_c^+$) it is expected that $R_n^2 \sim n^{2\nu}$, and $H_n \sim n^\nu$, where $\nu = 0.587\ldots$ [@C10] is the metric exponent. This scaling changes in the adsorbed phase (when $a>a_c^+$); in this phase it should be the case that $R_n^2 \sim n^{3/2}$ and $H_n \sim \hbox{constant}$, since an adsorbed walk in the cubic lattice should have the statistics of a walk in one dimension lower.

These expectations are supported by the data, as seen, for example, in figure \[figure8-3\], where data for the mean square radius of gyration are plotted as a function of $a$. These graphs clearly show two scaling regimes, namely a high temperature phase (when $a< a_c^+$) where the walk has bulk critical exponents and is desorbed, and a low temperature phase where the walk stays near the adsorbing boundary and has critical exponents of a walk in one dimension lower.

The metric exponent $\nu$ may be estimated from $R_n^2$ by examining the ratios $$2\,\nu_{n,m}(a) = \frac{\log ( R_n^2/R_m^2 )}{\log (n/m)}.
\label{eqn41-3}   $$ Here, $\nu_{n,m}(a)$ is a function of $n$ and $m$. By averaging over $m$, the estimate $\nu_n(a) = \LA \nu_{n,m}\RA_m$ can be determined. Taking the average for $100\leq m \leq 500$ in multiples of $5$ (and for $m\not= n$) gives an estimate for $\nu_n(a)$. The results are plotted in figure \[figure8-nu3\] for $n\in\{50,100,150,\ldots,500\}$. The data for $a\leq 1.2$ give $\nu \approx 0.592$, and for $a \geq 1.7$, $\nu \approx 0.740$.

The scaling of $\nu_n(a)$ as a function of $\tau=n^\phi(a\minus a_c^+)$ can be uncovered by plotting the data in figure \[figure8-nu3\]. This gives a set of curves which are very close to one another, uncovering a scaling function where $\nu_n(a) =\hbox{\Large$\nu$}(\tau)$.

figure30-.tex

The average height of the endpoint of the walk is plotted as a function of $a$ in figure \[figure8HH3\]. The left panel displays the height normalised by division with $n^\nu$ and gives a set of curves which increase with $n$ for $a< a_c^+$, and decrease with $n$ for $a>a_c^+$. The curves intersect close to the critical adsorption point, and the limiting curve (in the $n\to\infty$ limit) should be a step function with critical point at $a=a_c^+$.

figure31-.tex

The vertical metric exponent $\nu^\perp$ can be estimated from $H_n$, by using a method similar to equation , namely an approximation by examing the ratios of $H_n$: $$\nu^{\perp}_{n,m} (a) = \frac{\log (H_n/H_m)}{\log (n/m)}.
\label{eqn42-3}   $$ The exponent is approximated by $\nu^{\perp}_n(a) = \LA \nu_{n,m}(a)\RA_n$. Taking the average for $100\leq m \leq 500$ for fixed $n$ gives esimates for $\nu^\perp_n(a)$. If $a<a_c^+$, then $\nu^\perp_n(a)$ should have value approximately equal to $\nu$; that is, $\nu_n (a) \approx 0.58\ldots$, and for $a>a_c^+$, $H_n \simeq \hbox{const}$ so that $\nu_n^{\perp}(a) \approx 0$ in this phase. The results are plotted in figure \[figure6-nuH3\] against $\tau = n^\phi(a\minus a_c^+)$.

figure32-.tex

Since $R^2_n \sim n^{2\nu_a}$ where $\nu_a = \nu \approx 0.588\ldots$ if $a<a_c^+$, and $\nu_a = \sfrac{3}{4}$ if $a>a_c^+$, the ratio of $R^2_{2n}$ and $R^2_n$ is given by $$\frac{R^2_{n}}{R^2_{2n}} \approx 2^{-2\nu_a} ,\q\hbox{or}\q
\frac{2^{2\nu}R^2_{n}}{R^2_n} \approx 
\cases{
1, & if $a< a_c^+$; \\
2^{2\nu - 2\nu_a} \approx 2^{-0.32}, & if $a>a_c^+$.
}$$ In figure \[figure8RR3\] this is plotted as a function of the rescaled variable $\tau = n^{1/2}(a\minus a_c^+)$ for $n$ from $25$ to $250$ in steps of $25$. The curves coincide well with increasing $n$ and signals a transition when $a=a_c^+$ from the desorbed scaling regime into the adsorbed scaling regime. When $a>a_c^+$, $2^{-0.32} = 0.801\ldots$, as shown in the graph. A similar approach using the heights of the endpoint would involve plotting $$\frac{H_{n}}{H_{2n}} \approx 2^{-\nu_a} ,\q\hbox{or}\q
\frac{2^\nu H_{n}}{H_{2n}} \approx 
\cases{
1, & if $a< a_c^+$; \\
2^{\nu - \nu^\perp} \approx 2^{0.588}, & if $a>a_c^+$,
}$$ where $\nu^\perp$ is the vertical metric exponent. When $a>a_c^+$, $2^{0.588} = 1.503\ldots$, as shown in figure \[figure8RR3\].

figure33-.tex

### The generating function:

The generating function of adsorbing walks in the cubic lattice is given by equation , where $c_n^+(v)$ is again the number of walks from the origin of length $n$ in $\mathL^3_+$, and with $v$ visits to the adsorbing boundary $\partial\mathL^3_+$. Approximations to $G(a,t)$ are given by $G_N(a,t)$ in equation , and $G_{500}(a,t)$ was calculated using the approximate values of $c_n^+(v)$ obtained by sampling with the GAS algorithm. The critical curve is given by equation (see figure \[figure4-6\]). $G(a,t)$ is singular when $t=t_c^+(a)$, and if $t>t_c^+(a)$, then $G(a,t)$ is divergent. In figure \[figureAdsrb-3\] the approximation $G_{500}(a,t)$ is plotted, with the location of the critical curve, and critical point $(a_c^+,t_c^+)$ indicated (where, as before, $t_c^+ = t_c^+(a_c^+)$). The critical curve is similar to the critical curve in figure \[figure4-6\], and the critical point divides the critical curve into two curves, namely a curve where the transition is a high temperature or desorbed walk marked by $\tau_0$, and a curve where the transition is to a low temperature or adsorbed walk marked by $\lambda$. Along $\tau_0$ the critical curve is given by $t_c^+(a) = 
\sfrac{1}{\mu_3}$, for $a\leq a_c^+$ and where $\mu_3$ is the growth constant of self-avoiding walks in the cubic lattice.

![$G_{500}(a,t)$ as a function of $(a,t)$ for adsorbing walks in $\mathL_3^+$. The critical curve $t_c(a)$ is the black curve, and the location of the critical point at $a=a_c^+$ is indicated. Below the critical curve $G(a,t)$ is finite, and approximated well by $G_{N}(a,t)$ (for large $N$ and not too close to the critical curve). Above the critical curve $G(a,t)$ is divergent, while $G_N(a,t)$ is finite.[]{data-label="figureAdsrb-3"}](figure34-.jpg)

The critical curve is parametrized by the scaling fields $(\sigma,g)$ as shown in figure \[figure4-6\]. Here, the scaling fields are given by $g = t_c^+\minus t$, and $\sigma=a\minus a_c^+$. The singular points in $G(a,t)$ along $t_c^+(a)$ are described by the scaling assumptions shown in equation . The exponent $\gamma_1$ can be estimated by putting $a=1$ so that $G(1,t) \sim g^{-\gamma_1}$. By using the model in equation , the estimate $$\gamma_1=0.725\ldots$$ is obtained. This result is close to the estimate $0.697(2)$ in reference [@HG94]. A similar analysis with $a=a_c^+$ gives the estimate $$\gamma_s=1.203\ldots
\label{eqn71}   $$ for the surface exponent at the critical adsorption point. This is slightly smaller than the estimate $1.304(16)$ in reference [@ML88A].

figure35-.tex

The situation is less clear in the adsorbed phase. The adsorbed walk should have the scaling of a self-avoiding walk in two dimensions, so that $\gamma_+$ is given by the entropic exponent of walks in $d=2$ ($\gamma_+=\sfrac{43}{32}$). Putting $a=4$ and plotting $Z_n^{1/n}(4)$ against $n$ gives an estimate for the critical value $t_c^+(4)$. In this case the data quickly converges to $t_c^+(4)=0.0926\ldots$ (to four decimal places). Assuming that $t_c^+(4)=0.0926$ and choosing the scaling field $g=(0.0926 \minus t)$ gives a model similar to equation . Plotting the left hand side as a function of $\sfrac{1}{\log g}$, and fitting it to a quadratic for $t\in[0,0.06]$, gives the estimate $\gamma_+ \approx 1.2\ldots$, still well below the expected result $\sfrac{43}{32} = 1.34375$. Examination of the data shows strong dependence of this result on the range of $g$ in the model. For example, a fit with $t\in[0,0.09]$ gives a smaller value $\gamma_+ \approx 1.1\ldots$. These variable results indicate that $G_{500}(a,t)$ is not a good approximation to $G(a,t)$ near the critical curve $\lambda$ in figure \[figure4-6\] for adsorbing walks in the cubic lattice.

In the vicinity of the critical point $(a_c^+,t_c^+)$ the generating function should exhibit scaling given by equation . Plotting $g^{\gamma_s}\,G(a,t)$ (with $\gamma_s=1.203$) against the combined variable $g^{-1/2}\sigma$ should expose the scaling function $f$. In figure \[figureGscale-3\] this is done by plotting $\log (g^{\gamma_s}\,G(a,t) )$ against $g^{-1/2}\sigma$ for $g=\sfrac{1}{\mu_3}\minus t \in [0.01,0.05]$ and $\sigma
=(a\minus a_c^+) \in [-\sfrac{1}{2}g^{1/2},\sfrac{1}{2}g^{1/2}]$.

The partition function (see equation ) also exhibit scaling for large $n$. The scaling assumption in equation applies here as well, where $\log \mu_a = \C{F}(a)$ and $\log \mu_a = - \log t_c(a)$. As before, the exponent $\gamma_t$ is related to the $\gamma_s$-exponent as in equation : $$\gamma_t-1 = \gamma_s - 1 \approx 1.203 - 1 = 0.203.$$ That is, when $a$ is close to $a_c^+$ (and $a<a_c^+$), then the partition function has asymptotic behaviour $$Z_n(a) \simeq n^{0.203} h(n^\phi (a_c^+\minus a))\, \mu_a^n .
\label{eqn73}   $$ This result may be tested by plotting $n^{0.203} Z_n(a_c^+) \, (t_c(a))^n$ against $\tau = n^\phi (a\minus a_c^+)$. This scaling is seen in figure \[figureZscale3\] for $a<a_c^+$ and $n\in\{50,100,150,\ldots,500\}$; all the data accumulate along a single curve, exposing the scaling function $h$.

figure36-.tex

Conclusions {#section4}
===========

The adsorbing self-avoiding walk is a classical model in rigorous and numerical statistical mechanics, and have received considerable attention in the physics and mathematics literature [@deG79; @HTW82; @LM88A; @HG94; @JvRR04].

In this paper the feasibility of collecting data in the microcanonical ensemble on adsorbing walks using a flat histogram implementation of the GAS algorithm [@JvRR09] was considered. This is an approximate enumeration algorithm, and the data can be used to directly estimate partition and generating functions, from which thermodynamic functions such as the free energy and specific heat can be determined (see, for example, reference [@JvR15]).

The implemementation of the algorithm was done using endpoint elementary moves on half space self-avoiding walks, and the algorithm sampled from a flat histogram with reasonable success in both length and energy in the square and cubic lattices. Analysis of the data gives good results, better than previous Monte Carlo simulations in, for example, references [@HG94; @JvRR04], but not as good as exact enumeration data in references [@BWO99; @BGJ12]. A significant advance of this algorithm is that its produces a large amount of microcanonical data. Modifications to obtained data with respect to other quantities are trivial, and can easily be implemented. The simulations reported here were done on a Dell Inspiron 530 desktop machine, but note that the algorithm can be implemented in parallel on a cluster with each cluster generating an independent sequence. This should give radically improved statistical data.

The success of the implementation suggests that this numerical method may be used on other models (collapsing self-avoiding walks [@ML89; @TJvROW96], for example). However, it may be necessary to extend the method by introducing, in addition to the sets of parameters denoted by $\{\beta_{\ell,u}\}$ and $\{\gamma_{\ell,u}\}$, additional sets of parameters which are conjugate to classes of elementary moves. For example, in the model of collapsing walks (see figure \[figure3\]), the energy may be changed by $\Delta u \in \{-2d\plus 1,-2d\plus 2, \ldots,2d\minus 1\}$ by an elementary move, and parameters may be introduced for each value of $\Delta u$ to achieve flat histogram sampling (in a way similar to the introduction of $\gamma_{\ell,u}$ for elementary moves increasing the energy of the walk. This will increase the complexity of the implementation, but with the result that flat histogram sampling will be easier to achieve.

In the models of square and cubic lattice adsorbing walks, the algorithm produced data which gave good estimates of the locations of the critical adsorption point. The best estimates are obtained from equations and , namely $$a_c^+ =
\cases{
1.779 \pm 0.003, & \hbox{in the square lattice}; \\
1.306 \pm 0.007, & \hbox{in the cubic lattice}.
}$$ These results can be used to estimate the crossover exponent $\phi$ association with the adsorption transition, and our best esimates are seen in equations and : $$\phi =
\cases{
0.496 \pm 0.009, & \hbox{in two dimensions}; \\
0.505 \pm 0.006, & \hbox{in three dimensions}.
}$$ In addition, other quantities from which $a_c^+$ and $\phi$ can be estimated were examined, and results largely consistent with the above values were obtained (see, for example, equations , and for square lattice results, and equations , and for cubic lattice results). These numerical estimates are in good agreement with those presented in reference [@JvRR04], and also in reference [@BGJ12] in the case of the square lattice. The estimate in this reference, obtained from exact series data, namely $a_c^+ = 1.77564$, agrees with the estimate above to two decimal places. In the cubic lattice the estimate for $a_c^+$ above is slightly smaller than the estimates $a_c^+=1.338
\pm 0.005$ [@ML88A] and $a_c^+ = 1.334 \pm 0.027$ in reference [@JvRR04] (rounding up of this last error bar gives a confidence interval which includes $1.306$).

The signature of the adsorption transition in the metric quantities of the model was also examined. The scaling of these quantities with $\tau = n^{\phi}(a\minus a_c^+)$ were plotted in figures \[figure6-D\], \[figure6-nuH\] and \[figure8RR\] in the square lattice, and in figures \[figure6-D3\], \[figure6-nuH3\] and \[figure8RR3\] in the cubic lattice. These results show a transition strongly characterised by changes in metric scaling and verify the value of the metric exponent and its finite size scaling through the critical point.

Finally, the scaling of the generating function and partition function in these models were examined. Our results strongly supports the conventional properties of the model, and the values of the exponents $\{\gamma_1,\gamma_s,\gamma_+\}$ estimated here are consistent with exact values and other estimates in the literature.

The results in the square lattice are consistent with the exact values of $\gamma_1$ and the surface exponent $\gamma_s$, and the generating and partition partition function exhibit scaling consistent with the value of $\gamma_s$, as shown in figures \[figureGscale\] and \[figureZscale\]. In the cubic lattice our data gave the estimates $\gamma_1 \approx 0.725$ and $\gamma_s \approx 1.203$. These values are in addition to estimates elsewhere in the literature (see references [@HG94], [@ML88A]), and although the estimates here may be improvements on previous esimates, they remain uncertain. However, scaling of the generating function in figure , and of the partition function in figure \[figureZscale3\], is some evidence that the esimate for $\gamma_s$ is at least consistent with the scaling in the model.

[**Acknowledgements:**]{} EJJvR acknowledges financial support from NSERC (Canada) in the form of a Discovery Grant.

[**References**]{}
