---
abstract: 'In this paper, we discuss possible qualitative approaches to the problem of KPZ universality. Throughout the paper, our point of view is based on the geometrical and dynamical properties of minimisers and shocks forming interlacing tree-like structures. We believe that the KPZ universality can be explained in terms of statistics of these structures evolving in time. The paper is focussed on the setting of the random Hamilton–Jacobi equations. We formulate several conjectures concerning global solutions and discuss how their properties are connected to the KPZ scalings in dimension 1+1. In the case of general viscous Hamilton–Jacobi equations with non-quadratic Hamiltonians, we define generalised directed polymers. We expect that their behaviour is similar to the behaviour of classical directed polymers, and present arguments in favour of this conjecture. We also define a new renormalisation transformation defined in purely geometrical terms and discuss conjectural properties of the corresponding fixed points. Most of our conjectures are widely open, and supported by only partial rigorous results for particular models.'
author:
- Yuri Bakhtin
- Konstantin Khanin
bibliography:
- 'Burgers.bib'
title: 'On global solutions of the random Hamilton–Jacobi equations and the KPZ problem'
---

Introduction {#sec:intro}
============

The problem of the KPZ phenomenon and universality has been one of the most active directions in statistical physics in the last decade, see  [@Alberts-Khanin-Quastel:MR3189070], [@Amir-Corwin-Quastel:CPA20347], [@Baik-Deift-Johansson:MR1682248], [@Balazs-Cator-Seppalainen:MR2268539], [@Borodin-Corwin:MR3152785], [@Borodin-Corwin-Ferrari:MR3207195], [@Borodin-Corwin-Ferrari-Vetho:MR3366125], [@Borodin-Ferrari:MR2438811], [@Borodin-Ferrari-Sasamoto:MR2430639], [@Borodin-Gorin:MR3526828], [@Calabrese-Le-Doussal-Rosso:0295-5075-90-2-20002], [@Calabrese-Le-Doussal:PhysRevLett.106.250603], [@Cator-Groeneboom:MR2257647], [@CaPi-ptrf], [@Corwin:MR2930377], [@Corwin-Quastel-Remenik:MR3373642], [@Dotsenko:0295-5075-90-2-20003], [@Gubinelli-Perkowsky:MR3592748], [@Hairer:MR3274562], [@Hairer:MR3071506], [@Le-Doussal-Calabrese:1742-5468-2012-06-P06001], [@Moreno-Remenik-Quastel:MR3010188], [@Prahofer-Spohn:MR1933446], [@Quastel-Spohn:MR3373647], [@Sasamoto-Spohn:MR2628936], [@Sasamoto-Spohn:PhysRevLett.104.230602], [@Seppalainen:MR2917766], [@Tracy-Widom:MR1385083], and multiple other contributions. A fascinating feature of the problem is a combination of two factors: exact solvability and universality. On the one hand, one can write exact formulas for the limiting objects of some particular models. On the other hand, these formulas are supposed to describe the limiting behavior of a huge class of systems that are not integrable. The universality is so global that in a certain sense we do not know how far it stretches. One can say that any large scale “directed” variational problem in 2-dimensional disordered media is expected to belong to the KPZ universality class. In this paper, we concentrate not on exact solutions but rather on the universality phenomenon.

Another remarkable aspect of the problem is the fact that it is tightly connected to diverse areas of mathematics and theoretical physics: PDEs, stochastic analysis, dynamical systems, statistical mechanics, random matrix theory, stochastic geometry, representation theory, to name a few.

We want to emphasize the connection with the problem of global solutions to random Hamilton–Jacobi equations. We consider both cases — classical Hamilton–Jacobi equations that correspond to the study of minimisers of Lagrangian action also called geodesics and their parabolic regularizations where action minimisers are replaced by directed polymers. In our approach, geometrical properties of action minimisers and polymers play an important role. We believe that these geometrical properties make random Hamilton–Jacobi equations a better playground than random matrices where no geometric aspects are currently known. We should warn the reader that the number of rigorous results in the area is rather limited. In most cases we provide explanations and conjectures rather than theorems. We believe that many problems that we discuss can me attacked mathematically. Others are more challenging. However, we see value in presenting the general picture and the set of ideas describing the phenomena.

The structure of the paper is the following. In Section \[sec:HJ-eq\], we define random Hamilton–Jacobi equations in any dimension and discuss their global solutions. We consider the general case of convex Hamiltonians rather than only quadratic ones.

To discuss properties of solutions, we introduce the notion of directed polymer that generalizes the usual one defined through the Hopf–Cole transformation and Feynman–Kac formula in the quadratic Hamiltonian case. Such generalized polymers are discussed in Section \[sec:HJ-polymers\]. We postpone the discussion of the equivalence of two notions of polymers in the quadratic Hamiltonian case to Section \[sec:equiv-of-poly-for-Burgers\] playing the role of an appendix.

In Section \[sec:shape\], we discuss the phenomenology of KPZ scaling exponents. We introduce the notion of shape function and demonstrate how its strong convexity property is related to $1:2:3$ KPZ scalings in dimension $1+1$.

In Section \[sec:min-shocks-1d\], we discuss the properties of minimisers and shocks in the 1D case. We present rigorous results in the compact case of periodic forcing potentials, and discuss how the behaviour of minimisers and shocks changes in the non-compact setting. We also discuss hyperbolic properties of minimisers playing an important role in the picture.

In Section \[sec:renorm-point\], we discuss point fields that play the role of structural backbones of the global geometry of minimzers and directed polymers. These point fields correspond to locations of concentration of minimisers/polymers and shocks separating the domains of attraction to those locations. We then define the renormalisation transformation acting on these point fields and formulate several conjectures related to fixed points of this transformation and their stability. The transformation is defined in purely geometrical terms without involving action values. This is a reflection of monotonicity which is present only in dimension 1. The scheme we discuss is more general than the KPZ phenomenon, with a 1-parameter family of fixed points, where the parameter is the exponent describing the rate of decay of density of the point field with time. We conjecture that the statistics of these fixed points is universal and reflects only the properties of interlacing between concentration points and shocks.

The KPZ case corresponds to the density decay rate exponent being equal to $2/3$. Another special value is $1/2$ which is related to stochastic flows of diffeomorphisms and coalescent Brownian motion. This is the simplest instance of the fixed point, and we discuss it in Section \[sec:colescing-BM\]. We also discuss the correlation functions of the fixed point which have been shown to exhibit Pfaffian structure. It is natural to ask whether a similar Pfaffian property may hold for other density exponents.

Another renormalization scheme based on Airy sheet is discussed in Section \[sec:renorm-airy\].

We finish with concluding remarks in Section \[sec:concluding\].

[**Acknowledgements.**]{} Large parts of this paper were written in KITP in Santa Barbara and CIRM in Luminy, and we are grateful to these centers for hospitality. The work of Yuri Bakhtin was partially supported by NSF through grant DMS-1460595. The work of Konstantin Khanin was supported by NSERC Discovery Grant RGPIN 328565.

Hamilton–Jacobi equations {#sec:HJ-eq}
=========================

We will consider the following randomly forced Hamilton–Jacobi equation:

$$\label{eq:general-Hamilton--Jacobi}
 \partial_t \Phi +  H(\nabla \Phi)={\nu}\Delta \Phi - F,$$

Here $\Phi=\Phi^\nu(t,x)$, $t\in{{\mathbb R}}$, $x\in{{\mathbb R}}^d$, is a scalar function. The Hamiltonian $H=H(p)$ is assumed to be a convex function of momentum $p=\nabla\Phi\in{{\mathbb R}}^d$; $F=F(t,x)$ is the external force potential, i.e., $f(t,x)=-\nabla F(t,x)$ is the external force. The viscosity parameter $\nu\ge 0$. We shall consider both viscous case where $\nu>0$ and the inviscid case where $\nu=0$. In the former case, solutions are smooth in space while in the latter, they typically devolop shocks, i.e., gradient discontinuities. Although there are many weak solutions with discontinuities, we will consider only the special one called a viscosity solution that can be defined as the limit of smooth solutions $\Phi^\nu$ as $\nu\downarrow 0$, but better understood via the Lax–Oleinik variational principle described below.

In this system, the randomness comes into play via the forcing that is assumed to be random, $F(t,x)=F_\omega(t,x)$, where $\omega$ is an element of a probability space $(\Omega,{\mathcal{F}},{\mathsf{P}})$. We will assume that $F$ is space-time translation invariant and, without loss of generality, represented via a flow of time shifts $(\theta^t)_{t\in{{\mathbb R}}}$ that are ${\mathsf{P}}$-preserving transformations of $\Omega$ such that $F_\omega(t,\cdot)=F_{\theta^t\omega}(0,\cdot)$. We will also assume that $F$ has smooth in $x$ realizations with fast spatial decay of correlations but may be white in time. This setup does not create any difficulty in mathematical treatment of the model, and the solutions are well-defined, see [@ekms:MR1779561], [@Gomes-Iturriaga-Khanin:MR2241814].

Clearly, if $\Phi$ is a solution of , then, for any constant $c\in{{\mathbb R}}$, $\Phi+c$ is also a solution. We often do not distinguish solutions that differ by a constant. Moreover, often we take the point of view that only spatial increments of potentials $\Phi$ are important and work in the space of potentials modulo additive constants.

We will often use the random Hamiltonian $H_\omega(p,x,t)$ that includes the random potential: $$H_\omega(p,x,t)=H(p)+F_\omega(t,x),$$ and the random Lagrangian that is defined as the Legendre transform of the random Hamiltonian:

$$L_\omega(v,x,t)=\sup_{p}( p\cdot v-H_\omega(p,x,t))=L(v)-F_\omega(t,x),$$ where $$L(v)=\sup_{p}( p\cdot v-H(p)).$$ Here $p$ and $v$ form a pair of dual variables with $p=\nabla_v L(v)$ and $v=\nabla_p H(p)$. We will always assume that the Lagrangian $L(v)$ grows superlinearly as $|v|\to\infty$.

Let us describe the Lax–Oleinik variational principle for the solution of the initial-value problem for the inviscid Hamilton–Jacobi equation, with a continuous initial condition $\Phi_0(\cdot)=\Phi(0,\cdot)$ satisfying a growth assumption at infinity, e.g., at most linear growth. If $\nu=0$, then the solution $\Phi(t,x)$ is given by the following formula:

$$\Phi(t,x)=\Phi^0(t,x)=\inf_{\gamma:\gamma(t)=x}\left[\Phi_0(\gamma(0))+\int_0^t L_\omega(\dot \gamma(s), \gamma(s),s)ds\right],$$ where the infimum is taken over all absolutely continuous paths $\gamma:[0,t]\to{{\mathbb R}}^d$ satisfying $\gamma(t)=x$.

In the viscous case $\nu>0$, this variational formula is replaced by a stochastic optimal control representation: $$\label{eq:stoch-control-problem}
 \Phi(t,x)=\Phi^{\nu}(t,x)=\inf_{u} {\mathsf{E}}\left[\Phi_0(\gamma_t)+\int_0^t L_\omega (u(s,\gamma_s), \gamma_s, s)ds   \right],$$ where $u(\cdot,\cdot)$ is a Markov control and $\gamma$ is the solution of the following SDE $$\begin{aligned}
\label{eq:controlled-sde}
 d\gamma_s&=u(s,\gamma_s)ds+\sqrt{2{\nu}}\,dB_s,\\
 \label{eq:controlled-sde-initial}
\gamma_t&=x,
 \end{aligned}$$ with respect to the natural filtration that increases in the reverse time.

We stress that the standard Brownian motion $B$ is not related to the disorder in the potential $F$, and is used solely as an auxiliary object in this representation. It is intuitive that in the limit ${\nu}\to 0$, two variational formulas match. It can be shown that the stochastic optimal control $u$ is equal to the velocity field corresponding to $\Phi(t,x)$, see [@Gomes-Iturriaga-Khanin:MR2241814], [@Fleming-Soner:MR2179357].

Namely, $u$ can also be represented as the Legendre conjugate $$u(t,x)=\nabla_p H(p,x,t)\Bigr|_{p=p(t,x)},$$ of $$p(t,x)=\nabla\Phi(t,x).$$ Thus, in addition to its variational meaning, representation – can be understood as a self-consistency condition.

The most studied example corresponds to quadratic Hamiltonian $H(p)=|p|^2/2$. In this case, $L(v)=|v|^2/2$, and the momentum and velocity are equal to each other. The Hamilton–Jacobi equation is then equivalent to the Burgers equation $$\label{eq:Burgers}
 \partial_t u + u\cdot \nabla u={\nu}\Delta u + f$$ on the velocity field $u=\nabla\Phi$, where $f=-\nabla F$. Here we use the traditional notation $u$ for velocity.

The quadratic case is special in many ways. In particular, if $H(p)=p^2/2$, then the Hopf–Cole transformation $$\label{eq:Hopf-Cole-0}
\Phi(t,x)=- 2 {\nu}\ln Z(t,x)$$ reduces equation  to the following linear heat equation: $$\label{eq:parabolic-model}
\partial_t Z(t,x)={\nu}\Delta Z(t,x)+\frac{F(t,x)}{2{\nu}}Z(t,x),$$ that is often called the parabolic Anderson model and can be solved using the Feynman–Kac formula: $$\begin{aligned}
 \label{eq:Feynman-Kac}
  Z[Z_0](t,x)&={\mathsf{E}}e^{\frac{1}{2{\nu}}\int_0^{t}F(t-s,x+\sqrt{2{\nu}}B_s)ds}Z_0(x+\sqrt{2{\nu}} B_t)\\
\notag           &={\mathsf{E}}e^{\frac{1}{2{\nu}}\left[\int_0^{t}F(t-s,x+\sqrt{2{\nu}}B_s)ds-\Phi_0(x+\sqrt{2{\nu}} B_t)\right]},\end{aligned}$$ where $Z_0(\cdot)=Z(0,\cdot)$ and $\Phi_0(\cdot)=\Phi(0,\cdot)$ are the initial conditions and $B$ is a standard Brownian motion in ${{\mathbb R}}^d$.

It is natural to interpret the Feynman–Kac formula as averaging with respect to the Gibbs state with free measure given by the Wiener measure on continuous paths and Boltzmann weights obtained from the potential energy of the path. Namely, we can introduce the distribution ${\mathsf{P}}_{t,x}$ on continuous paths on $[0,t]$ absolutely continuous with respect to the Wiener measure in reverse time emitted at point $x$ at time $t$ with variance $2{\nu}$, with density $$p_{t,x}^{\omega}(\gamma)=\frac{1}{Z[{{\mathbf{1}}}](t,x)}e^{\frac{1}{2{\nu}}\int_0^{t}F(t-s,\gamma_s)ds},$$ where ${{\mathbf{1}}}$ is the initial condition identically equal to $1$. Then one can rewrite  as averaging with respect to the polymer measure $P_{t,x}$: $$\label{eq:FK-via-polymers}
 Z(t,x)=\langle Z_0(\gamma_0)\rangle_{{\mathsf{P}}_{t,x}}Z[{{\mathbf{1}}}](t,x).$$

In case $d=1$ and the potential $F$ being space-time white noise, equation  with quadratic Hamiltonian is known as the KPZ equation. In this case making sense of solutions of this equation that have to have very low regularity is a nontrivial problem related to recent work of Martin Hairer, see, e.g. [@Hairer:MR3071506]. By assuming that our potential is smooth in $x$, we avoid this problem altogether. This is a natural assumption since the universality phenomena related to the KPZ equation concern the large scale picture and microscopic details of the setup should not be essential.

In the definition of polymer measures as Gibbs distributions, the role of temperature is played by the viscosity parameter $\nu$. In particular, the zero-viscosity limit for Hamilton–Jacobi equations can be understood via zero-temperature limits for the associated polymer measures. This connection also extends to generalized polymer measures arising in the stochastic control construction of solutions of general Hamilton–Jacobi equations discussed above.

We now turn our attention to the problem of global solutions. We are interested in the behaviour of solutions over long time intervals. In order to see statistically stationary behaviour, we must consider the dynamics on equivalence classes, so that at any given time two functions that differ by a constant are considered identical.

There are two ways to look at this problem. The first one is more traditional. Due to the nature of white noise, one can view the solution as a Markov process in appropriate functional space ${\mathcal{L}}$ and study the long-term statistical properties by looking at stationary distributions of this Markov process. Another way is to describe the evolution by a skew-product structure on the space $\Omega\times{\mathcal{L}}$. One can define a nonrandom flow on this space by $$\Theta^t(\omega,\Phi)=(\theta^t\omega, S^t_\omega\Phi),$$ where $S^t_\omega$ is the solution operator for the Hamilton–Jacobi equation from $0$ to $t$ defined by variational formulas. One can study invariant measures of $\Theta^t$ with fixed marginal distribution ${\mathsf{P}}$ on $\Omega$. Any such measure $\mu$ can be represented as $$\label{eq:disintegration-into-sample-measures}
 \chi(d\omega,d\Phi)={\mathsf{P}}(d\omega)\mu_\omega(d\Phi).$$

The measure $\chi$ is called physical and measures $\mu_\omega$ are called sample measures if the dependence of $\mu_\omega$ on $\omega$ is measurable with respect to ${\mathcal{F}}^{-\infty,0}$ which is generated by $(F(t,x))_{t\le 0, x\in{{\mathbb R}}}$, i.e., it is determined by the history of the forcing from $-\infty$ to the present $(t=0)$, and $\mu_{\theta^t\omega}= S^t_\omega\mu_{\omega}$.

Ledrappier–Young [@Ledrappier-LSY:MR968818] proved that that there is a natural one-to-one correspondence between invariant distributions for Markov processes and physical invariant measures for the skew product. Namely, every invariant measure $\mu$ of the Markov process can be represented as the ${\mathcal{L}}$-marginal of a measure $\chi$ from and thus admits a representation $$\mu(\cdot)=\int_{\Omega} {\mathsf{P}}(d\omega)\mu_\omega(\cdot).$$

What is described above is a general setting for random dynamical systems. In the case of random Hamilton–Jacobi equation, the situation is more special. It turns out that in this case, the conditional distributions $\mu_\omega$ are delta-measures concentrated on particular global solution $\Phi_\omega(t,x)$ defined almost surely. Obviusly the uniqueness of global solution can be valid only up to an additive constant. It also depends on an additional parameter $b\in{{\mathbb R}}^d$ that can be viewed as the average momentum. Namely, under mild conditions on the stationary potential $F(t,x)$, the class of function of the form $\phi(x)=b\cdot x+\psi(x)$ where $\psi(x)$ has sublinear growth as $|x|\to\infty$ is invariant under the Hamilton–Jacobi dynamics. Although we consider global solutions without initial conditions, they still “remember” the parameter $b$. In other words for every $b\in{{\mathbb R}}^d$, there is a unique global solution $\Phi_{b,\omega}(t,x)=b\cdot x+\psi_{b,\omega}(t,x)$, where $\psi_{b,\omega}$ is sublinear in $x$.

The property where a unique point $\Phi_{b,\omega}(t,\cdot)$ is compatible with the history of the forcing is called One Force — One Solution Principle (1F1S). In other words, for fixed $b$, the global solution $\Phi_{b,\omega}(t,\cdot)$ is a deterministic functional of the forcing up to time $t$. This property immediately implies uniqueness of invariant distribution $\chi$ for the skew-product system and hence uniqueness of a stationary distribution $\mu$ for the associated Markov process.

The real meaning of 1F1S is that the global solutions play the role of one-point random pullback attractors. Namely, solutions of the Hamilton–Jacobi equation with initial data of the form $b\cdot x+\psi(x)$ given at a time $-T$ converge to the global solution $\Phi_{b,\omega}(t,\cdot)$ as $T\to\infty$.

1F1S has been established for random dynamics defined by equation  in several settings.

In [@ekms:MR1779561], 1F1S was obtained in the inviscid setting for quadratic Hamiltonian and space-periodic forcing potential under some nondegenracy conditions. These results were clarified and expanded to the multidimensional periodic situation and other convex Hamiltonians in [@Iturriaga:MR1952472], [@Gomes-Iturriaga-Khanin:MR2241814], [@Boritchev-Khanin], [@KeZhang-Khanin2017]. In all these cases, the function $\psi_{b,\omega}$ introduced above is periodic in space.

Convergence of positive viscosity invariant measures to zero viscosity invariant measure for the torus case was also obtained in [@Gomes-Iturriaga-Khanin:MR2241814]. In these situations, the problem is effectively compactified since one works on the circle or torus. Another similar result in compact setting is 1F1S for Burgers equation with random boundary conditions [@yb:MR2299503].

In the noncompact case, 1F1S has been established in 1D for quadratic Hamiltonian in the viscous and inviscid situations under certain assumptions on the forcing [@Bakhtin-quasicompact], [@BCK:MR3110798], [@kickb:bakhtin2016], [@Bakhtin-Li:2016arXiv160704864B], [@Bakhtin-Li:preprint]. It also can be proved in higher-dimensional ($d\ge 3$) quadratic Hamiltonian case with positive viscosity and weak forcing. This special situation is called weak disorder and was studied starting with 1980’s [@Imbrie-Spencer:MR968950], [@Bolthausen:MR1006293], [@Sinai:MR1341729], [@Kifer:MR1452549], [@Comets-Shiga-Yoshida:MR1996276].

One of the main conjectures we want to formulate in this paper is that 1F1S is valid in full generality:

\[conj1\] For any $d\in{{\mathbb N}}$, and any $F_\omega(t,x)$ with exponential decay of space-time correllations, 1F1S holds, i.e., for every $b\in{{\mathbb R}}^d$, there is a unique time-stationary (modulo time-dependent additive constants) global solution $\Phi_{b,\omega}(t,x)=b\cdot x+\psi_{b,\omega}(t,x)$ where $\psi_{b,\omega}$ has sublinear growth. These solutions are continuous in $\nu\ge0$. In particular, global solutions are preserved under the zero-viscosity limit.

Due to randomness, the role of $b$ is not essential in this picture in contrast with the Aubry–Mather theory where arithmetic properties of $b$ play a crucial role. Often, if the value of $b$ is not indicated, we assume $b=0$.

The uniqueness of global solution of the Hamilton–Jacobi equation is tightly related to the problem of existence and uniqueness of one-sided minimisers or geodesics in the inviscid case and one-sided directed polymers in the viscous case.

Below we discuss this connection in the inviscid case. The Lax–Oleinik variational principle deals with minimisers defined on finite time intervals. For global solutions, one has to consider minimisers on intervals of the form $(-\infty,t]$. A curve is called a one-sided minimiser if any compact perturbation of the curve that does not affect the endpoint and the infinite tail, can only increase the action. The counterpart of the conjecture above is the following conjecture.

Under the same assumptions as in Conjecture \[conj1\], almost surely the following holds:

(1) Every one-sided minimiser $\gamma$ has an asymptotic slope $$a=\lim_{s\to-\infty}\frac{\gamma(s)}{s}\in{{\mathbb R}}^d.$$

(2) For fixed $a\in{{\mathbb R}}^d$ and all $(t,x)$, there is a minimiser with endpoint $(t,x)$ and asymptotic slope $a$.

(3) For fixed $a\in{{\mathbb R}}^d$ and Lebesgue-a.e. $(t,x)$ a minimiser is unique.

(4) For every two minimisers $\gamma_1,\gamma_2$ with the same slope, $$\lim_{s\to-\infty}|\gamma_1(s)-\gamma_2(s)|=0.$$

(5) Let us fix $a\in{{\mathbb R}}^d$ and define $$\Phi^a_\omega(t,x)=\int_{-\infty}^t L_\omega(\dot\gamma_{t,x}(s),\gamma_{t,x}(s),s)ds-
    \int_{-\infty}^0 L_\omega(\dot\gamma_{0,0}(s),\gamma_{0,0}(s),s)ds,$$ where $\gamma_{t,x}$ is a minimiser with slope $a$ and endpoint $(t,x)$. Then $\Phi^a$ is a global solution of Hamilton–Jacobi and there is a uniquely defined $b=b(a)\in{{\mathbb R}}^d$ such that $\Phi^a=\Phi_b$.

(6) The function $\Phi^a$ defined above is continuous and locally Lipschitz. Points $(t,x)$ of non-uniqueness of one-sided minimisers are called shocks. They are discontinuity points of $\nabla\Phi^a$.

The solution $\Phi=\Phi^a$ is stationary in time if we consider it modulo additive constants. For fixed $t$, the field $\Phi(t,x)$ is not stationary in space but it has stationary spatial increments. In 1D, this allows to formulate the following conjecture:

\[eq:CLT-conjecture\] Assume that $a=b=0$. Then there is a constant $\sigma>0$ such that the rescaled process $(\Phi(0,sx)/(\sigma\sqrt{s}))_{x\in{{\mathbb R}}}$ converges in distribution to the standard two-sided Wiener process as $s\to+\infty$.

If we assume that the Hamiltonian $H$ is an even function of $p$, then, by symmetry $b(0)=0$. In general, the connection between $b$ and $a$ is determined by the effective Hamiltonian which is the Legendre transform of the shape function. We will discuss the shape function and its connection to the effective Hamiltonian later. Notice that in the general case, one should subtract the linear form $bx$ in the above conjecture.

We finish this section with an argument explaining why minimisers with different endpoints must be asymptotic to each other in the reverse time. The argument can be applied in any dimension.

Assume again that $a=0$ and consider “point-to-line” minimisers on a finite time interval $[-T,0]$, where $T\gg 1$. Consider points $x$ and $y$ at distance of order $1$ apart. With large probability the difference in action of minimisers with enpoints $(0,x)$ and $(0,y)$ is of order 1, since otherwise the action of one of them can be decreased by merging with the other one. On the other hand the variance of action of each minimiser grows with $T$. Hence, the difference of actions is of order $1$ only if these minimisers coincide or get close to each other for large negative times.

Generalized directed polymers {#sec:HJ-polymers}
=============================

In the previous section, we defined directed polymers associated with the Hamilton–Jacobi equation with quadratic Hamiltonian and positive viscosity.

Recall that the polymers were defined as paths on interval $[0,T]$. Similarly to the situation with one-sided minimisers, it will be convenient to consider polymers to evolve backwards in time on intervals $[-T,0]$.

The main goal of this section is to extend this notion to more general Hamiltonians where the Hopf–Cole transformation is not available. Asymptotic properties of the classical directed polymers were studied in [@Carmona-Hu-2004:MR2073415] and [@Comets-Shiga-Yoshida:MR1996276]. Their findings can be summarized in the following statement.

[**Proposition.**]{} [*In dimensions $d=1$ and $d=2$, the directed polymer is localized. In dimension $d\ge 3$, there is a transition between diffusive behavior for small forcing potentials and localization for large forcing.* ]{}

Rigorous mathematical methods allow to prove only weak forms of localization, namely, that there is a constant $\alpha>0$ not depending on time and a family of arbitrarily large times $t$ and intervals $I_\omega(t)$ of length $1$ such that the probability to find the endpoint of the polymer in $I_\omega(t)$ is at least $\alpha$. However, the localization picture to have in mind is that for typical $\omega$ the variance of the endpoint of the polymer with respect to the polymer measure is bounded. A subsequential version of this claim was recently established in [@Bates-Chatterjee:2016arXiv161203443B]. This is in sharp contrast with diffusive behavior where the variance of the endpoint is of order $T$.

Let us now define generalized polymers for general Hamiltonians. For a fixed $\omega$ at $T>0$, the directed polymer on time interval $[-T,0]$ is the stochastic process $X$ that solves the following SDE in reverse time $$\begin{aligned}
\label{eq:gen-poly}
dX_s&=u(s,X_s)ds+\sqrt{2\nu}dB_s,\\ 
X_0&=x,\notag\end{aligned}$$ where $u(\cdot,\cdot)$ is the velocity field corresponding to the solution $\Phi$ of the Hamilton–Jacobi equation on $[-T,0]$ with $\Phi(-T,x)\equiv 0$: $$u(s,x)=\nabla_p H(\nabla\Phi(s,x)).$$ We call the distribution of $X$ the generalized polymer measure. Naturally, one can consider polymers associated to more general initial conditions $\Phi(-T,\cdot)$.

In the quadratic Hamiltonian case, the generalized polymer measures coincide with the usual Burgers–Hopf–Cole–Feynman–Kac–Gibbs polymer measures. This is an important fact that justifies the new notion. We postpone a detailed explanation of this connection to Section \[sec:equiv-of-poly-for-Burgers\].

It is also possible to state the positive viscosity versions of conjectures in Section \[sec:HJ-eq\] in terms of generalized polymers. The role of one-sided minimisers in this positive temperature situtation is played by one-sided polymer measures that can be viewed as thermodynamic limits of finite volume directed polymers. These notions are classical in the case of quadratic Hamiltonian, see [@Bakhtin-Li:2016arXiv160704864B], and can be naturally extended to general Hamiltonians and associated polymers. Moreover, we conjecture that in the zero-temperature limit $\nu\to 0$, the generalized polymers converge to their respective ground states, i.e., one-sided Lagrangian minimisers, see [@Bakhtin-Li:preprint] for the quadratic Hamiltonian case.

Also based on the analogy with the quadratic case, it is natural to state the following conjecture:

In dimensions $d=1$ and $d=2$, the generalized directed polymer is localized for any convex Hamiltonian with superlinear growth of the associated Lagrangian. In dimension $d\ge 3$, the generalized polymer undergoes a transition from diffusive behavior for small forcing potentials to localization for large forcing.

Although the description of the generalized polymers through SDEs is less explicit than the one provided by the Feynman–Kac formula, it has significant advantages as well. First, the drift $u$ in  is the optimal control in the variational principle . Secondly, this drift is given by the velocity field which has a clear physical meaning and provides an intuitive explanation of the localization phenomenon.

It is well-known that in the inviscid case, the velocity field pushes points towards shocks. When viscosity is positive, the velocity still pushes them towards compact shock-like regions working against the diffusion effect provided by white noise. If the shock-like domains are periodic in space (say, in the periodic case), the polymer would still have diffusuve behavior since transitions between those regions, although rare, happen with positive rate, so over large times the process can be viewed as a random walk over those regions. However, if, due to fluctuations, the polymer reaches a domain with isolated shock-like region evolving in time, then the velocity field will provide a potential barrier that the polymer will not be able to escape through. Hence the polymer will be localized near a time-dependent zone concentrated around the shock. So, in contrast to the periodic case where the transition rates between neighboring shock-like regions are constant which leads to diffusive behavior, in the nonperiodic case the typical escape times vary from trap to trap thus resulting in an situation similar to Sinai’s random walk in random environment where deep traps create a localization effect.

In dimensions starting with $3$, however, there is also another effect that has to be taken in consideration. A small potential barrier cannot trap diffusing particles. Hence for small velocity fields generated by small potentials the polymer will have diffusive behavior. This explains the transition from weak disorder to strong disorder in high dimensional case.

Shape function and KPZ scalings {#sec:shape}
===============================

In this section, we discuss the shape function which plays an important role in all the discussions below. To define it one has to consider either point-to point minimisers or directed polymers conditioned to end at a certain point. In other words, consider a minimiser $\gamma$ on a time interval $[-T,0]$ with boundary conditions $\gamma(0)=0, \, \gamma(-T)=-aT$, or a directed polymer corresponding to $\nu>0$ with the same boundary conditions. We start with the case of quadratic Hamiltonian. We denote the Lagrangian action of the minimiser by $A_\omega(a, T)$ and the corresponding partition function by $$Z_{\nu,\omega}(a,T)=\int e^{\frac{1}{2\nu}\int_{-T}^0 F_{\omega}(s,B_s)ds}P_{\nu,aT,0}^{-T,0}(dB),$$ where $P_{\nu,-aT,0}^{-T,0}$ is the distribution of the Brownian bridge of variance $2\nu$ connecting points $-aT$ and $0$ between times $-T$ and $0$. Then the following general statement follows from the sub-additive ergodic theorem.

For any $\nu\ge 0$, there is a nonrandom convex function $S_\nu(a)$ such that for every $a\in{{\mathbb R}}^d$, almost surely $$\begin{aligned}
&\lim_{T\to \infty} \frac{A_\omega(a, T)}{T} = S_0(a),\\
&\lim_{T\to \infty}\frac{-2\nu\log Z_{\nu,\omega}(a, T)}{ T} = S_\nu(a),\quad \nu>0.\end{aligned}$$

The problem with the above statement is that it is very difficult to control the shape functions $S_\nu(a)$. In most applications, one needs to know that they are strictly convex and have a non-degenerate minimum at the origin. Although there are no doubts that this should be true in a very general situation, including general Hamiltonians, at present we don’t have proper technical tools to prove such a statement. In all the cases where this can be done, one uses special symmetries which allow to get an exact formula for $S_\nu(a)$. One such special property is the so called shear-invariance. The random potential $F_\omega(t,x)$ is said to be shear-invariant if for all $a \in {{\mathbb R}}^d$, the processes $F_\omega(t,x+at)$ and $F_\omega(t,x)$ have the same distribution. This property is satisfied for many natural models. In the case of quadratic Hamiltonian one can calculate $S_\nu(a)$ explicitly provided the shear-invariance holds. It turns out that $$S_\nu(a)= S_\nu(0) + \frac{|a|^2}{2},\quad \nu\ge 0,\ a\in{{\mathbb R}}^d.$$ Below we deal with the case of minimisers. The polymer case is similar. Notice that any curve $\gamma_a$ going from the origin to the point $-aT$ on the time interval $[-T,0]$ can be obtained from a curve $\gamma$ starting and terminating at the origin, by adding a linear function $at$. By the shear invariance the distribution for the action of $\gamma$ and $\gamma_a$ is the same apart from the kinetic term. Since $$\int_{-T}^0\frac{|\dot \gamma_a|^2(s)}{2}ds= \int_{-T}^0\frac{|\dot \gamma (s)+a|^2}{2}ds=
\int_{-T}^0\frac{|\dot \gamma(s)|^2}{2}ds + \frac{a^2}{2}T,$$ it follows that $S_0(a)= S_0(0) + |a|^2/2$.

For general Hamiltonians, the definition of shape function for positive viscosity must be modified and can be based on the linear growth of solutions of the associated Hamilton–Jacobi equation with initial condition concentrated near $-aT$ at time $-T$. Even for shear-invariant potentials strict convexity of the shape function is an open question.

Notice that the shape functions are closely related to the concept of homogenisation for random Hamilton–Jacobi equations. Since the slope $a$ equals the average velocity, the shape function can be considered as the effective Lagrangian. Then its Legendre transform is the effective Hamiltonian $H_{{\mathrm{eff}}}$. To clarify this connection, consider the inviscid Hamilton–Jacobi equation with initial condition given at $-T$ by $b\cdot x+\psi(x)$ for a a fixed $b\in {{\mathbb R}}^d$. To find the slope $a\in {{\mathbb R}}^d$ corresponding to $b$, notice that for a given slope $a$ the action on the time interval $[-T,0]$ will be $(S_0(a) - a\cdot b)T$ plus terms sublinear in $T$. Minimizing $\min_a[S_0(a) - a\cdot b]$ we obtain $a(b) = {\mathop{\mathrm{argmax}}}(a\cdot b - S_{0}(a) )$ and $H_{{\mathrm{eff}}}(b)= \max_a[a\cdot b - S_{0}(a)]$. This also implies that $a(b)=\nabla H_{{\mathrm{eff}}}(b)$.

We now switch to the one-dimensional case and demonstrate how the KPZ scaling exponents follow from our conjectures and the quadratic behaviour of the shape function. What makes the one-dimensional case so special is the property of monotonicity. Minimisers cannot cross each other and maintain the same order at all times. On the other hand, all one-sided minimisers are asymptotic to each other backwards in time. This suggests the following picture. Consider $T \gg 1$ and all one-sided minimisers on time interval $(-\infty, 0]$. By time $-T$ the minimisers will concentrate in small (exponential in $T$) intervals separated by large intervals. Each small interval of concentration generates a large interval of minimiser endpoints at time $t=0$. We will discuss this picture in more details in the next section. Considering the small concentration intervals as points we get two random point fields, one at time $t=0$, and another at time $t=-T$. The first point field consists of points of separation between long intervals corresponding to neighbouring concentration domains. Those separation points are, in fact, points of shocks, since each of this points has at least two minimisers going to different concentration domains (see Figure \[fig:shock-minimiser-duality\]).

![Long-time behaviour of one-sided minimisers. The time axis is directed upward. The dots repesent locations of high concentration of minimisers. The crosses represent shocks separating the minimisers.[]{data-label="fig:shock-minimiser-duality"}](x-0.pdf){width="8cm"}

Denote by $L(T)$ a characteristic length corresponding to time scale $T$. It can be defined as a typical distance between neighbouring points of any of the two point fields described above. It can also be viewed as a typical displacement of the one-sided minimiser on time interval of length $T$. Denote by $\xi$ the corresponding critical exponent, namely $L=L(T)\sim T^\xi$. The second critical exponent $\chi$ corresponds to typical fluctuations of the action of one-sided minimisers on time intervals of length $T$ which are assumed to be of the order of $T^\chi$. We shall show below that $\xi=2/3$, $\chi=1/3$. Sometimes these scalings are referred to as 1:2:3 scalings.

Let us compare the one-sided minimiser with endpoint $(0,0)$ to the optimal path $\bar\gamma$ on $[-T,0]$ beginning and terminating at the origin. If the displacement of the one-sided minimiser is of the order of $L$, then, compared to $\bar\gamma$, its action increases by a term of the order of $L^2/T$ due to the quadratic behaviour of the shape function, but it can also decrease due to the fluctuations of the action on time interval $[-T,0]$ and fluctuations of the tail action on the interval $(-\infty, -T]$. The first of these fluctuations is of the order $T^\chi$, while the second is of the order of $\sqrt{L}$ due to the conjectured diffusive scaling for the stationary solutions. The scale $L$ is determined by either assuming that $$\label{eq:first-term-dominates}
L^2/T \sim \sqrt{L}\quad \text{and}\quad T^\chi\lesssim \sqrt{L},$$ or, in the opposite case, $$\label{eq:second-term-dominates}
L^2/T \sim T^\chi,\quad \text{and}\quad \sqrt{L}\lesssim T^\chi.$$ In the case $\chi > 1/3$ the first assumption of leads to $L\sim T^{2/3}$ and, hence, $T^\chi \gg \sqrt{L}$ which contradicts the second condition of . It follows that the first assumption of  holds, which implies $L\sim T^{(1+\chi)/2}$ and $\sqrt{L}\sim T^{(1+\chi)/4}\ll T^\chi$. This, however, contradicts the predicted diffusive behaviour for the global solution. Indeed, suppose that points $x$ and $y$ are at the distance of the order of $L$ apart, then, since the fluctuations of each $\Phi_\omega(0,x)$, $\Phi_\omega(0,y)$ are of the order of $T^\chi$, the difference $\Phi_\omega(0,x) -\Phi_\omega(0,y)$ will be at least of order $T^\chi$ which is much larger than $\sqrt{L}$. We conclude that $\chi\leq1/3$. It is easy to see that in this case both   and lead to $L^2/T \sim \sqrt{L}$, and, hence, $L\sim T^{2/3}$ and $\xi=2/3$.

Let us show that $\chi = 1/3$. Suppose that $\chi< 1/3$ and consider two points $x$ and $y$ at distance of the order of $L$ apart. On the time scale of the order of $T\sim L^{3/2}$ their minimisers will almost merge with positive probability. On a slightly larger time scale $T\sim L^{3/2+\epsilon}$, the minimisers will merge with large probability. Then the difference $\Phi_\omega(0,x) -\Phi_\omega(0,y)$ will be determined by two contributions: one from the shape function, and another from the fluctuation of action on time interval $[-T,0]$. The position of the concentration point at time $-T$ will be of the order of $T^{2/3}\sim L^{1+\frac{2}{3}\epsilon}$. Hence the shape function contribution will be of the order of $L\cdot L^{1+\frac{2}{3}\epsilon}/T\sim  L^{2+\frac{2}{3}\epsilon}/T\sim L^{\frac{1}{2}-\frac{1}{3}\epsilon}$. The second contribution will be of the order of $T^\chi\sim L^{(3/2 +\epsilon)\chi}$. Now one can take $\epsilon$ so small that the exponent satisfies $(3/2 +\epsilon)\chi< 1/2$. Thus $|\Phi_\omega(0,x) -\Phi_\omega(0,y)|\ll
\sqrt{L}$ which again contradicts the diffusive behaviour assumption.

minimisers and shocks in the 1D case {#sec:min-shocks-1d}
====================================

In this section, we consider in more detail the behaviour of one-sided minimisers and shocks in the one-dimensional case. In the previous section, we already mentioned the monotonicity property that means that minimisers do not intersect. Here we present a more complete picture.

Let us consider the field of one-sided minimisers with fixed, say, zero slope.

Let us consider two times $t_1<t_2$ and denote space variables at these times by $y$ and $x$, respectively. If there is a unique one-sided minimiser with the end-point $(t_2,x)$ then there is a unique $y(x)$ such that this minimiser passes through $(t_1,y(x))$. On the other hand, if $(t_2,x)$ is a point of shock so that there is more than one minimiser at this point, then there is a closed interval $[y_l(x),y_r(x)]$ of points at time $t_1$ associated to $(t_2,x)$. Namely, these are points absorbed into the shock $(t_2,x)$. Here the point $(t_1,y_l(x))$ corresponds to the left-most minimiser with endpoint $(t_2,x)$, while $(t_1,y_r(x))$ corresponds to the right-most minimiser. The map from $x$ to $y$ is monotone, injective, but, in general, not single-valued. One can also construct the map from $y$ to $x$. Namely, if a minimiser with endpoint $(t_2,x)$ passes through $(t_1,y)$ then $x(y)=x$. Also, for any shock point $(t_2,x)$, all points $y$ from $(y_l(x),y_r(x))$ are mapped into $x$. This map is also monotone, single valued, but not injective.

Denote by $A(t_1,t_2)$ the set of points $(t_1,y)$ reachable by minimisers with endpoints $(t_2,x)$, $x\in{{\mathbb R}}$. It is easy to see that the sets $A(t_1,t_2)$ are decreasing in $t_2$, i.e., $A(t_1,t'_2)\subset A(t_1,t_2)$ as long as $t'_2>t_2$. If there is $(t_1,y) \in \cap_{t_2>t_1} A(t_1,t_2)$, then there is a global minimiser passing through $(t_1,y)$. That means that the one-sided minimiser for $(t_1,y)$ can be extended indefinitely forward in time keeping the property of being a one-sided minimiser.

If $(t_1, y)$ is a point of shock, then no one-sided minimiser defined at $t_2>t_1$ can pass through it. Therefore, this point is covered by a closed interval generated by some shock at time $t_2$. In other words, for every $t_2>t_1$ there exists a shock at $(t_2, x(y))$. That means that there is a curve of shocks $\{(x(y),t_2), t_2\geq t_1\}$, where $x(y)=y$ for $t_2=t_1$. In other words, every shock evolves forward in time, while one-sided minimisers evolve backward in time. One-sided minimisers are asymptotic to each other in backward time but do not intersect. The curves of different shocks can merge.

In a certain sense, minimisers and shocks are dual objects. However, there is also a significant difference. Namely, shocks can be created at certain space-time points, called pre-shocks. Pre-shocks correspond to singularities of the velocity field. The space derivative of the velocity field at the point of pre-shock is equal to $-\infty$. The shock curves originating at a preshock point evolve in time and merge with other existing shock curves while new shocks are being formed. It follows from the results of [@BCK:MR3110798], [@kickb:bakhtin2016] for the quadratic Hamiltonian case and is conjectured for the general Hamiltonian case that the processes of eliminating shocks through their merging, and creating new shocks through the pre-shock phenomenon are balanced, so that they produce an equilibrium corresponding to a stationary distribution of shocks. At a given time, shocks form a stationary point field. In addition there is another parameter attached to every shock, namely, its age, indicating for how long a shock can be traced in the past until the original pre-shock event. In a stationary regime, the age of a shock is just a random variable with distribution given by some density $q_s(a)da$, where $a$ is the age value. Although the density $q_s(a)$ is not universal, the behaviour of its tails reflects the universal KPZ scalings. We have seen above that one-sided minimisers after long time $T$ concentrate at certain random locations, with a point field of shocks separating intervals of minimisers corresponding to a given location. These shocks are separated by intervals of the length of order $T^{2/3}$, hence the density of the corresponding point field is of the order of $T^{-2/3}$. It is easy to see that this sparce separating point field corresponds to shocks of age $a$ of the order $T$ and larger. This implies that the density $q_s(a)$ must decay as $a^{-5/3}$.

Similarly to the discussion of separating shocks, intervals of concentration of minimisers correspond to locations where minimisers can be extended in the future for time order $T$ or larger. We shall explain below that in the general case global shocks do not exist. In other words, every one-sided minimiser will terminate being absorbed by some shock at some future time. Hence, one can define a random variable of life expectancy of one-sided minimisers. In the stationary regime this random variable has a density $q_m(a)da$. An argument similar to the previous one implies that the density $q_m(a)$ also decays as $a^{-5/3}$ as $a \to \infty$.

There is one case when a rigorous mathematical analysis of the behaviour of shocks and minimisers was carried out. This is the so called compact case. In the simplest situation, the random potential $F_\omega(t,x) = \sum_{i=1}^N F_i(x)\dot W_i(t)$, where $F_i(x), 1\leq i\leq N$, are non-random smooth $1$-periodic functions, and $\dot W_i(t), 1\leq i\leq N$, are independent white noises. Periodicity condition implies that the configuration space for the problem is the unit circle.To avoid degenerate behaviour we also assume that the map $(F_1,\ldots,F_N)$ from ${{\mathbb T}}^1={{\mathbb R}}^1/{{\mathbb Z}}^1$ to ${{\mathbb R}}^N$ is an embedding. Then the following theorem holds ([@ekms:MR1779561],[@Boritchev-Khanin]).

For every $a\in {{\mathbb R}}^1$, almost surely there exists a unique global minimiser with asymptotic slope $a$. It is a hyperbolic trajectory of the random Lagrangian flow. All other one-sided minimisers with slope $a$ asymptotically approach the global one backward in time. Moreover, the rate of convergence is exponential and is given by a non-random positive Lyapunov exponent.

In fact, this theorem also holds in multiple dimensions under the periodicity assumption, but the mechanism of hyperbolicity is very different and the proof in that case requires additional ideas, see [@KeZhang-Khanin2017].

![Minimisers and shocks on the universal cover in the periodic case.[]{data-label="fig:universal-cover"}](universal-cover.pdf){width="8cm"}

To better understand the structure of minimisers and shocks it is convenient to lift the picture to the universal cover ${{\mathbb R}}^1$ (see Figure \[fig:universal-cover\]). Different bold curves are different copies of the global minimiser. All one-sided minimisers approach the global one but they have to choose its particular representative to follow. Hence there are separation points that correspond to the global (or, topological) shock. This shock can be traced indefinitely in the past, and hence it is infinitely old. All other shocks are local in nature. This means that although there are two minimisers meeting at a local shock, they are asymptotic to each other and the same global minimiser in reverse time. All these shocks have finite age. However, the density $q(a)$ of their age decays exponentially fast as $a \to \infty$. This is in contrast with the behavior conjectured above for the nonperiodic case where $q(a)\sim a^{-5/3}$. The unique global minimiser and the unique global shock exist due to purely topological reasons.

It is expected that in the general non-periodic case there are no global minimisers and no infinitely old shocks. The reason for this is easy to see. Recall that the sets $A(t_1,t_2)$ defined above shrink as $t_2$ increases. The complementary set ${{\mathbb R}}^1 \setminus  A(t_1,t_2)$ of locations forbidden for one-sided minimisers grows as $t_2 \to \infty$. It is natural to expect that any compact subset of ${{\mathbb R}}^1$ will belong to the forbidden set after some large time $t_2$. So, as $t_2$ increases there are simply no place available for a global minimiser.

The conjectured behaviour of minimisers and shocks is somewhat similar to the periodic case, but has significant differences as well. All one-sided minimisers are asymptotic to each other with the exponential rate of convergence. However, two minimisers can be separated for a long time by an old shock. Eventually they will come close and will start to converge exponentially, see Figure \[fig:clustering-of-minimisers\]. The time of separation $T$ and the original distance between minimisers $L$ are connected by the KPZ scaling relation $T\sim L^{3/2}$. It is natural to expect that for large $L$ the distance between two minimisers backward in time decays as $\exp(-\lambda (|t| -T(x,y))_+ )$, where $\lambda$ is a non-random Lyapunov exponent, and $T(x,y)\sim L^{3/2}$.

![Minimisers and shocks in noncompact case[]{data-label="fig:clustering-of-minimisers"}](long-minimizers.pdf){width="10cm"}

The picture described above is mostly conjectural. Only few results in this direction have been proved for few concrete models. We wanted to present it in order to emphasise two aspects that are not always discussed in the literature. The first one is the structural interaction between minimisers and shocks. The second one is the hyperbolicity of minimisers. Although the hyperbolicity is not uniform, it is still a very essential feature of the problem. It is important that two different phenomena are present simultaneously: the separation by old shocks whose density decays algebraically, and the exponential convergence when minimisers are not separated anymore.

Renormalization for point fields of concentration and separation {#sec:renorm-point}
================================================================

In this section, we look more closely at the point fields of concentration and separation defined above.

Let us fix $T\gg 1$ and consider a sequence of times $-iT, \, i\geq0$. For each time $-iT$, we consider all one-sided minimisers with endpoints $(x,-iT), x\in {{\mathbb R}}^1$. As before, we record small intervals of concentrations of these minimisers at time $-(i+1)T$ and the separating point field at time $-iT$. As we saw above, these point fields will have density of the order of $T^{-2/3}$. Let us rescale them by $const\,T^{2/3}$ so that the resulting point fields will have density $1$. We also rescale time by a factor $T$ so that we have two point fields at every strip $[-i, -i-1]$ (see Figure \[fig:points-and-crosses\] where shocks are denoted by crosses, and concentration points by dots).

Another important piece of this structure is the connector field. We can enumerate intershock intervals by integers ${{\mathbb Z}}$, from left to right, assigning $0$ to the interval containing the origin. Similarly, we can enumerate the concentration points assigning $0$ to the point closest to the origin. Every interval between two neighbouring shocks corresponds to a particular point of concentration, thus defining a map ${{\mathbb Z}}\to{{\mathbb Z}}$. Since this map is monotone, it is enough to determine the label of the point which corresponds to $0$-interval. We denote this integer number by $\xi$, and define a related sequence of random variables $\xi_i$, called connectors, for every strip $[-i, -i-1]$. Notice that the sequences of point fields and connectors for different strips form a stationary but strongly correlated sequence. In addition, two point fields corresponding to any strip are also strongly correlated.

![The renormalisation transformation on point fileds of crosses, dots, and connectors. The solid lines represent connectors before the transformation. The dashed lines represent the connectors after the transformation. The crosses and dots with no adjacent dashed lines are eliminated under the renormalisation.[]{data-label="fig:points-and-crosses"}](renorm-x-0.pdf){width="12cm"}

We next define a renormalisation transformation corresponding to rescaling the time by the factor of two. The transformation can be defined separately on individual strips of width 2 in time units of the previous step. Namely, out of a configuration of crosses, dots, and connectors in each strip of the form $[-2i,-2i-2]\times{{\mathbb R}}, \, i\ge 0$, we will obtain a new configuration of crosses, dots, and connectors in the strip $[-i,-i-1]\times{{\mathbb R}}$.

After the renormalisation transformation, a cross can be present at $(-i,x)$ only if there was a cross at $(-2i,x)$ before the renormalisation. Some crosses survive the renormalisation, and some get eliminated. Also, a dot can be present at $(-i-1,x)$ only if there was a dot at $(-2i-2,x)$ before the renormalisation. Some dots survive the renormalisation, and some get eliminated. The point fields of dots and crosses at odd times $-2i-1,$ $i\ge 0$, will be discarded but they are used to decide which crosses on $\{-2i\}\times{{\mathbb R}}$ and dots on $\{-2i-2\}\times{{\mathbb R}}$ survive the renormalisation and which ones get eliminated. The elimination procedure that mimics the behaviour of shocks and minimisers is the core of the transformation. It can be described as follows.

We start with a point field of crosses at time $-2i$, both point fields, of crosses and dots, at time $-2i-1$, and a point field of dots at time $-2i-2$. In addition, each interval between two crosses at time $-2i$ is mapped to a particular dot at time $-2i-1$, each interval between two crosses at time $-2i-1$ is mapped to a particular dot at time $-2i-2$. Consider now an interval between two neighbouring crosses at time $-2i-1$.

If there are no dots inside this interval, then the dot corresponding to this interval at time $-2i-2$ should be eliminated. Indeed, no one-sided minimisers at time $-2i$ will reach the interval between crosses, so the concentration point to which the interval is mapped, also cannot be reached.

Now, suppose that there are $k$ dots inside the above interval between crosses. At time $-2i$, there are $k+1$ neighbouring crosses corresponding to these $k$ dots. It is easy to see that for consistency with behaviour of shocks and minimisers, we have to eliminate all but the first and the last cross so that the whole interval gets mapped into one dot at time $-2i-2$. The other crosses do not separate anything anymore and have to be eliminated. Note that if $k=1$, we will have only two crosses upstairs, the first and the last, and we don’t eliminate anything at all (see Figure \[fig:points-and-crosses\]). This elimination procedure should be carried out for all strips. After that, one should rescale time by the factor $2$, and rescale point fields so that they will again have density $1$. Note that the new connectors are uniquely defined by the above procedure. The result is one step of the renormalisation transformation $R$. Note that the rescaling can be done after several steps since it commutes with the renormalisation procedure. Moreover, due to this commutativity, one can naturally define renormalisation in the space of point fields modulo rescaling.

Since at each step we double the time interval, iterating the transformation we defined above results in exponentially growing time intervals. One can also define it for linearly growing time intervals. In this case one should add strips one by one.

Suppose we applied the renormalisation by adding together $n$ strips of the initial width one. The resulting fields of dots and crosses, before rescaling, will have a density which decays as $n\to \infty$. The rate of decay is not defined by the procedure. For example, in the KPZ case it decays as $n^{-2/3}$. Later we will see cases where the density behaves as $n^{-1/2}$. In general the density will decay as $n^{-\alpha}$. A critical exponent $\alpha \in (0,1)$ is a dots-crosses counterpart of the critical exponent $\xi$ introduced above to characterize large time displacements of one-sided minimizers, i.e. $\alpha=\xi$. It is an interesting question whether values of $\alpha$ different from $2/3$ and $1/2$ can appear in physical problems.

For every critical exponent $\alpha\in(0,1)$, there is a fixed point for the renormalisation transformation $R$ with density scaling as $n^{-\alpha}$. The fixed points are stable except for one neutral direction corresponding to the parameter $\alpha$. It follows that if a random process has density decay $n^{-\alpha}$, it will converge to the $\alpha$-fixed point under the action of $R$. In other words, the $\alpha$-fixed point provides universal asymptotic statistics for all systems with exponent $\alpha$.

In the above conjecture, the stability of fixed points is suggested when the point-connector configurations to which the renormalisation transformation is applied have fast decay of space-time correlations. Otherwise, one cannot expect convergence to $\alpha$-fixed points.

If the conjecture is correct, then it provides an insight on how far the KPZ universality extends. What is actually required is monotonicity and the scaling $L\sim T^{2/3}$. The condition of exact monotonicity can be probably replaced by a weaker assumption of approximate monotonicity.

Let us stress again that the renormalisation transformation above deals only with geometric data. In other words, we record only the positions of dots and crosses but not the values of action. Thus it is very different from the renormalisation transformations based on so called Airy sheet that have been discussed in the literature previously. The geometric objects like concentration dots and separation crosses that we use can be naturally defined for minimisers and directed polymers. It is an interesting problem to find a counterpart of these objects in the setting of random matrices.

Coalescing Brownian Motions and Stochastic Flows {#sec:colescing-BM}
================================================

There is one case where the renormalisation conjecture can be studied rigorously. This is the case of the density scaling exponent $\alpha=1/2$. Then the configurations of dots, crosses, and connectors, are independent for different strips. Essentially this case was studied in [@Piterbarg-thesis],[@Piterbarg:MR1626162],[@2Piterbarg:MR1686850]. Below, we briefly discuss the setting and present the main results.

Consider the stochastic flow $$\label{eq:SF}
dx(t)=f_\omega (t,x(t))dW(t),$$ where $W(t)$ is the standard Wiener process, and $f_\omega (t,x)$ is a random function similar tho the random potential $F_\omega (t,x)$. We also assume that $f_\omega (t,x)$ space-time stationary, smooth in $x$ and white in $t$. The randomness of $f_\omega (t,x)$ and the one provided by the white noise are completely separate, i.e., independent. The stochastic flow (\[eq:SF\]) is defined for a fixed realization of $f_\omega (t,x)$. It corresponds to solving the SDE (\[eq:SF\]) for all initial conditions $x(0)$ simultaneously using the same white noise.

It is well known that maps $S_W^t$ that assign solutions $x(t)$ to the initial conditions $x(0)$ form a family of diffeomorphisms almost surely in $W$. These diffeomorphisms are monotone which also means that trajectories $x_1(t)$ and $x_2(t)$ for two different initial conditions cannot intersect. Also, it is easy to see that $x_2(t)-x_1(t)$ is a positive martingale provided $x_2(0)>x_1(0)$. Hence there exist a $\lim_{t\to \infty}(x_2(t)-x_1(t)) = c(x_1,x_2)$. If the function $f_\omega (t,x)$ is aperiodic, then $c(x_1,x_2)=0$ and all trajectories are asymptotic to each other as $t \to \infty$. On the other hand, if $f_\omega (t,x)$ is space-periodic, say, $1$-periodic, then $c(x_1,x_2)$ can take any integer value. In this case the diffeomorfism $S_W^t$ can be considered as a diffeomorfism of the unit circle ${{\mathbb T}}^1$.

In the latter case, there is a strong similarity between the behaviour of one-sided minimisers and trajectories of the stochastic flow (\[eq:SF\]). Similarly to the case of Lagrangian minimisers, the family $(S_W^t)$ is hyperbolic. There is also a unique random point similar to the global shock. Namely, an exponentially small neighbourhood of this point gets expanded by $S_W^T$ onto almost the entire unit circle for large times $T$. At the same time, the rest of the unit circle contracts into an exponentially small interval at time $T$. To find this random point of instability one has to take initial conditions at a time $T\gg 1$ and run the stochastic flow backwards in time until time $t=0$. The concentration domain at time $t=0$ converges to the instability point in a limit as $T\to \infty$. In a same way one can find a unique random point similar to the position of the unique global minimiser. In this case one has to put the initial conditions at time $-T$ and then run the stochastic flow forward in time until time $t=0$. In the limit as $T\to \infty$ the concentration interval shrinks to this special random point. The rate of exponential contraction and expansion is determined by a non-random Lyapunov exponent. The behaviour is similar to the behaviour of one-sided minimisers in the compact case (see Figure \[fig:universal-cover\]). The only difference is that the trajectories of the stochastic flow evolve in forward time.

The situation changes in the non-compact setting. As we explained above, all trajectories are asymptotic to each other. Now there are no points that are unstable for all times. However, for a given time scale $T$, there are small intervals of instability corresponding to this time scale. Trajectories from these exponentially small intervals are expanding to intervals of the length of order $\sqrt{T}$ at time $T$. Nevertheless, if we wait for long enough time, these intervals will start to contract with exponential rate. Again, the picture is similar to Figure \[fig:clustering-of-minimisers\] representing one-sided minimisers in the noncompact case. These intervals of instability are similar to old shocks of the age of the order $T$ or larger. The only difference is the characteristic length scale for a large time $T$, which is $L\sim T^{1/2}$ now instead of $T^{2/3}$. Indeed, $T^{1/2}$ is a typical scale for the diffusion process associated with the SDE (\[eq:SF\]). This is exactly the difference between the density scaling exponents which we discussed in the previous section. The behaviour is qualitatively similar. However the scaling limit of asymptotic statistics is different since it is determined by the exponent $\alpha$.

Since the length scale is $T^{1/2}$, one should expect that two trajectories originating from distinct points $x$ and $y$ at distance $L$ apart will converge exponentially fast but nonuniformly. Namely, $|x(t)-y(t)|\sim \exp(-\lambda (t -T(x,y))_+ )$, where $\lambda$ is a non-random Lyapunov exponent, and $T(x,y) \sim L^2$. This is also similar to the behaviour of one-sided minimisers apart from the different scaling exponent. It was shown in [@Piterbarg-thesis],[@Piterbarg:MR1626162],[@2Piterbarg:MR1686850] that in the diffusive rescaling, the stochastic flow converges to the coalescing Brownian motion. However, the convergence has been proved only on a topological level. The hyperbolicity picture presented above was not established rigorously, although we believe it can be done.

We next discuss the construction of the point fields and connectors. Essentially it is similar to the construction in the case of one-sided minimisers. We first fix a large time scale $T\gg1$, and consider the time strips $[iT, (i+1)T], \, i \geq 0$. Then for each strip we consider exponentially short intervals of instability at time $iT$ which extend into long, of the order of $\sqrt{T}$, intervals at time $(i+1)T$, and exponentially short intervals of concentration of mass at time $(i+1)T$. We then apply the diffusive scaling rescaling time by $T$ and space by $const\,\sqrt{T}$. In the limit $T\to\infty$ we obtain for each rescaled time strip $[i,i+1]$ two point fields of density $1$: the field of crosses at time $i$ and the field of dots at time $i+1$. Constant in the space scaling is needed only to ensure that the density of limiting point fields is equal to $1$. The connectors between two point fields in each strip can be defined exactly in the same way as before. Indeed, the whole interval of points between two crosses is mapped into a particular dot. Then by enumerating intervals between crosses and dots we can define the connectors. We note that in the case of stochastic flows, the point fields and connectors in different strips and independent, although two point fields in a particular strip are still strongly correlated. Hence, it is enough to describe statistics in a given strip. Since the stochastic flow converges to the coalescing Brownian motion, the statistics is completely determined by the latter, and, hence, is universal.

Coalescing Brownian motion is a process defined by a collection of independent Brownian motions starting from every point $x \in {{\mathbb R}}^1$ at time $t=0$. It is also assumed that after two different Brownian motions meet at certain point of space-time they both continue to move together in forward time. The system was studied by R.Arratia in the late 70s ([@Arratia:MR2630231]). It was shown that after any positive time $t$ only countably many of distinct Brownian motions will be present, and their locations at time $t$ form a locally finite point field of positive density. Now, let us consider time $t=1$ and define a point field of positions of Brownian motions at this time (dots), and a point field of crosses at time $t=0$. As usual each interval between crosses correspond to a particular dot at time 1. A connector field is also naturally defined. It follows immediately from the scaling properties of the Brownian motion that these two point field and connector chosen independently for all the time strips $[i,i+1]$ form a fixed point for the renormalisation scheme introduced in the previous section. Moreover, it follows from [@Piterbarg-thesis],[@Piterbarg:MR1626162],[@2Piterbarg:MR1686850] that this fixed point is stable.

We finish this section with another interesting property of the above fixed point. It turns out that it carries certain exact integrability features. Namely, as it was shown in [@2Piterbarg:MR1686850], see also [@Zaboronski:MR2259212], [@Zaboronski:MR2851057] the correlation functions of the fixed point fields are expressed by Pfaffians. More precisely, consider $2n$ ordered points on the real line $x_1<x_2< \dots<x_{2n-1},x_{2n}$. They define $n$ intervals $I_1=[x_1,x_2], I_2=[x_3,x_4], \dots, I_n=[x_{2n-1}, x_{2n}]$. Consider now the event that all of these intervals are empty, that is there are no crosses at time $t=0$ inside these intervals. Then the probability of this event is given by the Pfaffian of the following $2n\times 2n$ matrix $A$: $a(i,i)=0, \, 1\leq i\leq 2n, \, a(i,j)=-a(j,i), \, i\neq j, a(i,j) = G(x_j-x_i), \, 1\leq i<j\leq 2n$, where the kernel $G(x)$ is defined by: $$G(x) =1 - \frac{1}{\sqrt{4\pi t}}\int^x_{-x}\exp\left\{-\frac{y^2}{4t}\right\}dy.$$ It is natural to ask whether similar property holds for other fixed points corresponding to different values of the parameter $\alpha$, in particular in the KPZ case.

Renormalisation for Airy sheet {#sec:renorm-airy}
==============================

In Section \[sec:renorm-point\], we constructed a renormalisation scheme based on geometrical coding. A more traditional approach is based on so called Airy processes. We define the Airy sheet in the following way. Consider a problem of point-to-point minimisers on a large time interval $[0,T]$ and define the following random process: $$\label{AI-1}
\bar A_{\omega,T}(x, y) = \frac{A_{\omega,T}(\mu(T) xT^{2/3}, \mu
(T) yT^{2/3}) - C(T)T}{\delta (T)T^{1/3}} -
\frac{\mu^2(T)}{2\delta(T)}(x-y)^2.$$ Here $A_{\omega,T}(\mu (T)T^{2/3}x, \mu (T)T^{2/3}y)$ is the minimal action between points $\mu(T)T^{2/3}x$ and $\mu (T)T^{2/3}y$ on the time interval $[0,T]$, and the quadratic term subtracted above is the compensation due to the shape function $S_0(a)$ which is assumed to be quadratic: $ S_0(a)=C+a^2/2$. We have seen above that this assumption is satisfied for systems in the case of quadratic Hamiltonians which satisfy the property of shear invariance. The constants $C(T), \mu (T)$, and $\delta(T)$ are determined by two conditions. First of all we choose $C(T)$ and $\delta(T)$ in such a way that the first two moments of $\bar A_{\omega,T}(0, 0)$ match the first two moments of the GUE Tracy–Widom law. Then we choose $\mu (T)= \sqrt{2\delta (T)}$, so that the coefficient in the quadratic term in equals 1. Now, taking the limit in distribution as $T\to \infty$, we obtain the Airy sheet ${\mathrm{Ai}}(x, y)$. The constants $C(T), \mu (T)$, and $\delta(T)$ will converge to non-universal positive constants $C, \mu, \delta$ where $C$ is the value of the shape function at $a=0$, that is $C= S_0(0)$. Note that the Airy sheet has not been constructed rigorously yet. However, the process ${\mathrm{Ai}}(0,
y)={\mathrm{Ai}}_2(y)$ known as $\mathrm{Airy}_2$ process was rigorously constructed by M.Prähofer and H.Spohn ([@Prahofer-Spohn:MR1933446], see also [@Johansson:MR1737991]). Airy sheet ${\mathrm{Ai}}(x, y)$ is translation-invariant on the $(x, y)$ plane. Its one-point marginal distributions are given by the Tracy–Widom law for the GUE ensemble of random matrices with the standard normalisation. As we have seen above, this requirement determines the choice of constants $C(T)$ and $\delta (T)$.

One can define the renormalization transformation $R$ associated to doubling the length of the time interval. The idea is that we have two independent copies of the same stationary process $A(x, z)$ and $A' (z, x)$ corresponding to two time intervals $[0,T]$ and $[T,2T]$. Then a new stationary process is defined by gluing these intervals (or the corresponding space-time strips) together and minimising over a common variable $z$. One also has to do rescaling since the time interval is of length $2T$ now. As before, we assume that the marginal distributions of the process $A(x, z)$ share the first two moments with the GUE Tracy–Widom law. The total action between points $x$ and $z$ is given by $A(x,z) + (x-z)^2$. Define now the following process: $$\label {RGAI}
B(x,y)= \min_{z}{[A(x,z) + A'(z,y) +(x-z)^2 +(z-y)^2]} -\frac{1}{2}(x-y)^2.$$ One can show that by subtracting the last term we make $B(x,y)$ a stationary process. We can now define a new stationary process $R(A)$ which is the image of the process $A$ under the action of the renormalisation transformation $R$ acting in the space of stationary processes. Namely, $R(A)(x,y)= \frac{1}{\delta
(A)} [B(\mu (A)x, \mu (A)y) - C(A)]$, where $C(A), \, \delta (A)$ are determined by the requirement that the first two moments of $R(A)(0,0)$ are the same as before, that is the same as for the GUE Tracy–Widom law. The constant $\mu(A)=\sqrt{2\delta (A)}$. It is easy to see that up to the shift and scaling the total action between points $x,y$ on the double time interval is given by $R(A)(x,y) +(x-y)^2$. Notice that due to the choice of $\mu(A)$ the coefficient in front of the quadratic term is equal to 1.

The above formula can be considered as a renormalisation transformation acting on the space of translation-invariant random fields. It is believed that this transformation has a unique fixed point which is exactly the Airy sheet ${\mathrm{Ai}}(x,y)$. Moreover this fixed point is stable in the space of translation-invariant random fields with certain condition on decay of correlations. Note that at the fixed point $A={\mathrm{Ai}}(x,y)$ the constants $C(A),\, \delta (A), \, \mu (A)$ are given by $C({\mathrm{Ai}})=0, \, \delta ({\mathrm{Ai}})=2^{1/3}, \mu({\mathrm{Ai}})= 2^{2/3}$.

The conjecture about the existence of a unique stable fixed point obviously implies universality of the KPZ phenomenon. However at present there are no rigorous approaches to the analysis of this renormalisation transformation. A nice feature of (\[RGAI\]) is that it is based on gluing together two independent copies of the same process. However, the analytical structure of the right-hand side of (\[RGAI\]) is very singular, which makes the analysis of fixed points, and, especially, their stability rather awkward.

Some connection between the Airy sheet process and point fields which we considered in this paper can be viewed in the following construction. Take a realisation of the Airy process and for a given $x$ define $y(x) = {\mathop{\mathrm{argmin}}}_{y} [{\mathrm{Ai}}(x, y) + (x -
y)^2]$. It turns out that the function $y(x)$ is locally constant. It means that it does not change when $x$ changes locally. In other words, there is a point field of points $y$ where the minimum can be attained. This point field of special values of $y$ is similar to the point field of dots that we discussed before. The end-points of intervals for different dots form a point field of crosses. Note that this dots-crosses point field is similar in spirit to the one constructed above for the one-sided minimisers but they are not the same. In particular, the former cannot be used for a simple geometrical renormalisation procedure. It is tempting to glue together two independent copies of the dots-crosses fields and define the renormalisation procedure similarly to the construction in Section \[sec:renorm-point\]. However, this will not be a self-consistent renormalisation scheme. Selection of the new dots-crosses field for the double strip will require also information about the realisation of ${\mathrm{Ai}}(x, y)$.

Concluding remarks {#sec:concluding}
==================

In this paper, we presented arguments in support of several conjectures. Some of them were related to the existence of global solutions to the random Hamilton–Jacobi equation and one-sided minimisers. The case of positive viscosity corresponds to directed polymers. We discussed why the results in both cases are supposed to be parallel. Indeed, in the case of strong disorder, directed polymers are localised and, hence, the situation is not much different from the inviscid case of one-sided minimisers. In the case of weak disorder, the mechanisms are much simpler and rather well understood. The above arguments apply in the case of quadratic hamiltonians. In the general case we define the process which can be considered as a generalization of directed polymers. We argue that the behaviour of the generalised directed polymers must be similar to the one in the quadratic case. Hence the above arguments can be applied to the general case as well.

We then concentrate on the 1D case and derive the KPZ scalings exponents from the predicted diffusive behaviour of the global solutions. After introducing the system of point fields corresponding to locations of concentration of one-sided minimisers and shocks separating domains concentrated at neighbouring locations, we define a renormalisation transformation acting on the space of such systems. It is important that the renormalisation transformation is defined in pure geometrical terms. We predict that our renormalisation scheme has family of fixed points parametrized by the scaling exponent $\alpha$ of the decrease of density of point fields with time. We also conjecture that every fixed point is stable apart from the obvious neutral direction corresponding to a change of the parameter $\alpha$. Hence systems with the same scaling exponent will have the same universal asymptotic statistical properties which are determined by the corresponding fixed point. Based on the above picture the following approach to the problem of KPZ universality looks very natural. One has first to establish the scaling exponent $\alpha=2/3$, and then prove stability of the fixed point for this exponent. The main idea here is that the exponent is determined by the physics of the problem, and then monotonicity of interlacing between minimisers and shocks requires particular statistical behaviour. In other words, the situation is extremely rigid. Basically, there exists only one statistics compatible with both interlacing and monotonicity, and given scaling exponent $\alpha$.

Let us make a couple of remarks related to the Pfaffian property of the dots-crosses fields. We have seen that this property holds in the case $\alpha=1/2$, and suggested that this may also be true for other values of $\alpha$. Notice that the kernel $G$ in the case $\alpha=1/2$ is simply a probability that two independent Wiener processes started from two points distance $x$ apart will intersect until time 1. A similar kernel can be defined in the general case. For the dots-crosses field constructed for the Airy sheet the role of the kernel $G$ would have been played by the kernel $G_{KPZ}(x), $ which is the probability that $y(x) = {\mathop{\mathrm{argmin}}}_{y}
[{\mathrm{Ai}}(x, y) + (x - y)^2]= y(0) = {\mathop{\mathrm{argmin}}}_{y} [{\mathrm{Ai}}(0, y) + y^2]$. However, apart from the analogy with the case $\alpha=1/2$ and the exact integrability associated with the Airy sheet we don’t have a serious argument on why the dot-crosses field may satisfy the Pfaffian property with this kernel.

Another remark concerns the limiting distribution for the end-point of one-sided minimisers or directed polymers rescaled by the factor $\mu T^{2/3}$. Consider the one-sided minimiser from the point $(0,0)$ and denote its position at time $-T$ by $\mu
(T) T^{2/3}y(0)$. The total action consists of two parts corresponding to time intervals $[-T,0]$ and $(-\infty,-T]$. The first one is given by $$\delta(T) T^{1/3}[\bar A_{\omega, T}(0,y) +
y^2] + C(T)T].$$ The second part is given by the global solution $\Phi_\omega(-T, \mu (T)T^{2/3}y)$. Due to the diffusive scaling it can be presented as $\sigma \sqrt{\mu (T)} T^{1/3}W(y(0)) + \Phi_\omega(-T,
0)$, where $W(y)$ is the standard Wiener process statistically independent from the Airy process ${\mathrm{Ai}}(x,
y)$. Adding two terms together and taking limit as $T \to \infty$ we get $$y(0)= {\mathop{\mathrm{argmin}}}_{y}{\left[ {\mathrm{Ai}}_2(y) + \frac{\sigma \sqrt{\mu}}{\delta}W(y) + y^2\right]}.$$ Here $\mu, \delta$ are limiting values of $\mu (T), \delta (T)$, and $\sigma$ is a normalizing constant in Conjecture \[eq:CLT-conjecture\]. The constants $\mu, \delta$, and $\sigma$ are not independent. In fact, the coefficient $\frac{\sigma
\sqrt{\mu}}{\delta}W(y)$ is matching the diffusion constant for the small increments of the ${\mathrm{Ai}}_2$ process which are known to be diffusive with the diffusion constant 2. Namely, $Var({\mathrm{Ai}}_2(y)-{\mathrm{Ai}}_2(0))\sim 2|y|$ asymptotically as $y\to 0$. Hence, $\frac{\sigma \sqrt{\mu}}{\delta}W(y)=\sqrt{2}W(y)$. Substituting into the above formula we get a universal probability distribution for the rescaled end-point of the one-sided minimiser $\gamma_{\omega,0,0}(t),
t\in (-\infty, 0]$ corresponding to space-time point $(0,0)$. Namely, as $T\to \infty$ $$\frac{\gamma_{\omega,0,0}(-T)}{\mu T^{2/3}} \to {\mathop{\mathrm{argmin}}}_{y}{\left[
{\mathrm{Ai}}_2(y) + \sqrt{2}W(y) + y^2\right]}.$$ The convergence above is in the distributional sense. Notice that one can naturally define the asymptotic distribution for the end-point for any value of the critical exponent $\alpha$. It is enough to run the renormalisation scheme until the strip of time length $n$ is reached. Then we take the concentration dot corresponding to the interval between two crosses containing the origin, rescale its position by $n^\alpha$, and take limit as $n\to \infty$.

We finish with two questions which we find interesting and challenging. We have seen before that dots-crosses point fields can be constructed starting from the Airy sheet process. Going in other direction, is it possible to reconstruct Airy processes starting from the dots-crosses point fields?

Another question is related to the scaling parameter $\alpha$ which determines different fixed points of the geometrical renormalization transformation. We have seen that $\alpha=2/3$ corresponds to the KPZ phenomenon, while $\alpha=1/2$ is connected with the stochastic flows of diffeomorphisms. One can ask whether other values of $\alpha$ can appear in some natural physical or mathematical settings. It is also interesting whether one can define an analogue of the Airy processes for the values of $\alpha$ different from $2/3$.

Appendix. Generalized polymers in the quadratic Hamiltonian case {#sec:equiv-of-poly-for-Burgers}
================================================================

The goal of this section is to explain that in the case of quadratic Hamiltonian, the generalized polymers that we introduced in Section \[sec:HJ-polymers\] coincide with the usual polymers defined via Gibbs modifications of the Wiener measure.

We first explain that polymer measures arising in the Hopf–Cole–Feynman–Kac solution of the Burgers– Hamilton–Jacobi equation define diffusion processes with drifts given by the velocity field solving the Burgers equation. Then we recall a general variational principle for Gibbs distributions and use it to derive a variational characterization of the Burgers polymer measures, thus showing that the Hopf–Cole–Feynman–Kac polymer measures coincide with generalized polymer measures constructed via stochastic control. Our results are similar to existing variational characterizations of Wiener functionals in [@Boue-Dupuis:MR1675051],[@Ustunel:MR3255483],[@Eyink-Drivas:MR3299883], see also [@Chen-Georgiou-Pavon:MR3489825], but present a different point of view.

Throughout this section we fix ${\nu}>0$.

Burgers polymers as diffusions
------------------------------

The Feynman–Kac formula  solving the linear heat equation  can be naturally interpreted as integration with respect to a Gibbs measure on paths constructed using the Wiener measure as the free measure and Boltzmann weight composed of contributions from the initial condition and the potential accumulated by paths. Let us define these objects more precisely.

For every $x\in{{\mathbb R}}^d$ and $t>0$, we define a version ${\mathsf{W}}_{t,x}$ of the Wiener measure to be the distribution of $\gamma_s=x+\sqrt{2{\nu}}B_s$ on $C([0,t])=C([0,t],{{\mathbb R}}^d)$, where $B_s$ is a standard Wiener process in ${{\mathbb R}}^d$. We define the energy function on paths by $$\label{eq:Energy}
 E(\gamma)=\frac{1}{2{\nu}}\left[-\int_0^{t}F(t-s,\gamma_s)ds+\Phi_0(\gamma_t)\right],\quad \gamma\in C([0,t]),$$ where we used a fixed realization of smooth potential $F$. Then the polymer measure ${\mathsf{P}}_{t,x}$ associated to $F$ is also a measure on $C([0,t])$, absolutely continuous with respect to ${\mathsf{W}}_{t,x}$, with density $$\label{eq:polymer-density}
\frac{d {\mathsf{P}}_{t,x}}{d{\mathsf{W}}_{t,x}}(\gamma)=\frac{e^{-E(\gamma)}}{Z[Z_0](t,x)},\quad \gamma\in C([0,t]),$$ where $Z_0=\exp\{-\Phi_0/(2\nu)\}$, and the partition function $Z[Z_0](t,x)$ is given by the Feynman–Kac formula that can be rewritten as $$Z[Z_0](t,x)=\int_{ C([0,t],{{\mathbb R}}^d)}e^{-E(\gamma)}{\mathsf{W}}_{t,x}(d\gamma).$$ For an arbitrary Markov control $v:[0,t]\times{{\mathbb R}}^d\to{{\mathbb R}}^d$, we denote by ${\mathsf{Q}}_{t,x,v}$ the distribution of solutions of the following SDE on $C[0,t]$: $$\begin{aligned}
\label{eq:controlled-diffusion} 
d\gamma_s&=-v(t-s,\gamma_s)ds+\sqrt{2{\nu}}dB_s,\\
 \gamma_0&=x,
 \label{eq:initial-condition-for-controlled-diffusion}\end{aligned}$$ which is simply the forward version of the SDE in –.

\[thm:Gibbs-coincides-with-u-controlled\] The distribution ${\mathsf{P}}_{t,x}$ of paths $\gamma$ coincides with ${\mathsf{Q}}_{t,x,u}$, where $u(\cdot,\cdot)$ is the solution of the Burgers equation .

[[[Proof: ]{}]{}]{}Recalling that due to , $$u(s,y)=\nabla\Phi(s,y)=-2{\nu}\nabla \ln Z(s,y)=-2{\nu}\frac{\nabla Z(s,y)}{Z(s,y)},$$ we use Girsanov’s theorem to write $$\begin{aligned}
\frac{d {\mathsf{Q}}_{t,x,u}}{d {\mathsf{W}}_{t,x}}(\gamma)&=\exp\left\{\frac{1}{2{\nu}}\left[\int_0^t -( u(t-s,\gamma_s)\cdot d\gamma_s) -\frac{1}{2}\int_0^t |u(t-s,\gamma_s)|^2 ds\right]\right\} 
\\&=\exp\left\{\int_0^t \left( \frac{\nabla Z(t-s,\gamma_s)}{ Z(t-s,\gamma_s)}\cdot d\gamma_s\right) -{\nu}\int_0^t  \frac{|\nabla Z(t-s,\gamma_s)|^2}{ Z^2(t-s,\gamma_s)} ds\right\}. \end{aligned}$$ Here $\gamma$ is a Brownian motion under ${\mathsf{W}}_{t,x}$, with $d\langle \gamma^i,\gamma^j\rangle_s=2{\nu}\delta^{ij}ds$ and $\gamma_0=x$. The integral with respect to $d\gamma_s$ is an Itô integral. Comparing this to , we only need to check that $X_t=-\ln Z(t,x)$, where for $r\in[0,t]$, the process $X_r$ is defined by $$\begin{gathered}
\label{eq:X-in-proof-of-Girsanov-FK}
X_r= \int_0^r \left( \frac{\nabla Z(t-s,\gamma_s)}{ Z(t-s,\gamma_s)}\cdot d\gamma_s\right) -{\nu}\int_0^r  \frac{|\nabla Z(t-s,\gamma_s)|^2}{ Z^2(t-s,\gamma_s)} ds
\\+\frac{1}{2{\nu}}\left[-\int_0^{r}F(t-s,\gamma_s)ds+\Phi(t-r,\gamma_r)\right].\end{gathered}$$ Since due to , $$\frac{1}{2{\nu}}\Phi(t-r,\gamma_r)= -\ln Z(t-r,\gamma_r),\quad r\in [0,t],$$ we have $X_0=-\ln Z(t,x)$, and it suffices to prove that $dX_r=0$. The rest of the proof is a standard stochastic calculus calculation. By the Itô formula, we have $$\begin{gathered}
 d\left[\frac{1}{2{\nu}}\Phi(t-r,\gamma_r)\right]= -d\ln Z(t-r,\gamma_r)
 \\=
 -\frac{\bigl(-\partial_tZ(t-r,\gamma_r)    +   {\nu}\Delta Z(t-r,\gamma_r)            \bigr)dr + (\nabla Z(t-r,\gamma_r)\cdot d\gamma_r)}{Z(t-r,\gamma_r)} \\
                           +\frac{{\nu}|\nabla Z(t-s,\gamma_s)|^2}{ Z^2(t-r,\gamma_r)}dr.\end{gathered}$$ Plugging this into  and using equation , we obtain $dX_r=0$, which completes the proof. [[[$\Box$ ]{}]{}]{}

The variational principle for Gibbs measures
--------------------------------------------

The following is a standard variational characterization of Gibbs measures that is sometimes called Gibbs inequality for Kullback–Leibler divergence.

\[lem:general-var-principle\] Let $\mu$ be a probability measure on a measurable space $({\mathbb{X}},{\mathcal{X}})$ and let $E:{\mathbb{X}}\to{{\mathbb R}}$ be a measurable function. Let $ \Delta$ be the space of measurable nonnegative functions $p:{\mathbb{X}}\to{{\mathbb R}}_+$ satisfying $\int_{{\mathbb{X}}}\mu(dx)p(x)=1$. We introduce the average energy functional $$I(p)=\int \mu(dx) p(x)E(x),\quad p\in\Delta,$$ the entropy functional $$h(p)= -\int_{{\mathbb{X}}}\mu(dx) p(x)\ln p(x),\quad p\in\Delta,$$ and the free energy functional $$G(p)=I(p)-h(p),\quad p\in\Delta.$$ Then the minimum of the free energy is uniquely provided by the Gibbs measure with free measure $\mu$ and energy $E$. Namely, introducing the the partition function $Z$ by $$Z=\int_{{\mathbb{X}}}\mu(dx)e^{-E(x)}$$ and the Gibbs density $q\in\Delta$ by $$q(x)=\frac{e^{-E(x)}}{Z},\quad x\in{\mathbb{X}},$$ we have $$\inf_{p\in\Delta} G(p)=G(q)=-\ln Z.$$

[[[Proof: ]{}]{}]{}We provide the well-known proof here for completeness since it is very short. Let $\varphi(x)=x\ln x$ for $x>0$ and $\varphi(0)=0$. For every $p\in\Delta$, we have $$\begin{aligned}
I(p)-h(p)+\ln Z&= \int_{{\mathbb{X}}}\mu(dx)p(x)E(x)+\int_{{\mathbb{X}}}\mu(dx)p(x)\ln p(x)+\ln Z 
\\&= \int_{{\mathbb{X}}}\mu(dx)p(x)\ln (p(x) e^{E(x)}Z) =  \int_{{\mathbb{X}}}\mu(dx)p(x)\ln \frac{p(x)}{q(x)}
\\&=\int_{{\mathbb{X}}}\mu(dx)q(x)\frac{p(x)}{q(x)}\ln \frac{p(x)}{q(x)}=\int_{{\mathbb{X}}}\mu(dx)q(x)\varphi\left(\frac{p(x)}{q(x)}\right)
\\&\ge \varphi\left( \int_{{\mathbb{X}}}\mu(dx)q(x)\frac{p(x)}{q(x)}\right)=\varphi(1)=0,\end{aligned}$$ where we used convexity of $\varphi$ and Jensen’s inequality. Also, if $p=q$, then this inequality becomes an identity, and the theorem follows, since Jensen’s inequality is strict unless $p(x)/q(x)=1$ for $\mu$-a.e. $x\in{\mathbb{X}}$. [[[$\Box$ ]{}]{}]{}

Stochastic control characterization of the Burgers polymer
----------------------------------------------------------

Let us state the main result of this section. Let us fix $\nu>0$, $t>0$, and the initial condition $\Phi_0$.

\[thm:FK-same-as-control\] There exists a unique Markov control $v:[0,t]\times {{\mathbb R}}^d\to{{\mathbb R}}^d$ realizing the minimum of the functional $$G(v)=\frac{1}{2{\nu}}{\mathsf{E}}_{{\mathsf{Q}}_{t,x,v}} \left[\frac{1}{2} \int_0^t v^2(t-s,\gamma_s)ds-\int_0^t F(t-s,\gamma_s)ds+\Phi_0(\gamma_t)\right].$$ This optimal control coincides with the solution $u$ of the Burgers equation . Moreover, ${\mathsf{Q}}_{t,x,u}$ is the polymer measure: it coincides with ${\mathsf{P}}_{t,x}$ defined in .

[[[Proof: ]{}]{}]{}We are going to apply Theorem \[lem:general-var-principle\] to $\mu={\mathsf{W}}_{t,x}$ and energy $E$ defined in . Taking any control $v:[0,t]\times{{\mathbb R}}^d\to{{\mathbb R}}^d$. and denoting by $p_v$ the Girsanov density $\frac{d {\mathsf{Q}}_{t,x,v}}{d {\mathsf{W}}_{t,x}}$, we see that, in the notation of Theorem \[lem:general-var-principle\], $$I(p_v)=\int {\mathsf{Q}}_{t,x,v}(d\gamma) E(\gamma)= \frac{1}{2{\nu}}{\mathsf{E}}_{{\mathsf{Q}}_{t,x,v}}\left[-\int_0^{t}F(t-s,\gamma_s)ds+\Phi_0(\gamma_t)\right],$$ so if we show that $$\label{eq:Girsanov-entropy}
  h(p_v)=-\frac{1}{2}\cdot\frac{1}{2{\nu}} {\mathsf{E}}_{{\mathsf{Q}}_{t,x,v}} \int_0^t\int_0^t |v(t-r,\gamma_r)|^2 dr,$$ then Theorem \[lem:general-var-principle\] will imply that the values of $G(v)$ are bounded below by the free energy of the Gibbs measure associated with energy $E(\cdot)$ and free measure ${\mathsf{W}}_{t,x}$. According to Theorem \[thm:Gibbs-coincides-with-u-controlled\] that Gibbs measure coincides with ${\mathsf{Q}}_{t,x,u}$, so our claims follow.

Let us check . We note that $p_v=e^{X_t}$, where $$X_r(\gamma)=\frac{1}{2{\nu}}\left[-\int_0^r \bigl(v(t-s,\gamma_s)\cdot d\gamma_s\bigr)
 -\frac{1}{2}\int_0^r |v(t-s,\gamma_s)|^2ds\right],\quad r\in[0,t].$$ Basic stochastic calculus implies: $$\begin{aligned}
  d e^{X_r}&= e^{X_r}dX_r+\frac{1}{2}e^{X_r}d[X]_r
          \\&=-\frac{e^{X_r}}{2{\nu}}\left[\bigl(v(t-r,\gamma_r)\cdot d\gamma_r\bigr)+\frac{1}{2} |v(t-r,\gamma_r)|^2dr\right]
+\frac{1}{2}e^{X_r}\cdot\frac{1}{2{\nu}} |v(t-r,\gamma_r)|^2dr\\ &
=-\frac{e^{X_r}}{2{\nu}}\bigl(v(t-r,\gamma_r)\cdot d\gamma_r\bigr).\end{aligned}$$

d(X\_re\^[X\_r]{})=&X\_r d e\^[X\_r]{}+e\^[X\_r]{}dX\_r+d\[X, e\^X\]\_r&\
=& - (v(t-r,\_r)d\_r)\
&- +|v(t-r,\_r)|\^2 dr&\
=& |v(t-r,\_r)|\^2 dr+dM\_t,&

where $M$ is a local martingale. Treating it as a martingale (in general, a more delicate argument is needed), we obtain $$\begin{gathered}
\label{eq:entropy-aux}
 h(p_v)= -{\mathsf{E}}_{{\mathsf{W}}_{t,x}}[ X_t e^{X_t}]=-{\mathsf{E}}_{{\mathsf{W}}_{t,x}}\left[\frac{1}{2} \int_0^t 
\frac{e^{X_r}}{2{\nu}}|v(t-r,\gamma_r)|^2 dr \right] \\
=-\frac{1}{2} \int_0^t {\mathsf{E}}_{{\mathsf{W}}_{t,x}} \left[\frac{e^{X_r}}{2{\nu}}|v(t-r,\gamma_r)|^2\right] dr. \end{gathered}$$ The martingale property of the Girsanov density process $(e^{X_r})_{r\in[0,t]}$ with respect to the natural filtration $({\mathcal{F}}_r)_{r\in[0,t]}$ of $\gamma$ implies $$\begin{aligned}
 {\mathsf{E}}_{{\mathsf{W}}_{t,x}} [e^{X_t}|v(t-r,\gamma_r)|^2]&= {\mathsf{E}}_{{\mathsf{W}}_{t,x}}{\mathsf{E}}_{{\mathsf{W}}_{t,x}} [e^{X_t}|v(t-r,\gamma_r)|^2|{\mathcal{F}}_r]
\\&={\mathsf{E}}_{{\mathsf{W}}_{t,x}}[|v(t-r,\gamma_r)|^2{\mathsf{E}}_{{\mathsf{W}}_{t,x}} 
[e^{X_t} |{\mathcal{F}}_r]]
\\&={\mathsf{E}}_{{\mathsf{W}}_{t,x}}[|v(t-r,\gamma_r)|^2e^{X_r}],\end{aligned}$$ so plugging this into , we obtain $$\begin{aligned}
 h(p_v)&=-\frac{1}{2} \cdot\frac{1}{2{\nu}} \int_0^t {\mathsf{E}}_{{\mathsf{W}}_{t,x}} \left[e^{X_t}|v(t-r,\gamma_r)|^2\right] dr
\\&=-\frac{1}{2}\cdot\frac{1}{2{\nu}} {\mathsf{E}}_{{\mathsf{W}}_{t,x}} \left[e^{X_t}\int_0^t |v(t-r,\gamma_r)|^2 dr \right]
\\&=-\frac{1}{2}\cdot\frac{1}{2{\nu}} {\mathsf{E}}_{{\mathsf{Q}}_{t,x,v}} \int_0^t\int_0^t |v(t-r,\gamma_r)|^2 dr,\end{aligned}$$ which completes the derivation of  and the entire proof. [[[$\Box$ ]{}]{}]{}
