So welcome everyone, thanks for joining us for this CSD speaker series seminar, part of our Monday Conflict Security and Development Research Group seminar series. So I'm delighted to welcome two guest speakers today. We have with us Christine Bakker, who's Professor of Political Science and International Relations at University College London, UCL. Christine's also Associate Research Professor at PREO in Oslo. She focuses on political violence and her research explores how states respond to opposition within their borders, the dynamics of violence and self-determination struggles, post-war state building and wartime legacies, and geopolitical orientations in Russia's near abroad. And she draws on both quantitative and qualitative methods, including surveys and fieldwork in Northern Ireland, India, Guatemala, Canada, and post-Soviet states and de facto states. And joining Christine is Kit Ricard, as a PhD candidate at UCL, and a research assistant also at the Peace Research Institute in Oslo and PREO. And his doctoral project focuses on how states affect foreign civil wars, with a specific focus on how diverging forms of external support shape relations among and within rebel groups. Now, both our speakers are going to be talking about a paper that they've recently authored, Legacies of Wartime Order, Punishment Attacks and Social Control in Northern Ireland. Really important, really interesting topic that I know many on our MA programme and many of my colleagues will be really, really keen to hear more about. So the way this will work is, Kristen will begin, she will share her screen in the presentation, and then hand over to Kit, and then afterwards we'll have a chance for questions from the audience as usual. If you have a question, there is no chat, the usual chat box instead, you put your question in the Q&A box. And if you put your question in there, when it comes to the question time, I will then read out your questions on your behalf and direct it to the speakers, and we'll get through all of those. So if you put questions in as we go through the talk, that's great, and then we'll have some ready for when we come to question time. So without further ado, thank you, Kristen and Kit for joining us, much appreciated, and over to you, Kristen. All right, there, thank you. Thank you for inviting us. We're really excited to share this work with you and we look forward to your questions and comments at the end of this. I'm just going to get rid of a little, okay, little comment box. All right, so our paper is called Legacies of Wartime Order, Punishment, Attacks and Social Control in Northern Ireland. And this slide here shows you two incidents that are described as paramilitary style attacks or so-called punishment attacks in Northern Ireland. And they happened in 2018 and 2019. These are just some examples drawn from random from news stories. And there are attacks in which paramilitary groups attack members of their own communities. Important here, these are attacks that have happened 20 years after the Good Friday Agreement was signed. The first story here happened in September 2018 in Larn, which is northeast of Belfast. It's a predominantly Protestant area. And the attack was most likely conducted by a loyalist group, such as the UDA or UVF. A man was attacked by sledgehammers in a brutal attack. The second story occurred last fall in Derry or London Derry, a predominantly Catholic city. And this attack was most likely conducted by a Republican dissident group, such as the real IRA, continue to IRA or new IRA. A man was walking with a friend through a residential neighborhood when two hooded men approached him and shot him in the legs. Now in this attack, it looks like the man was taken by surprise, but sometimes there's less surprise to how these attacks happens. The quote here is from a teenage boy in 2017. So my mommy visited me and said, listen, I've been talking to someone to try and sort it out to get someone to give you an easy shooting.  I only moved a bit. But the second time they shot me, I was screaming. It went right through and hit my main artery and busted my whole knee bone. So this is an arraign or sort of an agreed attack where there's an arrangement made that you're going to be subject to an attack as a punishment and show up at a certain place and then this attack will happen. And then these attacks, these examples are not one offs. The figure here shows to the data from the police service in Northern Ireland and it shows to paramilitary style attacks since the Good Friday grammar so from 1998, and then the data goes up till 2019. And then what we can see here is that these attacks have persisted in the so called post conflict or post war period, more in the immediate aftermath of the Good Friday agreement. But what's remarkable and what we are particularly concerned about in this paper or what motivates this paper is that the continuation of these attacks. So if you look at the last 10 years and these attacks have remained at the fairly, you know, steady rate in fact with some, you know, increase in later years. And these attacks can come in the form of beatings as we saw in the sledgehammer attacks or in the form of shootings, and they happen in both communities so both by in the name of Republican paramilitary groups and in the name of loyalist paramilitary, paramilitary group. The shootings are sort of locally known as kneecappings. They can be more or less brutal, pending on the crime that these sort of offense that these attacks are meant to police. So they can sort of be clean in the sense that they're not busting someone's artery or kneecap and they're called dirty if they do. If it's a particular severe type of offense that someone is being punished for, then, you know, the person could get what's called a six pack where you know the knees ankles and elbows are are shot. If it's a less severe attack, it might be something that's called a clipping. So only and I say only in quotation marks a flesh wound rather than, you know, through the kneecaps, for example. And what we see, so that's what these, you know, these attacks here in this, these figures are covering. And what we see is that there is a justice system that range or justice system in quotation marks a punishment system that ranges from verbal warnings to beatings and shootings of various forms. Some then sort of further up the scale potentially is that you're expelled from the community and then ultimately death. Military groups across Northern Ireland target people accused of anti social behavior drug dealing and sexual offenses within their communities that's typically the justification for conducting these attacks. And this is this is certainly the story that we see in in the media when these attacks are covered. So, you know, the question then that motivates our paper is, you know, why did these attacks persist. So we see that Republican loyalist paramilitary groups or individuals acting in their name are trying to exercise social control within their communities through so called punishment attacks. And the question is, you know, how are they able to do so why do these practices persist. And, you know, this is perhaps particularly puzzling in a case like Northern Ireland which is a relatively strong and stable post conflict post war state. There was a peace agreement that many see as a fairly successful peace agreement as there hasn't been a return to the levels of violence we saw during the 30 years of the troubles. And then there was a six, you know, there was the commissioning and the military DDR process as part of the peace agreement there was a security sector reform as part of the peace agreement to make the police more representative and accepted in both communities. So it's a peace process that has been backed by in a significant financial resources service show that, you know, it's a peace process that overall has quite high levels of support so in many ways it's puzzling that these practices social control by non state armed actors is continuing in this post conflict society. So that's the motivation of, of this study. In order to answer this question, we, we go to the literature on rebel governance, we also look at literatures on criminal governance and literatures on wartime socialization. So, and many of, many of your kings know this very well that there's a, there is a growing literature on on rebel governance and how armed actors are not just agents of violence but they are also, they also engage in governance in wartime. And armed actors do this for a variety of reasons they want to solidify their group strength and territorial control by taking on governance over the course of the conflict. It might signal their strength vis-a-vis the state in the sense that they're able to govern and they're not just rotation marks arm actors, it gives them perhaps an aura of legitimacy, both towards their own community in the sense that they are taking on governance functions, policing for example, war also war and war provides them with an aura of legitimacy outwards towards the international community. And it might strengthen recruitment and popular support while preventing defection in the sense that you know they're controlling areas but not just controlling them by being violent actors and engaging in their counter insurgency campaign, but by providing governance. Now this literature on rebel governance suggests that there's, and there's a range of sort of public goods that armed actors are contributing to in wartime. At the minimal end they can be trying to, they can try to take on the policing or security functions of governance, which is primarily what we saw in Northern Ireland. But then in other settings we've seen armed actors take on a range of public goods provisions of education and healthcare and take on state-like functions in wartime. There's also a research in sociology on criminal groups and on gangs and mafia groups that suggests that, again, that these actors aren't just agents of violence, they may also offer protection to the communities in which they're operating to gain some level of legitimacy and support within their community. Now in this, these literatures, in particular in this literature on wartime governance, one of the things that scholars often point out that is that, you know, we need to focus on the fact that armed actors are providing or taking on these state-like functions in wartime, they're trying to govern in various ways, and that's going to have legacies into the postwar period. But what we don't have much research on is, well, how do, you know, how do these practices or do these practices end your? And so that's where we see our contribution in the sense of trying to explore what some of these legacies of wartime order might be. And we make an argument, a sort of two-step argument. So first, we argue that there are a top-down mechanism for why wartime orders might persist into the postwar period. Overall, our argument is that these wartime orders or wartime, wartime social control is sticky, and it's hard to dislodge these informal institutions that might emerge during years of armed conflict. Now top-down, there's an instrumental logic in the sense that armed actors are engaging in this kind of behavior in the spoiler, according to the logic of spoiling. Strategically, they may have political and operational reasons to maintain the orders that they created or were emerged or were created during the war. Politically, areas under social control by non-state actors serve to delegitimize and denormalize the state. And logistically, these areas and the people within them can provide these groups with safe haven, resources, and recruits. So groups might have these reasons, be they political or criminal, for wanting to keep these orders alive in the post-conflict period. And they're going to do so in areas where the predecessors controlled, exercised social control during the war, because this is easier than setting it up in new areas. So we would expect to see a continuity of where you had social control during the war into the post-war period, because armed actors have strategic reasons for wanting to control territory, for wanting to control certain areas. There's also a socialization mechanism here, potentially, on the part of armed actors in the sense that armed groups or members of armed groups, elites, might have become accustomed to positions of power during the war. So they had a certain role in their community during the war, and they might want to keep, you know, have that role into the post-war period. They see as their role, their purpose to uphold order, the order they were creating during the war, and to keep doing so into the post-conflict area. So those are two mechanisms sort of on top-down, and by top-down I mean sort of motives on the part of the armed actors for wanting to maintain social control into the post-conflict period, why we would see these practices endure. Then there's also a bottom-up, or potentially a bottom-up mechanism, based on socialization. So civilians may come to rely on informal institutions for their quote-unquote strategies of survival, and in areas where armed groups developed informal institutions during the war, and in the case of Northern Ireland during 30 years of conflict, people may have come to accept an internalized norm stipulating that governance or certain aspects of governance, in the case of Northern Ireland, community policing, was done and done better by non-state groups rather than the state. So there's a bottom-up socialization in the sense that this is not just driven by top-down with armed actors who want, there's also a demand bottom-up from the population. So that's the mechanisms that would underpin why these practices might endure from the war into the post-war period. And if this is right, there's a couple of things we expect to see. So one, we would expect that after the war, armed actors will seek to exercise social control in areas where there were well-developed informal institutions during the war. So we expect some geographic persistency of where these things are happening. And the second, in order to sort of differentiate, well, is this really driven by this top-down or the bottom-up mechanisms, we suggest that if the geographic persistence of informal institutions from the conflict into the post-conflict period is also driven by civilian socialization, then civilians living in areas controlled by armed groups are likely to rate informal authorities highly and they're likely to be skeptical of formal authorities. So we would expect to see, you know, there's something here about the attitudes of the population. So those are the two propositions that we then, we test in this study. So how do we do that? So our research design, there's lots of bits and pieces to it. We try to throw lots of things at it to see what's going on there. So we rely on secondary sources and also interviews and archival work from the from the troubles, what was going on during the troubles, particularly to establish where and how were wartime institutions developed. We've conducted interviews with stakeholders that was in 2018 and to establish where dissident groups are active, we map in-group killings during the troubles, and I'll talk, we'll talk more about, I'll talk more about this measure in a minute, but we try to map where groups had social control during the troubles and we rely on a measure, we use a measure for in-group killings to establish that and then we match that or try to see does that match where we're seeing these kinds of attacks happening in the post-conflict period. And then to assess the effectiveness of social control post-conflict or to see how do people rate informal versus formal authorities, we rely on the survey that we conducted in 2016. So let me talk a little bit about each of these bits and pieces of evidence here. In terms of the history, we outlined how areas in Northern Ireland came to be under the control of paramilitary groups. And there's some, so in the, when the trouble first began, of course, there's a sectarian conflict where you have these communities became pitted against one another. And the sectarian violence led to the development of so-called no-go areas. And the reason for this is that when the sectarian, in 1969, the sectarian violence became so severe that it was, that the British government decided that, well, we need an army on the streets here. The police can't handle this. Now, when the army entered, the police withdrew and withdrew from certain areas. And in these areas where the police, from which the police withdrew, there was a policing vacuum in the sense that, you know, ordinary or ordinary crimes, you know, antisocial behavior, these things continued, but there was no police for people to go to. So people started turning to the paramilitary groups. And when we spoke to members, sort of former members of paramilitary groups in Northern Ireland, and the way they described it was that, you know, they weren't, this wasn't something they set out to do. It wasn't the sort of purposeful plan from the beginning that we want to take on policing functions, but people came to them and they sort of reluctantly took on this role. And initially, you know, they saw this as, this is taking away from, you know, what the key struggle is about, the key struggle is about fighting in the Republican communities, fighting the British states. But, you know, we're also going to take care of this law and order problem, this more administrative, this administrative aspect of it. So they dedicated people to deal with community policing. And they came to realize over time that, well, actually this serves, you know, this has a purpose, it can help us by, you know, giving us support in the communities and, you know, establishing control within the communities. We also saw this develop in loyalist, loyalist areas, because also there there was a policing vacuum. And institutions emerged, you know, as a bottom up demand during the troubles where people came to the paramilitary groups and expecting them to take on policing function, particularly to deal with things like anti social behavior. Joy writing is one thing that is very often brought up as an example. So we see this development. And what's important about this historical bit of our analysis is that we're establishing what these institutions look like that this was mainly about the kind of public good that was provided was mainly about policing within the communities. It happened in both communities. And that it was a result of the troubles. This is these are not practices that were there before the conflict, but they emerged during the trouble. This is a historical bit of our analysis. Then, you know, having established, you know, that these practices emerged in certain areas during the troubles. Then we have this proposition so after the war we suggest armed active will seek to exercise social control in areas where they will develop informal institutions during during war. And this is where we expect this geographic persistence. So how do we go about figuring out this geographic persistent. So, in order to establish where groups are exercising social control after the good Friday agreement we rely on these data from the police service of Northern Ireland that I showed you early on about these paramilitary style attacks. So areas where you see Republican or loyalist groups carrying out attacks within their own communities. These can be both beatings and shootings. So that's and we have what we we got from the police service in Northern Ireland we got to geolocated data for where these attacks are happening so we can map them out. Now similar data does not exist for the period of the troubles itself. So what we rely on there is a measure of in group killing so reports of Republican paramilitary groups during troubles killing members of the Catholic community and similarly for loyalist groups killing members of the Protestant communities and we also geolocate those attacks. Now, of course, you know these attacks are at the tip of the iceberg in the sense that, as I said earlier there is this, you know, system of informal justice where it starts with a warning and then there might be another warning. So, you know, you don't get to killing until quite sort of high up in the system of punishment so in that sense it's a fairly conservative measure of where groups exercise social control during the troubles. Now, you know, relying on these data where groups exercise social control with this fairly conservative strict measure from the troubles and then our data on paramilitary style attacks. You can map where these things happen during the troubles and afterwards. And here you see a map of informal justice in Belfast and so it shows you data of where you are seeing attacks happening and on the first map here you see where in group killings were happening during the and then you see on the second map where paramilitary style attacks have happened after the troubles and the red ones are loyalist attacks and the red areas here are majority Protestant areas and then the green attacks are attacks carried out by Republican paramilitary groups in the green areas and predominantly Catholic areas. And there are two things, you know, you see here that these attacks are mainly in group, right, so that you have, you know, green attacks happening in green areas, red attacks mainly happening in red areas so they are in group policing. And then you see just by looking at this map, right, you see that there is an overlap of where these attacks were happening during the troubles and where they are happening afterwards. And we've, you know, we map this out for all of Northern Ireland, this figure here just shows shows you Belfast. Now, what we also do is to see, well, you know, of course there might be other reasons for why certain areas in the post conflict period is experiencing paramilitary style attacks it might be that this has to do with these attacks are more likely to happening in poor poor areas in urban areas in areas that are community strongholds. And we run a couple of regression analysis using data to try to control for the fact that there might be other things that are accounting for why these attacks are happening where they are happening today. And what we find and I'm going to show you the results here is that the numbers are these measures for in group killings from the conflict from the troubles will are still statistically significant and related to where attacks are happening today so that's our the first two variables you see here so you see that Catholic in group killings are positively and significantly related to where you're seeing paramilitary style attack attacks Republican paramilitary style attacks happening in the post conflict period and the same for Protestant in group killings. So that gives us some confidence that you know we're not just seeing this in this maps. This also you know shows up as being important when we control for a whole range of other factors that of course may also account for why we're seeing some neighborhoods be more likely to see paramilitary style attacks today. Now I'm going to give the word to kids because he will talk a bit more about so kid is the one who has created this maps and who's also run this analysis and has done a lot of work trying to assure us that this measures that we have are good measures I'm going to give the word to kid and then I'm going to come back to back to you and talk a bit more about well what do people who live in these areas where you see these attacks happening today, what do they think of informal and formal authorities to test the second proposition. Thanks Kristen. So, yeah, my name is kids. Thank you so much for for for inviting us here to talk today. So I thought it'd be interesting at this stage of the presentation to take almost like a side step and discuss our measure for social control. So, as you can see the measure for social control in the post conflict period here is the number of paramilitary style assaults in an area. And the areas that we use our census areas they're called super output areas they exist in the UK. And so they exist in Ireland. And I think this is important because it's an important feature of our paper. So it's the outcome variable for the first proposition which Kristen has just outlined, but it's also the main key independent variable and main predictor for a second question, which, which Christians Christians going to outline afterwards. So what I particularly like about the next few slides is that it shows the, you know, 80% of ideas research and work that never makes it into a journal article. But I do think it's interesting to discuss it, especially when you know we have 40 minutes or longer today to discuss this. And it kind of gives a kind of tick description. I think I learned a lot during it about Northern Ireland society today. So when we approached this project, our question from the start was this kind of puzzle why these attacks happening in certain areas in Northern Ireland. And what we thought those attacks were, we thought they were an indication of some kind of attempt to have social control over certain communities. And so we thought, okay, so the overall question is, you know, how do you measure parametric control? How do you know that these attacks are happening in places that parametric groups are trying to control. And so just a side note on the data that we use. It took us a year of emails, a meeting in person with the PSNI and non disclosure agreement to get the data for the article that you've just seen in the maps. And, you know, we can't share that data for a really good reason, in that it's highly accurate GPS coordinates, and it could be used to identify victims. I mean, an important point here and I think Kristen presented it quite nicely at the start of the talk is some of the attacks are surprising that happened in the street. Some of them will be by appointment, which is described in the quote also. But for those who don't turn up for appointments, a lot of them happen in the home. And when they happen in the home, they're particularly brutal. And so our data reflects that. So a lot of the data that we have will show you the home of the victim. So we were never sure if we're going to get this data, right, which is highly, highly accurate geographically, but also disaggregated over time. And so for a year, I spent which in hindsight, of course, was a complete waste of time, but I spent a year collecting alternative data that I thought we could use to measure social control. So I thought I'd discuss it now and it kind of opens a kind of discussion about how do you even measure these kind of things that we're interested in. And, you know, like I say, it's the 80% that never ever gets on print. So the first idea was to use news reports from, let's say 1998 till the present day, which at the time was 2017. But we quickly realized that they would over represent Republican attacks. Most Republican attacks, which you can see in the first few slides that Kristen presented, involve firearms, whereas loyalist attacks involve predominantly kind of blunt instruments. So, you know, baseball bats and sticks. So firearms are less discreet and they're less discreet and in urban areas, you know, they're more likely to be known and reported by the media. So we decided not to use the news reports. But, you know, if you can't get the data from the PSMI, this probably is the best kind of alternative to the data that we have. And there are people collecting this data. For example, the armed conflict location and event data project I collected has expanded has expanded to include Northern Ireland. And I think they're releasing their first kind of batch of information, which will be from 2020 onwards in January. We also thought of using hospital reports. But again, shootings are more obviously an attack than a beating. And so beatings might, might again be underreported.  You might not record it as a punishment attack. In any case, for the hospitals, there are few hospitals in Northern Ireland. So it wouldn't really give you an indication of where these attacks are occurring and the likelihood of getting, I mean, we've no interest in getting hyper personal information on patients, right? So it wouldn't give you an indication of where they're occurring, but it definitely would give you some kind of temporal variation. So you could see peaks in times when there are attacks. And so in light of those kind of issues, I actually started coding additional measures of control, which never made it into paper. I'm going to describe this one. So maybe Kristin, you can go to the next slide. So the thing behind the measure was, okay, what we could get is some kind of course measure. We could find a course measure of paramilitary violence from news reports and we could bolster it up by combining with other measures that might make an area that we think theoretically might make an area more susceptible to control by paramilitary groups. Other measures, you know, might include, you know, like the concentration of a certain community in an area. So whether it's 90% Catholic or 90% Protestant according to census data, but also deprivation in an area or in our case, what we try to do is what we would have used those, but collect geospatial data on community organizations, which may be used by paramilitary groups to kind of maintain social links. And the kind of obvious community organization that I didn't code was churches, but Joseph Brown at University of Massachusetts. He has coded those. Why I decided to collect was data on the locations of GA clubs. So the GA is a gay athletics association and the locations of orange halls, which are clubs associated to the orange border. So, you know, the GA is a sporting organization for Irish sports in Northern Ireland, most GA members are Catholic. That's reflected in national surveys in Northern Ireland, the life and times survey. It does have links to the conflict in the sense that police members from and, you know, members of security forces more generally could not become members of the GA up until I won't say a year because I can't remember but fairly recently let's say. The GA is also like this whole line in the association. There's no border in the GA. So it does symbolize this kind of United Ireland kind of political aspirations for for many kind of Republican members of the community. Many of the clubs also named after prominent Republicans. I think the kind of the most most obvious case of this is Casement Park in in Belfast, which is the GA stadium. And it's named after Sir Roger Casement who, you know, for Irish Catholic Republican people is, you know, the name for hero before British people. You know, even in the UK, you would you would associate him more as a as a villain if it's dark because, you know, he brought weapons to Ireland and was executed. And he was a Sir, so he was not a Queen mother. So it's a very symbolic right to name a club after someone like that. Then the other group, the other community organization which we thought would kind of capture whether community was acceptable right was the orange order. It's a brotherhood which is loosely based on the organizational structure of the Freemasons. It's a Protestant for fraternal order. And it plays a really important role in marching and parades in Northern Ireland and I think anyone who knows anything about Northern Ireland will know that that is incredibly important plays an incredibly important role culturally and historically for the Protestant community in Northern Ireland. And I wrote, when I was writing out what I was going to say, I wrote most members of Protestant, I think it's probably safe to say that all members of the orange order are Protestant. So, how was I going to get the data. So the GA was easy because it's online. So I just called it and based off the addresses and I got 261 GA clubs in Northern Ireland. And that excludes tree counters. So the way it's structured is the provinces of Ireland. It's Ulster. So it's just the six counties of Northern Ireland. The orange order, it was a lot more complicated and a lot more fun. And actually told me, I learned a lot more about Northern Ireland, tried to code the orange order, the orange halls than I did the GA clubs. So the orange order is a secretive and extremely decentralized structure where members can basically set up their own hall relative ease. So I could set up a hall in my in my shed, for instance. So I first contacted the Grand Orange Lodge of Ireland and I think they were somewhat surprised that I contacted them. And they told me that they didn't have a list of all the halls and the addresses, but they told me that they were, you know, in and around 600 halls in Northern Ireland. And so that was nice because I thought, okay, that's something to aspire to 600. I then did a freedom of information request to the Northern Irish Department of Justice, because I thought it might have a list of halls. And that was for security reasons, because I know from growing up that halls sometimes are targeted for, you know, sectarian attacks against the actual building that the halls are in. They sent me a list of halls, which were subjects, which were subject of compensation claims as a result of receiving criminal damage, but they never told me what the time span was. In any case, I got 50 halls that way. And the 50 that I got were in predominantly Catholic areas. So these were probably the result of kind of sectarian attacks on the halls themselves. So parading is a huge part in Northern Ireland, as I've said, and many parades, they start or end at an orange hall. So I know that from reading and if you read anything about the parades, this becomes quite clear quite early. I contacted the parades commission in Northern Ireland, and I got the routes for all the parades in 2016, which, which amounted to over 3000 routes. And this allowed me to identify a further 138 halls because I actually provide the address where it started and ended and whether or not it was an orange hall. And then finally I did a crawl through Google Maps and did this kind of Google map crawl and I got another 119 halls. And so in the end I had 260 trios, which is just over a third, and probably many more exist. And in hindsight, and I found all this out after having coded all the halls. Eric Kaufman at Birkbeck has a collection of the locations of halls and I think he's got somewhere near like 500 holes. So it was always a time. But mapping them out. Kristen, could you show the next slide? This is quite neat, right? So this is Belfast. The yellow lines show interface areas. The green dots are GA clubs. The orange dots are orange halls. The triangles are Sinn Fein offices. That was another thing I thought we could use as a kind of measure because you know, Sinn Fein office on the Falls Road, for instance, you know, it has a long history, a long association with the community. But the DUP offices was definitely a bad measure in hindsight because their relationship with the loyalist communities is not as clear as the Republican or the Republican communities relationship with Sinn Fein. But in any case, if you look at Belfast, what's obvious is that GA clubs probably do capture some kind of communal kind of strongholds that say like an association of places that you might think are susceptible if combined with other measures to kind of paramilitary social control, as in they're like rich fertile kind of ground to do that. But the orange halls don't seem to capture that in urban areas anyway, in Belfast anyway. And next slide please. Here is a south to our mass of New York. This is on the orange border. This during the troubles was referred to as a bounded country. It's a provisional IRA during the troubles and provisional IRA stronghold. And so what you can see here is you get a lot of these GA clubs as you would expect. You also get some orange halls, but you don't get very many orange halls. And what's interesting is, GA clubs are almost always. It's very hard to find GA clubs that are not in areas which are predominantly Catholic. But you do get these orange order orange halls in areas that are predominantly Catholic, and they're probably the ones I got from the Department of Justice. Next slide please. This is a north down which is a predominantly Protestant area. And so you can see they're like the GA clubs. There are just no GA clubs in the more Protestant areas, but you can see that the orange halls are well represented there. And then one final map, which is dairy. And you can see here that the Sinn Fein office is in, I think it's in Krogan. So, and Rose, I'd have to check. But you see, the measures that we were using, if combined, you may have kind of created some kind of index of social control, but I think they would definitely have been subject to much debate. But what I think was nice about this data is that it sheds light on the kind of communal, intra-communal relations in Ornald, and it shows just how segregated they are, but how easily it maps on to spatial segregation. And then the last thing is we use an indicator of social control during the troubles, which is the number of in-group killings. So that's a Catholic person being killed by a Catholic group, or a Republican group, and a Protestant person being killed by a Protestant or loyalist group. But as an alternative for these no-go areas, yeah, you can go to the next slide, this one. We thought, okay, so the no-go area, so as an alternative for this measure, we thought, why don't we use the no-go areas? Because we know that these no-go areas emerged, that they were controlled by these groups, that even when the army moved in in 1972 and it reoccupied these areas, that the permanent groups continued to exercise social control through these informal systems. And so I went to the National Archives in London, and I got the mission outline for Operation Model 1. And here are the photos of just some of the pages, I mean, just tons of pages. But this is the mission outline by Lieutenant General Sir Harry Tuzo, and based on this, I coded places that you could identify and associate with the unit, the kind of modern day census unit today. And so you can see that there are these, if you know Belfast, or if you know Northern Ireland, you've recognized these areas, Turf Lodge, Ardoin, New Lodge. These are areas that are, you know, predominantly, these are areas that we would expect to see. These are also predominantly Catholic areas. And you can see that they did this across Ireland on the right. And when you map these out, and that's the next slide please, you see these are those in the No-Go areas in Belfast in 1972. And again, if you know Northern Ireland or if you know Belfast, you'll see that Protestant areas are really underrepresented. This is not because they didn't have No-Go areas, but this is because they weren't considered a security threat and they weren't the target of Operation Model 1. I could potentially get in trouble for just saying that, right, with no evidence. So actually, it's quite obvious that this was the case in the Mission Outland. And in the next slide, if Kristen could just show, there we go. So this is just one excerpt, but you'll see plenty of excerpts like this, which shows this kind of partial nature of Operation Model 1. And I think generally, if you see a document that says appear impartial, it probably is partial. So I'll just read it out very quickly. Protestant barricades in their areas, so obviously there were some, are to arrange the removal of these barricades, which will have to be taken down very shortly after HR because of the need to appear impartial. Military assistance is to be offered in the unlikely event of Protestant refusal to remove barricades, there to be removed by force. However, every attempt is to be made through negotiations at ground level to ensure that the situation does not arise. What's really interesting in the Mission Outland is that they're told to tell, you know, the paramilitary groups to kind of local elites in these areas that they can continue what they've been doing. And they refer to like administering justice, but that they have to remove the barricades in order to appear impartial. So that's the end of my side note. In the end, I think we have a really good measure of social control, both during and after the conflict, because it's dynamic, right? I mean, it changes over time. The ones I've just outlined, they don't change over time. They, once they're built, they're there. In the same way as once a place is a no-go area, it always is a no-go area. Whereas ours, they change over time, and they're more fine grained with. It allows us to answer the first question about geographic persistence, which Kristin did a really job of describing. But they also, which I think would have, I think, you know, if we hadn't the data, we wouldn't be able to answer our question, the data that we have. The other thing is, you know, in this kind of wayspin of ideas, I learned a lot about Northern Ireland and its history and my attempts to find an alternative measure that I hope that, you know, some of you may have found interesting. So I'll hand it back to Kristin, if she's going to talk about the second proposition of our theoretical argument. And I look forward to hearing from her to the questions afterwards. Thank you. Thank you, Kit. I'm glad we had this opportunity to actually share with you some of the, all this data collection efforts we've gone through and Kit in particular has gone through to try to, you know, how do we measure social control to try to do that in different ways. So, but the measures we use our course in the post war period is measured to the paramilitary style attacks and in group killings from the troubles. Now the second proposition we had was about, well, is this bottom up socialization. So, you know, do these institutions persist due to civilian socialization. So how can we figure that out. And I'm showing you this slide is pictures in this slide here because these are public campaigns in Northern Ireland that are aimed at getting the public aware of that these attacks are happening and making the public stop turning to our militaries. So the first is just a picture for the stop attacks video from a public campaign, which sort of informs people about, you know, these practices are potentially dangerous for you and for the communities. And then you see the second one is a photo from a couple of years ago on a bus stop in Belfast and paramilitaries don't protect you they control you. So clearly these attacks are aimed at the public sort of recognizing that there are members of the public who do go to these groups and you do somehow. It's not just a top down thing, right. So, but beyond, you know, knowing that there are these public campaigns, which suggests that there is something about the public. Finding or there is something to this bottom up mechanism. How can we try to figure out more systematically. And sorry. I'm just going to make sure. Yeah, okay, I'm there. And so what we do is that we rely on survey data. And this is a reminder of our of our second proposition. So if the geographic persistence that we saw of informal institutions from the war time from war into the post conflict period is also driven by civilian socialization. We would expect civilians who live in areas that are controlled by armed groups. So where these attacks are happening today to believe likely to rate informal authorities, highly, and be more skeptical or formal authorities. So that's our expectation. And we get at this by using survey data but also based on our fieldwork. So let me talk a bit about our fieldwork highlights first so we spoke to a range of people when we were there we were only there for a very short fieldwork trip. And we're also relying again on secondary sources and reports and newspaper reports. But you know, from the people we spoke to we certainly got the sense that there is something to both the top down and bottom up mechanisms. There are top down there are armed groups at various motives you have dissident groups who want to delegitimize the state and remain relevant today. There are also criminal groups who are engaged in extortion they seek to control the drug business and the areas where they're going to choose to sort of set up shop our areas where their predecessors for paramilitary groups had control during the troubles. We can't really distinguish what whether what's happening is sort of based on political or criminal motives from this top in this top down mechanism. Bottom up, there is what some of the people we spoke to is a fault memory of going to paramilitary groups within certain communities and we were told several times that particularly among in processing communities that you you just do not go to the police. Right, so 30 years of conflict 30 years. People have been socialized here over 30 years of conflict not going to the police. And also another expectation that it's somehow quicker to go to paramilitary groups if there's a problem of antisocial behavior in your area. It might be quicker and there's sort of a swift or speedy justice by going to the paramilitary groups to get the problem so rather than going through formal policing channels. So that suggests you know just when people we spoke to that there is something to this bottom up mechanism to then more systematically we try to get this with survey data. And so we have a survey that is conducted with colleagues at the peace research Institute and also it's a survey called attitudes to peace. And it's asked people arrange your questions about the post conflict period in Northern Ireland it's a comparative surveys we did a similar survey. We did a survey on Nepal as well, but it asks people arrange your questions about the post conflict period and of course the conflict period. And then we pair what people think of formal and informal authorities I'll show you that in a minute with whether these people are living in areas that are controlled by paramilitary groups today so we used to data from the police to figure out if one individual is living in an area that is controlled by paramilitary groups and whether that then shapes his or her perception of formal and informal authorities. So you know how do we get at what people think of formal and informal authorities. So we asked people a question about antisocial behavior. This scenario we give them this following scenario a man lives in a neighborhood where there's a severe problem with antisocial behavior such as vandalism car theft. What would he do. And I'm sure many of you, you know, can recognize this scenario from where you're living in, like you're living in the UK. And for each possible solution presented the respondent is asked whether it would make no difference help a little or help a lot and solving the problem to go to a set of authorities. And in order to assess whether people would think it would be useful or help to go to the police we asked them, you know how helpful would it be to go to the police that's our measure for formal authorities. We did not ask them directly how useful it would be to go to paramilitary groups, we were told that, you know, that wouldn't necessarily give us we wouldn't necessarily be able to get on this answers to that. And but also in part what we're interested in is not sort of only that people are going to paramilitary groups whether people think it's more normal to go to informal authorities someone that isn't the state right. So we asked this question contact a member of the community who has influence that's our measure informal authority. And you know people could rate that from it would help a little or make no difference help a little or help a lot. And then based on these two questions, we create three dichotomous dependent variables. So we create a measure for whether people rate the police as more effective as a weather response rate the police as effective. So, basically whether it will help a little help a lot versus make no difference. We create a measure for whether respondents rate informal authorities as effective so help a little or help a lot versus make no difference. And then we also create a measure that we're based on these two questions whether people rate the effectiveness of informal authorities as higher than that of the police. So we have getting at you know what do people think of the formal and informal authorities. And then what we're really interested in seeing is their correlation between the how they rate formal and informal authorities, based on whether they're living in areas that are controlled by paramilitary groups today, based on the measure of paramilitary style attacks. I'm going to show you the main findings here we also in the models we run so we run six different models we run logit regressions. And we control again we control for a range of other things that might affect what people think of informal informal authorities but I'm just going to show you the findings for our main findings for to make it more readable and simple. These are the regression results logit regressions itself for the key for the key relationships that they find and the red here shows you the rating of the formal authorities as effective. The green shows you the rating of the informal authorities as effective so you know personal influence as effective and the blue is whether they rate informal authorities as more effective than the formal authorities. So that's just a coefficient here and then I'm going to show you the predictive probabilities which might be slightly more intuitive to look at. So what we see here for our Catholic respondents, we found that there as the number of paramilitary style attacks in the area where they live so the more through control by paramilitary groups there is in the area that they live, there's less likely they are to rate the police as effective right so there is a correlation this correlation that we would expect with if you're living in an area of paramilitary social control, you're going to be skeptical to formal authorities and we find that among our Catholic respondents. We don't actually find that there's a corresponding relationship where they see informal authorities as effective among the Catholic respondents. We do find that among our Protestant respondents. So Protestants respondents, if they're living in an area that is controlled by paramilitary groups, they're more likely to rate informal authorities as effective. And they're also the same. You see a similar finding when we asked them this variable to sort of ask them to rate informal authority as you know it's informal authority more high is informal authority more effective than the formal authority. And again, we see there's a relationship between here between living in an area that is controlled by loyalist paramilitary groups and then thinking that informal authorities are more effective than the formal authorities. And this suggests that there you know there is also something to this bottom up mechanisms that these, you know, the, the geographic persistence that we find from the post conflict from the conflict into the post conflict period is not driven. We think just by top downs and motives on the part of our groups. There is also something to what people actually think here that drives the persistence of these practices. We can talk more about in the Q&A about why are there these differences between the Catholic and Protestant areas. We speculate on that. We're not entirely sure why, but we speculate on that in our paper. To wrap up. So we began with, you know, this puzzle about the persistence of these practices in Northern Ireland and particularly perhaps, at least I we find it's quite puzzling at these practices persist in Northern Ireland and relatively strong state a peace agreement that has, you know, support financial support, public support, and you still see these practices persisting and we find that, you know, these practices persist from the conflict to the post conflict period that driven both by top down and bottom up mechanism in some ways in a mutually reinforcing way. This has implications for scholarship for how we think about the legacies of wartime governance and there is the legacy of wartime governance here that post war reform efforts are not happening in a governance vacuum. And this has implications beyond, beyond Northern Ireland. The next steps that we wanted to tackle in this project was that we wanted to ask, we have another survey that is going to happen and we actually we put it in the field. In January last year, but then it was stopped because of COVID and we're hoping we'll be able to pick it up but in this next story what we want to do, we want to ask you a ask people more directly about we'll give them a set of scenarios and ask them more directly whether they would go to paramilitary groups. And we want to try to match areas where we're seeing strong paramilitary social control with areas that are similar on socio demographic variables that we could expect paramilitary groups to have control, but that don't have control to see you know what is it that's driving differences here so that would be a more sort of close up analysis of what's driving these things where the story we're relying here is a nationally representative survey but it would allow it would allow us to explore these mechanisms more in that. I think that's all we wanted to say and we'd love to hear your questions and comments on any part of this.


So, another thing I'm commonly asked on courses is, should I make swales? So the first thing you need to know is, what is a swale? Perhaps you know already, but a swale is a ditch on contour with the idea that when water runs down the slope, it gets trapped in the ditch and soaks into the ground. So there's no liner here, we're not trying to hold the water, we're just trying to slow it, temporarily hold it so it can soak into the ground. And swales are best suited for dry landscapes, particularly if you've got a slope which has no vegetation on it, so there's no roots of trees and plants to slow that water and create their infiltration, no root systems to feed the life in the soil where it creates a good structure and allows the water to penetrate the ground, and particularly land that has no cover because land that's uncovered, when it's hot in a dry landscape, that land gets hot, the air in the soil gets hot, the air rises out and when you do get rain it pushes the rain away. So swales are good for dry lands, slopes particularly, I mean anywhere up to a metre of rainfall a year, which is quite a lot, but in Britain, the east side of Britain is drier and so you might have situations where slopes that have been eroded might need a swale system in order to allow that water to penetrate in and get the vegetation going. But very much a temporary, almost a one-off intervention in the sense that what you're trying to do is to establish plants and trees to stabilise that slope, you shouldn't be needing to dig swales on a regular basis. So here I'm standing in a ditch, it's actually not a swale, but it's here and I can talk about it in a way that perhaps it was, so if this was level on contour rather than a slope, then instead of the water on Dartmoor, instead of the water coming down, being collected here and draining gently away rather than going at speed, the water would be held and would soak into the ground. Now here on Dartmoor we don't need that, it's about two metres of rain a year, there's a lot of precipitation and there's no need for holding the water, there's plenty of water in the soil and here particularly there are trees in the landscape here and some vegetation behind. But what this does show is that when water comes into a place like this, so there's a fairly steep drop into the swale and then a gentle fall away, you'll get an accumulation of material like this, so a lot of leaves here from the trees, but also we get a bit further down, it's a very nice material and over time this will fill up, this whole area will fill up and we'll end up with a terrace, now that will take quite a lot of time and the thing is that a terrace is actually a good thing as well because then the water is slowing down on the terrace, it doesn't need to have this ditch in the long term. Now when we plant swales we also put trees on the bank, top of the swale and also often trees behind and those trees will be able to take advantage of that extra water that's being held here that seeps into the soil and into the water table and then they can support, the water supports the tree and in turn the trees support the structure that you've created, so when we make the swale we take a little bit of the soil from here and we put it slightly downhill okay and they can be, swales can be quite small or they can be really quite big if you have a lot of rainfall that falls in a short period of time, sort of monsoons and things then swales need to be able to accommodate that water without overflowing and then disappearing down the slope, so a swale has to be able to take the water from above and hold it for long enough for it to soak in, now if you're, if you have quite a long slope then you need a series of swales down the slope so there would be this one and then some distance down there'll be another one and so on so the water that falls between this one and the next one down is collected in that one and it has the ability to capture that and hold it and soak it in. How do you know how big your swales need to be? How far apart? Well you just need to know things like your geology and the slope of the and how much rainfall you have and there is actually a website called the swale calculator and you can just go there and type the numbers in it does all the sums for you which is great. So do you need swales? If you've got a fairly dry landscape you don't have a lot of rainfall and when it rains it runs off then almost certainly yes because that's about remediating your soil if you get plenty of rain then really there is no need you may even need to put in drainage but certainly not swales.


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This will be Janne's second lecture on the semi-circle loan. Hi everybody. Sorry, just a sec here. These are harder to get at. Let me just put it there. Okay, so first of all, I'm sorry. This is starting a little late. I got on the wrong bus and I was told that I had to get off in the middle of nowhere. Okay. So I'm a little short of breath too, because I ran here. Alright, so let's see where we were yesterday. In our exploration of the semi-circle laws. Recall the notation. So we have a matrix W, a random matrix W, defined as Wij. All the entries are centered. And the second moment of the off-diagonal entries is one. So the variance is one. Yesterday, the assumption was that there are moments of all kinds for all the variables. Okay, so the moments are bounded. Today, we will remove that. But let me remind you that we're not actually working with W. We're working with a scaled matrix W bar, which just represents essentially what it does is it puts the eigenvalues of W almost on a compact set, asymptotically on a compact set. The object of study is the empirical spectral distribution, which is defined as being the average of the delta functions at the eigenvalues of the matrix W bar. Sorry, W bar. Okay. So this is the notation that we used yesterday. And we proved yesterday the following two facts. We showed that if we average, so if you take the expectation, over both the distribution F of W, sorry, it's not F of W bar, I should have put F sub W bar. So let me just make a note of that.  It's a distribution. It's the aforementioned distribution. So if you take the expectation with respect to W bar and F of W bar of the monomial X to the K, that's the same thing as looking at the average one of the expectation over W bar of the trace of W bar to the power K. And that goes to zero if K is odd, and it goes to the Catalan number for K over 2 if K is even. This is what we did, the bulk, I should say, of what we did yesterday. We showed that the moments converge to those of the semicircle, and these are the moments of the semicircle. Moreover, also yesterday we showed concentration of F of W bar or the moments with respect to W bar of F of W bar around their expected values. So we showed the following thing. The variance with respect to W bar and F sub W bar of X to the K, which is the same thing as one over N squared times the variance over W bar of trace of W bar to the K. That object goes to zero for all K. That's what we showed. In fact, we showed that it's of the order of one over N squared. So in other words, if you want this piece here, just the variance bit, is order one. And that makes the whole thing order one over N squared, which goes to zero as N goes to infinity because K is fixed for any K. These are the facts that we showed yesterday. From this, we concluded that the moments of the SD, so F sub W bar, which are these objects, one over N trace of W bar to the K, is the same as the expectation, as its expectation. And it goes to zero or C K over two asymptotically as N goes to infinity, depending on the parity of K. So maybe I shouldn't have put exactly equal. I should have put similar as in its asymptotically the same. Concentration, that's what we showed. Okay. So today we will show that because the moments are concentrated, this means that actually F of W bar itself converges weekly in probability to the semicircular. We'll do this by a combination of via stress and bounding or doing some sort of a spilling bound. So we will bound the probability that the eigenvalues of the matrix spill over some compact set. And that's part of what we'll be doing today. Okay. So let's first establish a couple of very simple facts. One of them is that C sub K is less than or equal to four to the K for all K. I just realized I don't have, oh, I do have a watch, never mind. All right, we're good. So the Katelyn number for K is less than or equal to four to the K for all K. So how do we show that? It's actually not hard to do. It just follows from the definition of the Katelyn number. Remember, C sub K is one over K plus one, two K choose K. A very quick and dirty bound. We'll tell us the following things. It will tell us that two K choose K is smaller than the sum of all the binomial coefficients corresponding to power two K. And that is what? Two to the two K. So this whole business is actually less than or equal to two to the two K over K plus one. Okay. No big deal here. And of course, this is just four to the K over K plus one. So this is a very, very simple minded bound. It's going to be sufficient for our purposes. And the second thing that I want to recall, I was told to protect the microphone if I'm going to cough. I forgot it. Sorry about that. So the other thing to remember is a virus stress approximation theorem, which says the following thing roughly, given any epsilon greater than zero in some function that is continuous and has support on a compact interval i, there exists a polynomial p sub epsilon. Mostly I will talk of it as being p, but you have to keep in mind that it actually depends on the epsilon that we choose.  What happens outside of this, we don't really care. What happens in practice is that this polynomial is going to blow to infinity very quickly outside of the interval i, but what we're interested for it to happen is that on the interval i and on the interval i alone, the difference between the polynomial and this function which has compact support is very small, uniformly very small. That's a virus stress approximation or theorem. Oops, sorry. All right. So now let's go back to looking at f sub w bar and think of it as a distribution which is defined by the values that it takes on various functions, continuous functions, functions with compact support, lip sheets functions. So there's a number of classes of functions on which it is efficient to know what f of w bar is or that it converges to something in order to be able to conclude that it converges in distribution. So let's take a look at continuous functions g. Now I'm going to denote that by the inner product. So when I write f w bar g, this is going to be essentially if you want the expected value of g with respect to the distribution f of w bar, right? So you can think of it like this or you can think of it as being 1 over n summation i equals 1 to n g of lambda sub i of w bar, okay? So far so good? That's what it is. All right. So now let's look at the following function, which is not quite continuous, but we can consider f of w bar with respect to it. So suppose that instead of looking at the monomial x to the k, what I'm going to do is I'm going to take the absolute value of x and raise it to the power k and then multiply it with the characteristic function of the set. Absolute value of x is bigger than 5. So I just want to look at what happens when I want to sum essentially or do the average of just those eigenvalues or the powers of those eigenvalues of w bar, which are bigger than 5. Is that clear? That's what this is, okay? It's the average of those eigenvalues that are bigger than 5. Suppose now that's bigger than epsilon. I want to look at the probability of this event, the probability that this happens. Well, as we know, this is going to be immediately bounded, this is Markov's inequality, by 1 over epsilon times the expectation over w bar of the same thing, okay? And that should be clear because essentially what we're doing is we're lower bounding, we're saying that this expectation is actually going to be, because f of w bar is atomic, is going to be a sum and you're just taking certain terms and you're actually, moreover, you're lower bounding them, okay? So this is this inequality here. And now I claim that if I just forget about the 1 over epsilon and look for a bound for this expectation here, that is going to be less than this. And the reason is simple. So let me write the inequality that I want to show here. E w bar, f w bar, x, oops, to the k. I write very enthusiastically, which makes every board shake. Okay, x is bigger than 5. Oops, like this. I will write it as times 5 to the k. I want to show that this is less than or equal to the entire expectation, but now it's not the expectation of f w bar, absolute value of x to the k. It's the expectation of f w bar x to the 2k. And it should be easy to see why this is happening. This is essentially conditional expectation. I'm looking at the expectation of the whole thing. I split it into the expectation if the variable is smaller than or equal to 5. And I ignore that completely. And I add to it the expectation that the variable is bigger than or, sorry, conditional on the fact that the variable is bigger than or equal to 5. And I lower bound that. Is that clear? I've forgotten a term and I've lower bounded the other. I've used the fact that I have x to the 2k here and I take x to the k of it. Conditional on the fact that x is bigger than 5, 5 to the k is going to be less than x to the k. So this here is going to be less than or equal to the expectation over w bar of x to the 2k given this. And that means that I forget the rest of the expectation. The other term in the expectation. Is that clear? You can convince yourselves of it if you just write what these things are. Essentially. Integral with respect to w bar, whatever that means, summation of absolute value of x to the k times 5 to the k is less than or equal to x to the 2k. And you integrate that over the entire domain, not just over x to the k. It's positive. It's positive everywhere. Okay. So that explains the second inequality here. And now if you put these two together, what do you get? You get that the probability that f of w bar in a product with absolute value x to the k over the interval where absolute value of x is bigger than 5 is bigger than epsilon. So the probability of this event is less than or equal to 1 over epsilon. Times 1 over 5 to the k times this expectation. But this expectation, we know what it goes to. It roughly goes to c to the k. Sorry, to c sub k, the Katelyn number. So maybe I should have here a c sub k minus epsilon over 2 or something like that. Or plus epsilon over 2. I can certainly take n large enough for this to be as close as what I wrote as I want. And the bottom line is that that is essentially like 1 over epsilon times 4 to the k over 5 to the k. And what's the most important thing about this number here is that as k goes to infinity, this goes to zero. And no, I shouldn't have that here. You're right. Sorry about that. This is a mistake. Thank you. There should be no indicator function here. Very, very right. Sorry about that. Okay. So note that what this is saying is that f w bar is almost surely compactly supported in minus 5 5. Because if you had some remnants, some non-zero mass outside of minus 5 5, 5 is bigger than 1, you would see that. You would see that. You would see that here. Okay. Because it would be some large contribution here. And the probability of this being bigger than epsilon would not, as k goes to infinity, would not go to zero, would go to something else. So now let's take a look at the approximation that we're going to do. Suppose I take some g, a compact function. I have lots of typos here. There's no such thing as bigger, bigger. It's just bigger. There's only one inequality sign here. So for any g, compact function with support in minus 5 5, compactly supported and continuous, given some epsilon bigger than zero, there exists by wire stress. A polynomial p is equal to p epsilon. And I'm going to drop the epsilon dependence just because it makes my life easier. Such that the absolute value of p of x minus g of x is less than epsilon over 8 uniformly on minus 5 5. The degree of this polynomial is, of course, epsilon dependent, it's finite. So now suppose that I want to look at sigma being the semicircle distribution. And I want to look at, specifically, this difference. I want to look at the difference between the inner product of fw bar with g and sigma with g. I want to look at the probability that this difference is bigger than epsilon. There should be an absolute value here. It's bigger than epsilon. What's the probability that this is bigger than epsilon? If I can show that the probability that this is bigger than epsilon is less than epsilon, let's say, or some function of epsilon for any epsilon given n large enough. It follows that actually fw bar converges to sigma in distributions. So now let's split this up. How am I going to split it up? I'll split it up as follows. So I will assume that instead of g, I'm subtracting off p. Maybe I should do this step by step because there are three terms there, and it may be confusing. So let's see what I'm doing. So I would like to apply essentially what I'm going to do is I'm going to apply a triangle inequality. So the probability, I'm going to look at the following first.  Plus sigma p minus sigma g. Now, this is easy. This part right here is going to be majorized. It's going to be bounded from above by the biggest difference between p and g on the interval minus 5.5 because sigma is compactly supported. So this is going to be less than or equal to some function of epsilon. Epsilon over 8, right? So this part here is what gives us the epsilon over 8. Let's see. And now I want to do something else. I want to, instead of looking at this, I would like to look at the expectation of f of w. So instead of, I wanted to replace sigma with the expectation of f of w. If I can do that, no, rather, I should be able to do that because I know that in expectation, fw does converge to the semicircle distribution. So if I add a little bit more than here plus some other, let's say, O of epsilon, I can take this and transform it into expected of fw bar. There are bars all over here. Sorry about that. Is this clear? We can do this replacement because we know that in expectation, f of w bar converges in distributions to sigma. So the difference can be made on this p, can be made as small as, we please taking an appropriately large. So there's some other O of epsilon. So that explains how we get the first term. Now, let me see what else I want to do here. No, I think I confused myself. Oh, okay. Let's see. Okay, so that explains the first two terms, actually. The first two terms come from here. This term, and this term here splits. So the first two terms in there come from this term. And this term here splits into, let's see, the part of it, which is on x less than or equal to five, on which the difference between p and g is very small and because f of w bar is a distribution, by the same reason that gave me that this is less than epsilon over 8, that part will also be less than epsilon over 8. So let me write that down.  This is less than or equal to the first part. And then the second part, which is this difference, is going to be less than or equal to epsilon over 8, for the same reason that this is less than or equal to epsilon over 8. We're averaging with respect to some distribution, but we're taking an average of a function, which in absolute value is small. Therefore, its average is going to be small. Okay? So that expoops, how did I get back to there? Okay, so this explains maybe we don't have an epsilon over 8 here, maybe we have some other kind of epsilon, two epsilon, let's say. But the bottom line is that we have these three pieces here and the rest of it is small. So now let's see, I claim that the probability term on each right-hand side is going to be appropriately bounded by some function of epsilon, maybe not exactly epsilon over 4. So why is this true? First of all, why does the following, the first one, the first bound, why is this going to be small? Some O of epsilon. Well, it's very simple. P is a polynomial, P is finite degree, and we just showed that the last lecture actually, we showed that if P were a monomial, then this essentially converges to zero as n goes to infinity. So what all we're doing is we're just taking a linear combination of things that go to zero. It's a finite combination, therefore we can always make it by taking appropriately large n as small as we please. Is that clear? I'm sorry, because of the light, I can actually get a lot of feedback from you because I don't see you. So if you have questions, please ask. So anyway, this is why this being a linear combination of monomials, of monomial terms, each of which will go to zero, we can make it as small as we please by taking an appropriately large. The second bit is also going to be, can be made as small as we please by taking an appropriately large, because we know that the expected value of f of w bar with respect to w bar converges in distribution to sigma. Okay, we know that. And then the rest, for the rest, which I wrote wrong here, is the expectation that is the integral of this polynomial over x bigger than five. Suppose instead of taking five hours to take some larger value here, a larger value of five, like 10,000, okay? Then everything I wrote here can be made as small as we please. And everything that we made here is going to be made as I wrote it, except with that larger value of five plugged in. And so I can conclude that this is also as small as I want to make it by, because four over that larger value of five is going to be arbitrarily small. Okay, sorry, I used five, I shouldn't have used five, I should have used some other value. But the bottom line is that by choosing five appropriately large, we can make this probability as small as we please. So the probability that the inner product of f of w bar with g minus sigma with g is bigger than epsilon, the probability of that event happening is going to be, can be made as small as we please. Yes, no, I'm probably confused you, so. So you want me to go back? This, yes, why is this true? Oh okay, so this is the level, the distribution corresponding to the level density, okay? So this is the distribution whose moments are the expected traces of powers, which we showed that they convert to the Katelyn numbers and therefore in distributions it converges to sigma. Yeah, any other questions? Yes, yes, that's going to be a minus larger value of five, larger value of five. Why? Yes, so this is essentially that I'm doing, what I'm doing, yeah. So the probability, so no, it's not equal to zero, it's going to zero, right? It's going to zero as it goes to infinity. Sorry, I did not mean to say almost surely, I meant to say actually with probability one that it's supportly compacted. Well, it's actually almost surely, but I'm not showing that here, okay? So this is more information than I'm proving, okay? Yes, yes, but I did not actually show that. So I did not show that, I wrote, okay. What I wrote here can be used to prove that this is completely supported almost surely, but not directly, I haven't shown that. Yes, okay, let's, for simplicity, let's just consider that I never said this because it doesn't actually serve me any purpose in the proof and we can talk about it afterwards, okay? So I shouldn't have waited so long to do my slides. Strike this through and replace five by some larger value of five, like 10,000 throughout and you'll get something true, okay? The notes are going to be correct, but obviously these slides are not. I'm not using that anywhere and so we can just talk about it offline, okay? All right, so to come back, this part was easy, this part was easy, and this part was also easy if instead of five you used 10,000 or some large value thereof, okay? Appropriately chosen with respect to epsilon to make four over five or 10,000 to the k smaller than epsilon square, let's say, or something like that, okay? So the bottom line is that each and every one of these terms on the right can be bounded by some function of epsilon. Therefore, we can get some epsilon appropriately small such that for any epsilon smaller than that, the probability of this is less than, say, two epsilon. But that's the same thing as saying that in probability f of w bar converges to sigma because it says that the limb soup of the probability of this being bigger than epsilon is zero for any for any function g, and therefore the convergence in distribution happens. Yesterday this worked perfectly, today not so much.