Universality for mathematical and physical systems
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Universality for mathematical and physical systems

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33

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CC Attribution 3.0 Germany:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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2006

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English

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Abstract 
All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all governed by the same laws and formulae of thermodynamics. In this talk we describe some recent history of universality ideas in physics starting with Wigners model for the scattering of neutrons off large nuclei and show how these ideas have led mathematicians to investigate universal behavior for a variety of mathematical systems. This is true not only for systems which have a physical origin, but also for systems which arise in a purely mathematical context such as the Riemann hypothesis, and a version of the card game solitaire called patience sorting.

Keywords  random matrices universality RiemannHilbert problems 
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now I think it's it's that the said organizers older colubris does need to be my angel said the reason that he told Le Shuttle of reprogram beaches at BT continues who particularly that would leave users scheduled for the U. S. Shields metal amid the use Bright the talks and everybody the price not file a lawsuit mentioned that some the short communication stock also up to today although interpersonal the programmers to Wikipedia the probably after this lecture sell a around in the back of the printer but these little he'd AT jointly that while ready for next year In upstairs and next Philip of a professor lead couldn't Institute of Medical Sciences up but she's an amazing with magician keeps surprising to bright she keeps surprising a getting surprising results and being
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ready dead now he said the heady as the ambassador chemical engineer and then ready still realize what he's got keyed a key he said he was reached the Rockets study due to let it was let go and a days but volatile state that he'd the study of possibility that the result will be gracious up huge that incongruity problems that year then my us we make story rather that late that problems of this salad at that we really something she said because the of results and particularly the new method coming from pretty much the problem and she said she applies in an amazing ways that will and he steady now per se he's a very warm and then baby is what he has said he will personality and very excited about the that GM these excitement attracts a lot of young people and his remaining a few rules would be deal eventually he scored each that there would look more can you desire the level at a HUD Phillips said about you his title Federer broke rumors Sally 0 0 but will eat those whose ya bug but you how a give sanctuary Chris I would like to I thank the program committee for the opportunity to speak at a very appreciative also said like to complementary these Spanish Mathematical
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done in organizing this that Congress stood but or problem talk to the universality for mathematical and physical systems so I try
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and get this boy down shares outlines of my talk
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a cell 1st of all I'm going to be giving a very general description of the universality uh some ideas that from physics main thing and then I want to propose or speak about a mathematical model and the particular model with China I want to focus on is random matrix Inc after that I want to speak about some physical and mathematical systems which will illustrate the ideas behind this talk and I want to show to relate the problems which a
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currency particularly particularly 2 sections being witches random animated theory and now wants is a little bit about what the mathematical methods on behind these results and how they relate to a possible future developments key set To start over at full of physical systems equilibrium probably do about the lowest of them Max and be 1st draw them and Amex everybody knows this conservation of energy is 2nd law has many different former formulations and the 1 I want to mention here works in the farmland white suppose that we have a huge reservoir but some temperature T 1 and supposedly have if he at at lower temperature T 2 we have some heat urgent care in the middle and you you take an amount of heat Q 1 from the reservoir you exhaust and amount of heat cute to move into the sink and the amount of work which is done by the attention Hugh White minus Q took now what we're interested in is the efficiency of weak on version of heat and to work so the efficiency is given by W Q on Monday due to a divided about Q 1 the 2nd law makes a statement about the the maximum possible value on the efficiency so maximum efficiency which you could obtain presumably the process thoroughly friction issues site that the maximum efficiency is about T1
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minus to appoint T1 and nature's you just can't do any better than not that's on the 1 hand and on the other hand there is a very old idea going back to the Greeks for that matter is made up of the constituent elements we call them Adams and a each of these Adams has its own different set of lures of interaction so it's really juxtaposition of these 2 points of view the macroscopic world table and the microscopic world which we imagine that lies underneath it bad debt presumed this Long ongoing challenge which involves 70 people 2 tries understand the emergence of a macroscopic world of the
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microscopic world so how does 1 derived these macroscopic as remembering that lead should be different can't constitute elements may have different microscopic lures of interaction so V salient feature of this challenge to reduce the macroscopic world from the microscopic world that exactly the same laws of thermodynamics emerged independent of the details atomic interaction the same laws but chip now In the world of physics this is not a broadly speaking as universe Saturday although there is some have a because softening by universality some statements about different critical phenomena skates standing laws but nevertheless I think is a good way to described things now costs Major said along the way that there are certain sub universality classes which mention later for example liquids like water and vinegar you expect them to debate heaviest subsequent review looking at some heavy oils you'd expect them not to to do that some other lawyers various lubrication questions so there are no subclasses which satisfying and what we could cause the universality now until recently this way of thinking that visitors have not been this these ways have not been common amongst mathematicians mass additions tend to think over problems as being different until proved equal so each mathematical Michael from mathematicians think problems says as to be generous each on Sunday unless you can prove some explicit or implicit isomorphous and between the 2 kinds of problems at the idea that broad classes of problems on some skirt should look the same without producing some explicit mechanism or why some awesome between name has not been common idea with mathematics nevertheless what I want to speak about today and a report today is that this type of universality sums for the emergence of a macroscopic mathematics for want of away seems to becoming more common and I want to illustrate this way a variety of examples which I will get to the moment they are mathematical presidents of course for what I'm thinking about we all know this is the central limit theorem going back to the 18th century we take variables independent identically distributed means you're variants 1 we add them up makes 1 then we scale Peru and we ask What's the probability is of the sky some to be less fatigue and that can't village To the normal distribution so 1 sees that each of these variables could be completely unrelated to each other X 1 could be the temperature in Madrid next to be the temperature set in Barcelona at 6 30 the pressure in Madrid in Milan and so I knew my bedroom they had no physical relationship is nonmechanistic relationship nevertheless this broad Fareham makes an assertion of universality of these systems now of course with the problem of probability theory this is just the 1st amongst many such universe 7 set that is the context and which I'm not going to presented the rest of the talk so the question is whether these kinds of phenomena which a wellknown with and physics and if you think for about think about that for a moment if they they were not these universality goalless with and physics there really couldn't be any physical laws Randall said Lindsay beginning now with a mathematical model of ran the matrix there's not at this at this point there are many many different random metrics models which are of interest costs around the metric system matrix and by metrics and the entries have some randomness attached to them a different models which it place on them and we will be interested primarily in the short distance to different on songs of the 1st ensemble the Garcia unitary song which is GUE Nabi elements Beyond solidly and by an emissions matrices and equals M start with Cohen the J and the probability distribution you put on these just some kind of renormalize the debate measure so DM is a big measure on a diagonal introduced the debate measure the real part of the off diagonal elements and have a part of the matrix and that's the debate measure on the imaginary parts of the Matrix each of us this traders squared away of normalizing down there be debate distribution and at I 1 . se images and normalization can't constant they not as it were a little bit to get the ball running road running is that if we replaced traces of em Square but traces of theater for example that the event could be damned before we could replaced traces of him squared with the trace of into the 4th you get a general example of the unitary ensemble and uh is sitting in the whole structure a universe solidly win within this choice in other words what is true irrespective of which you the statistical properties of the matrices again be independent of their choice that is fairer and that sort of Osub universality result moving a lot so bad the unitary part of people think about it refers to the fact that such a distribution is not on unitary Majesty's distribution on mission but the distributions are invariant and the unitary can't conjugation not just as the matrices Our random and have this distribution their eigenvalues which we arrived 1 victory goats amounted to theories and will become rank random variables and that's In particular that's true and GUE end His 2nd ensemble is neat Garcia orthogonal ensemble orgy of Judson Armstrong the elements and and real symmetric matrices equals M chance with entries and I did probability distribution is very similar to the GUE cases except not beg measure Everything Israel's salads just be MKJ because the judge again you could replace trace of squared traces of say into the 4th all of the 6
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authorities said Colin again there will be universality results along the way which will tell you that interesting statistical Kwan quantities are independent of the choice of each of course values lender the 1 that day and also become random variables sounded G eat so just summarizing a little bit of what time at this point although I'm presenting GUE GOP models they could look much wider class Our songs and tank exactly the same results not comes a pen and important point is that what do we mean what we say the system is modeled by random metrics theory but we said it's model by random edits series even the heads the victory like I that is of some large DUE EAT random matrix have to make a little more precise along the way it is something which is known as the standard procedure so what you should have in mind is a situation a little bit like like the following a scientists is trying to investigate some phenomena and the scientists puts the phenomenon some sly which he or she then put into a microscope and can do to do things the 1 thing that can be done as 1 consent of the slide the other thing it you can do is also focus but once you've done that your set and you have to look and see what you get the analog of that it is what is what fun means but the standard procedure so what you have is a set of quantities little AK In the neighborhood of some 4 . 8 and you want to see if these quantities take a look like the eigenvalues of a matrix you now Majid you have argued that Islam decay of metrics in the neighborhood of some energy E they what we always do is center so you move the slide in the middle of the microscope so you move a a minus capital a new movie eigenvalues they came under city union scale both of them and the agreement what is meant by the standard procedure is you ensure that the expected number of AK tells instead OK cases per unit interval it it is the same as expected number of scaled eigenvalues a unit interval and in the bulk of its usually taken to be 1 so this is these away things operate whenever we want to compare winding phenomena man physical with the item values a random matrix we always understand that we're prepared discussions by following the the standard procedure now we're interested in 2 particular statistics for Duane and they're a similar statistics in form of formulae for G a week but I'm not going to write them down I'm just going to ask you to imagine that there there so say there is some positive number and defined the gap probability P which is the probability that the GUE metrics has no eigenvalues in the gap Miners stated they send gamma can be appropriate standing the standard procedure and that's a wonderful result from the sixties down and made which showed that for any positive number 1 if you ask what is the probability that there are no that in the scale interval figures given by an explicit formula which is to determine the oneliners KY with KY tricycle operator With socalled assigned personal and acting on held from minus why wine and what I would ask you to do it is perhaps not remember the details of this formula but that there is such an explicit formula and it's part of the year as it further the charm and the effectiveness of the subject that there are these beautiful form formulae which can be evaluated and that give you very precise information on the disco quantities looking at the sick and statistic that I want to bring to your attention it is the statistics of the largest eigenvalue lemon Lember 1 and watch we do again that's a similar business you look at them 1 and you stand here this century must be done by taking away square attempt to end and you scared some appropriate where this goes into the minus 60 and the severity of Tracy and whether that this distribution when the size and the matrices its largest given by an explicit formula called the Tracy with them distribution and has this absolutely wonderful form which is an exponential basically the square of a solution you need global solutions equal the Hastings McLouth solution on handover to equation which if you'd think think canceling the nonlinear please you see looks like a serious question and you choose your solution you to be the 1 which looks like Thierry function the classical theory function goes lesson from the beginning and ask you to remember the exact form but just that there are explicit form formulae for these 2 basic still statistics the 1st being the gap probability the probability that no eigenvalues in the gap scale and also the probability distribution for the largest eigenvalue around the metrics now 1 of these most important features characteristic features of GUE OGO your way nearly orthogonal unitary on songs is the notion of repulsion which I will come back to that quite a bit later on as we say this you have these random matrices you have derided values of the eigenvalues on themselves random variables and Yukon calm you can compute exactly the distribution function for the yacht and the feature it has Is this van demand raised to the power base if we dealing with GOE Bay there is 1 if we did with GUE and beta is June and 1 other distributions I'm just putting in beta forward something that simplex ensemble just a remark 90 is what this sad it's tell you that if 2 eigenvalues of the probability of that event is very small so what that means is that naturally speaking when you're looking at a that matrix display adopt line they by the natural repulsion which is built and the probability of being close together is small and this is a key feature of random matrix series this notion of repulsion so not I'm and to the point of told where I want to speak about some example is not the 1st example From physics and that's where random metrics 1st introduced into the theoretical physics world
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in and after that came into the mathematical world it was introduced by vignette and so it's appropriate to begin at the spot so what you should imagine as my 1st example you should imagine that you're scattering neutrons at some energy onto some very large nucleus which could be uranium to 238 thorium 232 now the picture is looking at the first one is for thorium Saskatchewan metric its peace getting diagram for thorium the 2nd 1 is full uranium among the axis along the xaxis is the energy and on the yaxis is loosely speaking the amount of skeptics stepping across crosssection the feature which I wanted to focus on this that are many many many lines and if I was to expand my xaxis you would see there would be hundreds of these socalled scattering resonances the meaning of the scattering residents if I pick energy which said at the speech that neutron their energy coming in hitting the storeroom would be mostly reflected but if I think an energy which is between 2 weeks this you try and will as it would go through the details of this Of course not important the question is how are you proceed to model such the physical situation a priori possibility of writing down some shredding time equations and in solving that numerically is clearly it was beyond the CompuAdd is that at this particular period in the 19 This is a 1972 certainly was it's beyond us now and it's inconceivable that 1 would actually be able to really put that on come computer and actually find these stepping raised some of them some other way had to be found bombmaking scientific sense of a diagram markets so the 1st question is how does 1 model is resonance peaks and the form of my talk understand posing for a while a variety of questions 1st all this 1 from physics and then some questions from mathematics the 1st question how does 1 model of these resident speaks the next question is subject which has put imagination of many people and it goes Back to the work of monk armory in the the early seventies he was interested in the zeros of every month say the function data of S and assuming agreement hypothesis Montgomery looked at the nontrivial zeros on in the lineup half look and he wrote rat in than usual way onehalf but sigh Gehman did he rescale again he had this standard procedure and what we would now call the center stage in the back of his mind is scale means basic 1 in the sense that the number of zeros scaled which listing of Conti goes 2 1 as he gets launched bin for any aid is being computed the twopoint correlation function for the gamma till this add a bun has another look at the details of the correlation function the loosely speaking it's sending you wind damage a 1 until then to their up close together but he then showed modulo certain technical restrictions show less correlation function for the zeros agreement there the functionaries scaled on line how often you divided by and then you took this limit this limit would exist and was given by explicit formula the question my 2nd question is what formula did Montgomery kind are a bit 8 not just problem I want to speak about comes from calm combatant Torre and it's a particular card game and you play the game and the farming where you have a date and cards which for convenience your number from 1 you shuffle that deck and they need take the top card and you put the card face on the table to my left take the next car if that card is less than the cards on the table and put it on top if it's bigger I make a a 2nd I take the 3rd part if it's less than either of these 2 cards put on top and I have the agreement that if it's less than both of them for the this part to the left as I can if it's bigger than a maker of file and so on until I don't tell the whole pack and be question which 1 asks is how many times do you get mathematically of course shuffle is just a choice Ave mutations fine June the highest number of files you get off to Europe led by the state the camps a more interesting version In the bar late at nite budget deck of cards question your bathing on how big a table to scan so there may give an example of how it works suppose we have 6 cards we shuffle them with take imitation time and we did the limitations 3 4 1 5 6 2 3 is my top card for underneath it 1 5 560 so I start again my talk goddess 3 I put it down my next it's bigger than for the time they arrived on my right 1 and 1 is listened those 3 and 4 and orders to put this far to the left aside I bring it down a fine 5 is now bigger than the top part 1 and the top court for supported 76 goes down have to is less than 4 5 and 6 figured in 1 go Of the 4 because of my room of bearing his father that as I can so the number of miles I get Cusick is equal to 4 when they corpse is with uniform measure and after question how of time very statistically as an gets well did say she is a major problem from transportation series so it's a problem about the the busses in the city of Cuernavaca In Mexico now a city is about half a million people they said they have a bus system but they don't have a Central Transportation Authority the end result is that there is no bus scared so what happens you get this typical plus unlike phenomena and that you would be standing at a bus stop and they will be big between 1 best export law the busses common there could be bunch now the busses are owned by typically individual operators they were facing a situation where they would come to a bus stop the
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Stewart says OK we are ready for boarding and people lined up at the deck and suppose that lineup the order 3 4 1 5 16 now these numbers refer to the ticket the person has so far the ticket number for citizens seat No. 4 and so on so the lined up at the gate In this sort of 3 is closest to the gate for right BBB behind themselves so that now file into the airplane that fire will free can go to busy but then 4 is
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blocked and cannot and must wait until 3 puts out the backs the Presidency No. 1 can that seat but then 5 6 and tumors white behind in their blood cell after 1 unit of time 1 and 3 sit down and now full 5 6 and who are free to move on for ghost the seat 5 and 6 of but to 2 in guarding foreign to put there but exempt after 1 year of time sounded 5 can go 5 takes 1 time finally 16 geared to see No. 6 and we see that this process is model process Bush take 4 units of time and the question which we are asking him it is assuming that passengers lineup randomly How long does it take 2 board such an aircraft so those of the 7 questions and now I want to start off buying he says know the remarkable factors that although all these problems come from extremely different areas of science mathematics physics plant applied mathematics all these systems are marble statistically by render metrics so Our a recall that there's something as a model but Brenda Magic City we have to go through the standard approach procedure and compare these statistics of these different random quantities with random so the 1st problem which I remind you Saskatchewan problem the neutron these baby nuclei scan scattering resonances after stance procedure the probability that there are no residences in interval minus why wine is given either by this formula which I ask you to recall of 1 minus take a wine which is the essence of again probability for GUE introduced about for it Jim both analog and you Rogelio GOV depending on some underlying symmetry and dished so In some very remarkable way v neutrons are behaving like eigenvalues overran the next the result was about this he runs the risk of theories of the remains the functions awarded Montgomerie found find after some technicalities founded the limiting twopoint function RAB for this zeros has this form explicit formula and related 1 minus signs try our squared 9 as noted by by Dyson and the famous story which I will not repeat this is precisely the limiting twopoint correlation function for the eigenvalues of a random GUE matrix not this as a basic video aired it has been taken up by many people people working in number and number theory next Sunday cats keeping many many people people but it's sort of these 2 examples now so that sets out the book of what we're talking about on the 1 hand we speak about this very explicit came a physical 8 experiment on the other hand we speaking about this very pure mathematical objects which of this era the agreement say the and somehow this commonality of description between so now what lies between these 2 extremes d 3rd example for the him it is a game of cards the station's sorting piano time which is the number of piles that fuel tank that turns up the have like the largest eigenvalue duly metrics other words if buys a number of power circuit again you gotta do some censoring instead but once you've done that you compute this and this ghost 2 therefore to witches exactly the Tracy where the distribution for the largest eigenvalue Jean duly metrics you may remember that that is something which involves the pound and the right to live tool question somehow in this very strange wages playing this game cards bringing in this esoteric from functions this is a fair amount of vaginal by myself and could Johansen and has been developed further behind many different people mention cocoa foreign many many people dressed and with them the 4th problem is the busses in Quetta Cuernavaca and the question was what did occur by electric and Shaver from welldefined found want remarkably that the spice things between the busses after the intervention of these hot observers sigh exactly like the eigenvalues around the GUE matrix so the formula is again you get this committee chairman of 1 take a 2nd derivative with respect to the length of the interval you integrate from sutures and that is what you get no kind of these things that I want to get across it's not as if these are so very approximate model the accuracy of the models is really quite astounding as I'm going to show you open In a moment it is quite remarkable 2 to me and nothing to edge everybody's thought for about just how good this random metrics model a new show you now what they actually found services taken paper of Kerbala Concerto in general math physics now what you're looking at heavy line is exactly this formula 2nd derivative integral the 2nd derivative of this Dean determined the crosses I want actually observed not if 1 is applied mathematician you get this kind of fit you quite a stone but the situation is even better than it looks at 1st glance this is an instead which is a blowup of the lefthand corner and then but you'll see if you look on the Indian said there is heavy beeline there over crosses and they are these lines now what these other lines Tuesday take into account that the bust rabbits are not Audi Our about observers are not recording all the information some of the information is being from away because what I believe is called the new problems then it so what they do do to overcome that drug problem dates sample these statistics of the eigenvalues by leaving some of them out when they leave some of them not to model the way that the actual observers operate get the star of the show which goes through these processes even better then um these at Andy original so the Fed is really quite extraordinary
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non the 1st problem Is the Walker problem was analyzed by Peter Peter Forrester and then found the question had random walkers how far does this guy on the did well the statistics there guys His exactly describe and dislodged by the largest on delivered G both tricks G I'm not their mission there the real symmetric matrices and again it's given by some explicit despair Tracy with distribution which is very similar to that if he charm right down before the 6 problem is as diamond is at take this squaring Utah that now it was a wonderful result of Nokia's problem and many of people add that there is something called objects circle phenomena we its larger scale 6 5 6 7 and did the socalled come them what you find and the circles call the Arctic Circle in the top regions the left to the bottom of the right to which a court polar regions you find that tiling is completely rated yeah it goes east west nor northsouth and inside its as it were she would ran Anderson and reaching the called tempered so bedsit result a variety of people so in southern polar region things of frozen and set the temperate regions things are not good Johansson proved an absolutely wonderful result you draw lines as a jaundiced parade line and sharing the edge of the circle crosses the line and 2 places and for any finite approximately described by this step for In general for finance and they will be flawed fluctuations solid results of Johansson as those fluctuations about the circle are exactly described by the Tracy wouldn't distribution so they behave the argued that is around the metrics not the airline boarding problem begin we find that under this model that the time it takes to board as people lined up at the gate randomly is again given by Tracy wouldn't distribution the largest size and so I just means examples just trying and spread out within mathematics away these phenomena are occurring in many many other problems this exceptional times can't condensation problems of percolation problems called theory developed by Peter Sonic and his collaborators and Keating's work connected to L L functions there are many many different receptors that is to give you some sense of how things work at the end of these mathematical there will be status of the problems are of course neutron scattering problem is experimental and numerical these data function is an actual mathematical theorem that behaves like that to the twopoint function on zeros had blocked twopoint function around the magic modulus certain for realtime 3 restrictions the patient's salting problem is fear on the busses and quiet Cuernavaca there is now a model for it which was developed by genetic by Alexi Borodin to fix Sudan and myself will be able to show the origin are the random metrics still statistics the problem that Fareham diamond problems of their own and rely boarding problem but cannot what is the kind of mathematics switches which is In both GM and In verbal systems is a key player in the coming into the analysis ideas from the scattering theory agreement with methods salary mansions and of a steering clear of DDE determinants of classical Henry Manitoba steepest descent method and also different many many different comminatory ideas people like Gisela's ideas and also short the idea is going back to shore is a kind of mathematical renounce its of course so that many lectures on son to really bring that about but that is the kind of mathematics which is which comes at him so on my last Friday I want to just raise a number of issues question is may be asking yourself underlying recognize that the system I'm interested behaves like Grand metrics and the 80 sign sign a more scientific study the statement devoid of a wouldbe is that In intrinsic probabilistic terms how they state affair which would be the analog of the central them the center limit there exist a but independent identically distributed variables I do a specific thing on I and scale them and then I get a normal distribution question 1 month's Austin purity probabilistic terms of sale but some independent identically distributed distributed bearable I do some operations Exxon they went to operate and some them render metrics that's the kind of intrinsic question which is being raised and work in this direction has been done by Michael Stewart Ugandan also independently but not Carter we think about what is still a question which is froze SAN more analytical terms is that what we used to be you want people believe is that the natural arena for thinking about these things is the of distribution so that the space initially it's something with Dr. structure with topography but we do know that is something special here a Garcia point like little Valley now we know that as you get near to it you can be and but now we understand there isn't just the Garcia point is also things like the trade Tracy where distribution among 1 somehow put some kind of magic down to understand how you flow in the space of probability distribution services a different direction finally said the question is to what extent always seeing an emergency what 1 might want to court macroscopic mathematics In 1 as microscopic physics and macroscopic physics which satisfies them from the Nymex and indoor front just want to presented picture which I would like to give to you said when should think as it were that 1 is in a valley and you walk around the valley and you see this thing and that's different from that thing but it's like this thing but it's different from that thing you begin to step down from the valley begin to walk away some of this the remarkable thing is that what happens is that the situation as you look back on does not just from the into
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some indistinguishable head picture what happens is that very clear picture begins to emerge with something very clear structure begins to emerge which is bad varied robust and contains a greater amount of detail and it is this distant picture that is so well around and the Matrix thank you