 Okay, so thanks everybody for making it out and thanks Lionel for giving the talk today. As I said, we've got Lionel Levine from Cornell University and he will be talking about Abelian sandpiles and Abelian networks, go ahead and take us away. Okay, thanks for inviting me and I wanted to say since it's a small crowd you should please stop me with any questions and another thing I wanted to say is it helps me give a better talk if you turn your video on so I can see people's reactions and whether I'm interesting or boring for you. Okay, thank you to those of you who turned it on. So I want to tell you today about mathematical love of mine, Abelian sandpiles, but I'm also in addition to telling you what I love about them, I'm going to tell you what I don't like so much about them, which is they're very fussy, they're very sensitive to the underlying graph and to the initial condition and we're going to make that statement precise in various ways and then we'll search through a broader space of models that I call Abelian networks for a more universal model which doesn't depend so much on the details of how it's set up. So before I tell you mathematically what a sandpile is, let me tell you the physical metaphor so you can imagine a big pile of sand and you're sprinkling additional sand grains on top of the pile and when you sprinkle each sand grain you can't control exactly where it falls on the pile so it falls in a random place and when it falls on the pile it may or may not dislodge sand grains nearby and so some sand grains might start to trickle down the pile as a result of the grain of sand you dropped and the model is going to be a kind of toy combinatorial model of these kind of avalanches of sand and there'll be a particular parameter that you're interested in namely the slope of the pile. So again before the mathematical details I'll tell you where this model came from it came from a physics paper 1987 by Bach, Tang and Wiesenfeld and the question they were interested in answering is why do you see so many heavy tailed probability distributions in nature or another way of asking this is why do you see long-range correlations in nature so many physical systems if I tell you what the system looks like in a particular place then that has a non-trivial correlation with what it looks like somewhere else far away so you know how do long-range correlations arise since the laws of physics are ultimately local there's no action at a distance and the mechanism that they came up with for why these things arise they gave this buzzword which people still use today they called itself organized criticality and so let me just illustrate it with the example of a pile of sand so if you actually form a pile of sand and do the physical experiment of sprinkling sand grains on the top what you'll find is that there is a critical slope of the pile which I call zeta such that if your pile is resting exactly at that slope then when you drop an additional sand grade on top of the pile and measure what's the probability that it will dislodge 10 sand grains as a result or 100 sand grains as a result this probability will decay like a power of the threshold so the probability that you dislodge at least t grains of sand will decay like a power of t and that power law only happens experimentally it only happens at this critical slope if you started with a pile that was flatter than the critical slope like this one here then when you drop sand on top of it you will hardly ever see big avalanches so the chance of seeing the chance of dislodging more than t grains of sand will decay exponentially in t and as a result of that exponential decay as you drop more and more sand on this flat pile it doesn't trickle very far down the pile and so the slope of the pile tends to increase conversely if you started above the critical slope then you formed a very very unstable sand pile and so unstable that if you drop just a small amount of sand on the top of it you'll create huge avalanches that will tend to flatten out the pile and so you see that dropping sand on a pile has this natural mechanism whatever slope you started with even if it was less than zeta c it will increase to zeta c if it was greater than zeta c it will decrease to zeta c and so so that was bak'teng and weisenfeld's idea that you know some physical systems even if they don't start at critical in a critical state they have a dynamics that tends to drive them toward that state okay so with that as motivation let's define a mathematical model of a sand pile this will take place on a finite connected graph g and i'll write the tilde for the adjacency relation of this graph and we'll define a sand pile on this graph is an integer valued function on vertices notice i allow this function to take either positive or negative integer values most of the time it will be positive but at a certain point it will be useful to take it negative so the way you should interpret this function is if you have a vertex i where s of i is positive you should think of there being that many sand grains at vertex i so the the grains of sand live at the vertices of our graph and since it's shorter to say i will sometimes call the sand grains chips or particles now how should you interpret a negative value of s you should rather than a number of sand grains you should interpret it as a whole that could be filled up with sand so that's the interpretation of negative values and in our toy universe of sand piles on graphs there is one law of physics and it's it's this one down here it says that vertex i is allowed to topple if the number of sand grains at vertex i is greater than or equal to the degree of i the number of of neighbors of i in the graph and what does it mean to topple you topple by sending one chip to each neighbor now it's useful to rephrase this toppling rule in terms of the graph Laplacian now rather than I can of course think of the graph Laplacian as a matrix but i'd like to think of it as an integer matrix and so it acts on integer valued functions on the vertices so so zv is the free-ability and group generated vertices i'll write delta i for its standard basis vectors so delta i is one at coordinate i and zero elsewhere and then um capital delta the graph Laplacian uh what does it do it acts on a function f and it produces another function and that function's value at vertex i is defined as the sum over neighbors of i of the difference fj minus f of i which you can also write this way um now what's the relevance of the graph Laplacian to sandpiles to see the relevance just apply the graph Laplacian to one of these standard basis vectors delta i and what you will get is the vector that is equal to one at neighbors j of i is equal to negative the degree of i at vertex i is equal to zero elsewhere so toppling vertex i transforms the sandpile s to the new sandpile s plus delta delta i now we want to capture the idea of avalanches in this model and those are sequences of topplings right so the idea is when i topple the vertex x1 that might cause some other vertices to be able to topple and then i want to topple those other vertices i could get a cascade of topplings but then you have the question well in what order should i perform these topplings so i'll define a sequence of vertices x1 through xm to be legal for a sandpile s0 if at each point so if it's i have enough chips at x1 to topple x1 and then after toppling x1 i have enough chips that x2 to topple x2 and so on all the way down the line so at each index i the number of chips at xi after toppling the previous i minus one vertices is at least a degree of xi okay so that's one particular legal ordering in which i could perform topplings the kind of dual notion to legal is stabilizing so i'll call a sequence stabilizing if uh after doing the topplings x1 through xm i can no longer topple anything else so if the final sandpile sm is point-wise less than or equal to degree minus one then i say the toppling sequence is stabilizing and now we get to the reason that this is called the abelian sandpile model so there's a certain sense in which order of topplings doesn't matter um so to make that precise here is a basic lemma that makes everything work in in this model let s be a sandpile on our graph and suppose we're making very weak assumption just that there exists a stabilizing sequence say y1 up through ym for this sandpile we're not even assuming that the sequence is legal just that it's stabilizing okay then we get a bunch of consequences first of all any legal sequence has to be a subsequence of a permutation of the stabilizing sequence y so the existence of a stabilizing sequence automatically will give you an upper bound on the length of any legal sequence secondly just the existence of a stabilizing sequence will actually give existence of a legal and stabilizing sequence simply by taking a legal sequence of maximal length so that the empty sequence is trivially legal and so there's at least one legal sequence and if you take a legal sequence of maximal length then it must be stabilizing otherwise you could extend it to a longer one and and finally this this last point is is sometimes called the abelian property any two legal and stabilizing sequences well one has to be a subsequence of a permutation of the other and vice versa so there they must just be permutations of one another so this limit took a while to state but the proof is actually very easy so the proof is just suppose x were a legal sequence and y were a stabilizing sequence then in order for y to be stabilizing we have to at some point get around to toppling x1 the first term of the legal sequence so yi equals x1 for some i so now you could cross off x1 from the legal sequence cross off yi from the stabilizing sequence and induct on the length sometimes this argument goes by the name diamond lemma so consequences of this every sandpile has what we call an odometer function which measures the total number of topplings to take place so for a sandpile s the odometer function u at vertex x is the number of occurrences of x in any legal and stabilizing toppling sequence this number doesn't depend on the legal stabilizing toppling sequence because they're all permutations of one another and the second basic object associated to s is the stabilization which i'll call s hat and to find the stabilization you just take any legal and stabilizing toppling sequence call it x1 through xl and you take s and you topple those sequences in that in that order now of course the Laplacian's linear so you can write this as s plus delta of the sum of these basis vectors and the sum of these basis vectors is nothing but the odometer function so this down here in the red box is the basic equation relating the sandpile its odometer function and stabilization so i want to pause at this point just to show you a picture that the stabilization even if s is a very simple object the stabilization s hat could be a very complex object so this picture that i'm about to show comes from uh my collaborator west pectin's website okay so hopefully you can see an image of a sandpile it's got some red some yellow and some blue okay good so um so what this picture is is you started with uh two to the power 30 chips in the middle at the origin and your underlying graph is z2 the square grid so with nearest neighbor adjacency so every vertex has four neighbors and um that means it takes four chips to topple and when you do a toppling you send one chip north one chip east one chip west one chip south to the four nearest neighbors okay and we've started with this large number two the 30 chips at the origin which means we can do a whole lot of top planes at the origin and then the neighbors of the origin will get a lot of chips and we can topple them a whole lot of times and so on and these chips will spread out to fill a large finite region of z2 everything in blue here was not reached by the chips and everything in a color other than blue was reached there there are uh four colors in this picture i'm zooming in now on you know a piece of the picture somewhere in the middle maybe i'll zoom in a little more so now you can see individual pixels of this picture and each pixel represents a vertex of z2 and the four colors indicate how many chips were at that vertex in the stabilization so this dark reddish color represents three chips um and yellow represents two light blue represents one and dark blue represents zero chips and what i find fascinating about this picture if you move around in it you could see various periodic patterns of sand emerging so here we have a little piece of the picture where you get this periodic pattern with kind of blue crosses with yellow in the center and these red and diagonal lines but if we zoomed in to a different portion of the picture well here's a very simple pattern it's just all red so here's a region of the sand pile that's completely filled up with the maximum number of three chips per vertex and here's another pattern of alternating threes and zeros and so on so this picture is quite uh is quite intricate and what i find really interesting about it is you know all these different periodic patterns that occur in the picture um while they occur they're predict there's a pattern to the pattern right the patterns occur in predictable places and there are different patterns in different places despite the fact that there's only one law of physics in the sand pile universe you know topplings are the same in this region um as they are in you know this region but the pattern of sand you see is is completely different so with a lot of work we were able to classify these patterns for the square grid there are countably many of them and they're classified by the the circles in an apollonian circle packing of the plane um so that was really cool however everything changes when you change the lattice so now i've changed to the triangular lattice and let me zoom back out so here's a sand pile of two to the 29 chips on the triangular lattice and you could see broadly it looks similar there are many different patterns and they occur um in different places but the you know the specifics are different so the patterns are different the places where they occur are different so even the limiting shape is different and every time you change the lattice you get sort of a variation on the theme so so what's interesting about this model is although it has a scaling limit so these pictures have a weak star limit as you know as you take the lattice finer and finer and increase the number of chips you started at the origin um they have a limit that is no longer a function on the lattice but a function on r2 however that function remembers the fact that it used to live on a lattice and it you know it behaves differently depending on what the lattice was even though the lattice spacing has shrunk to zero so it's a it's a lattice dependent limit and uh that's the first indication that this model well it's you know does fascinating things uh it's not very universal right it depends a lot on the underlying graph so what can we hope to say about this stabilization as hat we've seen that it could be a very complicated object but let's write down the simplest thing we know about it which is it's pointwise smaller than degree minus one simply because it's it's stable so given a sandpile s we'd like to say as much as we can about its stabilization s hat and because of this equation it's equivalent if you can describe the odometer function u then you can describe the stabilization s hat and vice versa so here's something here's a characterization of u which we call the least action principle um suppose s is a sandpile for which there exists a non-negative integer valued function w that satisfies this inequality that s plus delta w is pointwise smaller than degree minus one from that you can conclude that s stabilizes and the odometer of u the number of top links that occur is the pointwise smallest such function w so among all functions w that satisfy these constraints you take the pointwise infimum the one way to interpret this is depending whether you like negative or positive words that sandpiles are lazy if you like the negative words or they're efficient if you like the positive words so basically the sandpile s it wants to stabilize and it will do so in the smallest possible number of top links and not just the smallest possible total number of top links but actually the pointwise smallest number of top links each site topples the fewest number of times that it possibly can to convert s to a stable sandpile now one natural reaction when you see a problem like this it's an it's an integer programming problem right so we have this linear constraint on this unknown w and we have another linear constraint that just w is non-negative but the tricky thing about this problem is that w has to take values in the natural numbers so a natural reflex is to relax this problem replace natural numbers by the non-negative real numbers if you do that you get something called a divisible sandpile and if you want to think in terms of a particle system instead of having discrete chips performing topplings you have a continuous amount of mass at each vertex and the the law of physics in this divisible sandpile is each each vertex has a capacity say a capacity one and if it gets more mass than one it distributes the extra mass equally among its neighbors so it's allowed to distribute you know fractional amounts of mass that's what distinguishes it from the abelian sandpile turns out this totally changes the behavior so you might be surprised by that because like if you if you did an experiment with a large number let's say on the square lattice like we just saw if you start with a large number of chips at the origin then there's going to be a lot of topplings you can show that the odometer is order n squared typically you know near that you know when you're not too close to the edge of the picture it's order n squared so you know if n was a million then a typical site is going to topple a million squared like a trillion times and so what does it matter if it's a trillion or a trillion point five you know that seems like it would be a very very small difference but it turns out when you require everything to be integer valued then that somehow interacts with the with the linear constraints in it in an important way and you actually get a different limit shape so we saw with the abelian sandpile the limit shape was lattice dependent and it had all this internal structure all these different patterns at different points in divisible sandpile the limit shape is actually a disc in two dimensions or euclidean ball in in higher dimensions and you lose all the internal structure basically you just you fill up completely a very nearly disc shaped region of the lattice and the only sites that retain a fractional amount of mass are right on the boundary of that close to disc shaped region so that's a big difference between these models another experiment you can do in the sandpile universe is instead of starting uh with this deterministic finite configuration of n chips at the origin you could start with an infinite configuration it's stationary and ergodic so the simplest example of that would be let's take every site in z2 to have a random number of chips with some probability p it has four chips with some probability one minus p it has zero chips right and then you see can you stabilize this random sandpile or not well it turns out there's a critical mean you know there's a critical value of p such that if you're above that critical value then you cannot stabilize and if you're below it then you can stabilize but what happens is it that mean depends on the distribution so instead of fours and zeros if i took you know fours and ones and zeros are you know or Poisson distribution or geometric distribution each of those will have a different critical mean and you can actually get any critical mean between two and three depending on the distribution that's another example of kind of fussiness of this sandpile model it it sees the whole distribution of random chips not just the mean in contrast the divisible sandpile only cares about the mean so if the mean is strictly less than one then you almost surely will stabilize and if the mean is greater than or equal to one then you almost surely do not stabilize here's another experiment you can do you could fold up z2 to a torus discrete torus z1 lz squared so you're taking a piece of z2 and you give it periodic boundary conditions and then you can run a Markov chain where at each time step you drop a sand grain at a random vertex and then stabilize now there are two versions of this chain i call them the wire chain and the free chain the free chain is just what i said you drop sand at a random vertex and then stabilize problem with the free chain is there's no way for sand to escape the system you've got this finite system and at each time step you're adding one grain of sand topplings conserve the total amount of sand so at some point you will reach what i call the threshold state which fails to stabilize because there's too much sand in this finite system and then what you might hope is that this threshold state has some universal properties as l goes to infinity in particular you might hope that it is uniformly distributed on the set of recurrent states of the wire chain so what's the wire chain the wire chain is i do the same Markov chain but i fix a vertex here i call it z and declare it the sink so the sink is a vertex that just absorbs sand and it's not ever allowed to topple you think of the metaphor is you have a pile of sand on top of a table and as you're dropping more sand it gets bigger and eventually sand falls out the table disappears from the system advantage of the wire chain is you can show it always stabilizes so you can you don't have to stop when you reach this threshold state because you can run it as long as you like and it will always stabilize the sink is absorbing chips and this wire chain has some very nice properties in particular its stationary distribution is uniform on the set of recurrent states and these recurrent states have some nice combinatorial properties they're in bijection with spanning trees of the underlying graph so in this case spanning trees of the torus so you might hope that this threshold state will reflect some properties of this nice uniform distribution on the recurrent state however it doesn't and the reason it doesn't has to do with the mixing time of the wire chain wire chain actually has a rather large mixing time so its stationary distribution is this nice uniform thing but what we showed with Bob Huff and Daniel Jarrison is that it takes a long time to reach that stationary distribution you have to drop on order l squared log l chips in order to get close to the stationary distribution and notice that's bigger than the number of vertices which is only l squared so what's happening is if you start say from the all zero sand file and you run the free chain then you're going to reach the threshold state in time order l squared and that's not enough time for the wire chain to mix and that turns out to be the underlying reason why the threshold state doesn't have nice nice properties there's one limit where I was able to show that it does have some universal properties and that's the limit where the initial condition tends to minus infinity so if you think about this long mixing time one way to allow the sand file more time to mix is to start with a very negative initial condition that way it takes longer to reach this threshold state and in the limit as you take the initial condition to minus infinity you do recover some universality the way to do it is you should keep so you should start with this very negative initial condition and then keep track of when you finally reach the threshold state what was the last vertex where you added sand to cause the infinite avalanche they call that the epicenter of the infinite avalanche eyes of tau and if you then take eyes of tau as your sink vertex for the wire chain then the recurrent representative of the threshold state becomes exactly the uniform recurrent state so there's this one case when you recover the universality but you need to take this minus infinity limit somehow saying a typical sand file retains some memory the threshold state retains some memory of its initial state and so it's not universal but you can remove that memory by taking the initial state to minus infinity okay so that's that's what i want to say about sand piles and the remainder of my time i want to talk about the general class of abelian network so what we're going to do after i define abelian networks we'll search through this class to try to find a model with more universal properties than the sand file model so abelian networks the idea goes back to the physicist Deepak Dhar in 1999 and the type of network i'm going to describe i call a unary network with two stacks so the way it works is that basically we're going to change our viewpoint instead of the sand grains being the active participants in our model the sand grains will be passive and the vertices of the graph become the active participants and the way you should think of this is each vertex is like a little computer it's an automaton and it can change its internal state and so when it when it reads a letter it can change its internal state and it can send letters to its neighbors depending on what its internal state was and what letter it read so the way i want to encode the states of these automata are in what i call a movement stack and activity stack so these are infinite stacks of instructions associated to each vertex the movement stack is an infinite sequence of vertices row k of v indexed by natural numbers k and the activity stack is an infinite sequence of natural numbers and so again there's a single rule it might be harder to process than the toppling rule of the sand pile but what it says is that if vertex v receives a total of l letters then v will send out a total of alpha v letters and where does it send them the kth letter that it sends out is sent to the vertex row k of v so alpha tells you how many letters to send and row tells you where to send them and we're only going to require one axiom of this alpha and row which is monotonicity of alpha so alpha l is pointwise smaller than alpha l plus one this axiom basically says that you cannot take back letters once you've sent them so so this axiom is what will guarantee an abelian property in general if i have a network of automata they're communicating with each other it can be wildly non-abelian right so order in general will matter a lot but it turns out if your network of automata has this form that its behavior is governed by this movement stack and activity stack as long as you have monotonicity of the activity stack then you have an abelian property where the order in which the automata act doesn't affect the odometer function it doesn't affect the final state so it turns out a lot of familiar examples can be recast in this framework of a unary network with two stacks so let's consider a simple random walk so i'm going to have a sync so in this incarnation of random walk instead of having an active walker who makes a decision at every time step which neighbor to jump to i have a passive walker the walker is a letter that's getting passed between automata and my network and so i want to keep the system in a state where there's always exactly one letter and the way i do that is just by declaring alpha to be the identity function alpha of l equals l so that's saying if i received a total of l letters then i also emit a total of l letters and the rows the movement stacks i just take them to be independent uniform neighbors of vertex b that tells me the kth time i received a letter i should send a letter to a uniformly chosen neighbor independent of the path you could kill your random walk at some vertex z if you want just by declaring alpha of that vertex to be zero instead of l so that's saying vertex z becomes a sync it never emits any letters so when your letter reaches vertex z it just gets absorbed there you could do branching random walk in this framework by taking alpha suitably random you can do a process called internal dla where each each vertex absorbs the first particle it receives and then thereafter it sends it to a random neighbor so you do that by taking alpha l equals l minus one instead of l you can do the sandpile the way you do the sandpile in this framework is you take instead of taking the movement stacks random you take them periodic so the movement stack at vertex v will be periodic with period degree of v and one period will be a permutation of the neighbors of v and then you take the activity stack to be the closest multiple of the degree rounded down so the effect of these two choices is that each each automaton when it receives letters it saves them up until it has a multiple an integer multiple of the degree and then it sends them all out once each neighbor you can combine the periodic movement stacks with the alpha l equals l and then you get something called rotor walk which is a kind of a de-randomization of random walk so instead of going to a random neighbor at each time step you cycle through the neighbors in a predefined periodic order there's walks that are intermediate between rotor walk and random walk i call them random walks with local memory so you keep alpha l equals l so there's always exactly one letter in the system but you could take the movement stacks to be marcovian so at saying that when you arrive when you're when your walker arrives at a given vertex it decides what neighbor to jump to it draws from a distribution which depends on the past history of what the walker did at that vertex so that that history itself is a mark of chain you can do sandpiles with stochastic topplings instead of deterministic topplings so basically instead of sending one chip to each neighbor you can send each chip to an independent random neighbor you can do something called the rice pile which kind of models sand grains that have a very different aspect ratio they're long like rice instead of roughly spherical like sand and what i currently consider the most promising model for something that has universal properties is it's called activated random walk and what you do for activated random walk so if i describe it in terms of these stacks i take road to be independent uniform neighbors just like for random walk and instead of taking alpha l equals l i'll take it l with some probability and l minus one some probability so it's giving each vertex a little bit of stickiness so if you have an l for which alpha l equals l minus one then you have a particle that gets stuck at vertex v and it will get stuck there until another particle jumps on top of it in which case it can keep going so an equivalent description of activated random walk is each particle in your system it performs random walk but at each time step it has some probability of falling asleep and particles that fall asleep they remain asleep until another particle jumps on top of them in which case they wake up and keep walking okay this is right okay this is the last slide of this portion of the talk and maybe it's actually a good place to stop but i'll say that right now my collaborators and i have a lot of conjectures about activated random walk and well we have a few theorems but they're very weak so we conjecture that it's much more universal than the sandpile for example its limit shape should be a disc instead of a lattice dependent shape but we're far from proving that we also conjecture that if you look at its stationary distribution on finite pieces of for example the square lattice then those distributions will have a limit as the size of the finite piece goes to infinity we're also far from proving that recently with Fang Lian we found an upper bound on the mixing time so we saw that mixing for the abelian sandpile was kind of one of the key things that made it not universal so upper bounding the mixing time is one step in the right direction but still most of the interesting properties of this model are open and if you're interested in it there's a great survey by Leonardo Rola last year from last year you can find on the archive thank you for listening and i'm happy to take questions thanks Lionel if we could all thank Lionel in some way and we'll go ahead and open it up for questions as well do they have any questions for Lionel you don't happen to have any pictures that activated random walk do you so they're surprisingly boring right the problem is we don't know how to prove that they're boring so what they look like is so if i if i start end chips at the origin in z2 i fill up what looks like a perfect disc and inside the disc i see noise right i see very evenly spread out sleeping particles in fact they're more they're more evenly spread out than you might guess so one conjecture we have is called hyper uniformity it says that if you look at the variance of the number of particles in a box of volume v the variance grows sub-linearly in v which is implies some kind of well you need some long-range correlations for that to happen the particles the sleeping particles they kind of repel each other so they manage to spread out very very evenly and we don't i mean intuitively yeah it makes sense that they would not want to fall asleep right next to each other but having this variance be sub-linear in the volume is is a very to me surprising finding i don't know why it's true have any other questions for Lionel yeah what is a what is the long-range behavior of the activated random walk look like on a finite graph well it should look like so let's say we take our finite graph just to be an interval right so it's a very that ought to be a very simple graph but it turns out to be a complicated mark of change so i drop my particles say at the left end point of the interval and they do activated random walks so they fall asleep they wake each other up and they can fall off the edges of the interval say it's a path of length l so they can fall off the left endpoint or the right endpoint and eventually you'll reach a stationary distribution of sleeping particles on the interval and it's hyper-uniform in experiments so the sleeping particles are very regularly spaced okay it's still random so they're not exactly equally spaced but they're very close to it we don't know how to prove that and that implies some long-range correlation because if i told you there's a particle at vertex 5 and i know that in the stationary distribution the average spacing is 50 then i can go and look you know far away from vertex 5 and predict with high success rate whether or not there'll be a particle in that far away location does it appear that the sort of empirical frequencies of the when it stabilizes are proportional to the random walk limiting distribution so there's one parameter in the model it's this it's this p which is um the chance of falling asleep at each time step as you increase p the stationary distribution becomes denser so but we don't know an exact form for this stationary distribution one one thing that's not too hard to prove is you can exactly sample from the stationary distribution there's a simple algorithm which is you start with one particle at every vertex of your graph and then you stabilize so you let all your particles walk around reach the sink some of them reach the sink some of the rest of them fall asleep and then you'll exactly be in the stationary distribution so that doesn't take too long on a finite connected graph so you can sample in polynomial time from the stationary distribution but um somehow the sampling algorithm doesn't give us or we haven't been able to use it to get much insight into the distribution itself I guess yeah I was thinking without sinks what a finite graph without a sink yeah that's another thing you can do so then your mark of change should not be adding a particle because you don't want to and you know put too many particles in the system so that it fails to stabilize but your mark of chain can be you know you start with all a bunch of sleeping particles then you wake one up and stabilize they wake one up it walks around maybe it wakes up some other ones eventually they all fall asleep so that's another interesting mark of chain and we also don't have a good handle on its stationary distribution please any other questions okay if not thanks again Lionel and thanks everybody for making it out and have a good weekend everybody bye