[00:00:05]
>> Thanks a lot for the introduction so as if I decide yes I'm going to be talking about Iran walks this is don't work at my supervisor love to low. I forgot putting in the fire but he's also from University of Waterloo as you might have guessed. Ok.

[00:00:25]
Ok. This food is a bit like a parent you know. This is a period Ok. So I want to be going to hire to run walks. In the. All that Ok so what are these higher demand walks that I'm going to be talking about the setting is I have a collection of.

[00:01:23]
Containing some key elements a subset of a universe you. And I would like to some poor from the uniform distribution of the set My God So you know life gives me a distribution that I want to sample from I describe around walk and hope that everything goes well so this is the Runamuck I'm going to do I'm going to start from some arbitrary set of a one.

[00:01:45]
Every set at every step. I'm going to be picking an element from this that I have and I'm going to be removing from the sack. And then in the next step I'm going to be picking some of the ones from Monica uniformly random among all those that are current that are concerning that it's you know p.b.s. said to be God element typed.

[00:02:07]
So this is the process that I'm going to be looking not it's not hard to see that the uniform distribution is maybe the station distribution for this process. And what I'm going to be studying is going to be. The you know using the mixing time with this March Ok So before you get into anything exciting Let me try to convince you that this is an interesting run and want to look out so one example that you can you know phrase this framework would be to go with an image for grandfathering So the set on the guy that I have is a set of all proper q. colorings of a graph where you know I coloring is just a labeling afford to says.

[00:02:42]
No to a decent pair 1st this is of the same color so I start from arbitrary coloring and then every step what I do is. Pick some word takes uniformly it's random forgot its color and the next step callers available to this you know were attacked in this neighborhood I pick one from it's random Now repeat hopefully I reached in from stupid.

[00:03:05]
Another example the circle spanning tree exchange walk I have a graph. And the collection of that I want to look at is the collection of all spending fees of the scruff So what I do is I start with a spanning tree I'm not sure whether it's visible but like the doctored edges are the edges that I don't have in my spending tree the actual edges are the edges that I have in my spanning tree so at every step what I do is pick an edge uniformly it's random of the spanning tree I remove it.

[00:03:32]
I'm old the edges that do not close a cycle you know in the remaining graph. I pick one I had it I complete myself a spanning tree rinse repeat eventually a conversation distribution. Another example you want to sample independence that's this collection I got it you have is the collection of independent sets of side effect size in the graph and on the side is just you know a subset of verses and a graph that contains no edges so what I do at every step I sample or text uniformly a trend I'm I removed from my independence that.

[00:04:11]
And won't pull the word this is. You know no connecting I just The remaining in the months that I pick one I added to my independence that rinse repeat rinse repeat and hopefully eventually every station should be. Ok so I told you that I'm going to be studying argument at least so let me briefly justify why I'm going to be doing it maybe you know it's known to most people here but if I have run what magic speed it is no not you know because this is around and won't matter x. the top item 0 is going to be one.

[00:04:45]
And it is an interesting thing to study the magnitude of the remaining item out it's so in particularly going to be looking at lumped into a number and I'm going to be looking at the magnitude so this is just a 2nd single or value and we're going to relate it to the mixing thoughts around and walk and the mixing time is just a number of steps of the running world that I need to take to be sufficiently close to the stationary situation.

[00:05:07]
Ok it turns out the farther away the sequence of per meter is from one to foster my run in local merges so if I want to show that all the previous run walk to the station is commission being the uniform distribution in this case. I need to show that the sequence to is bonded away from so this is what they're going to do.

[00:05:27]
Ok so let me also beefy justify why I care about this right and walks hopefully as I might or might not have promised you this model of fundamental sculptures you know some processes that we care about. And c.n.c. like Yes For example you can think about problems of about systems in a small market change for something one colorings market chance for something that it bases that saturates.

[00:05:55]
And it also seems that there are some interesting things you can do with these run walks or rather the analysis of this run on walks in calling theory. Ok so by the way I am going to be phasing all these things for the uniform distribution on top of that but you know it's not much hard to change for arbitrary distributions.

[00:06:15]
Ok. You know I'm certainly not the 1st one to look at a run or walk in a 2nd when are you mixing there is a large role the if there is a large amount of previous work there is work days a multi commodity flows which essentially they would someone to come with the flow and then they look at the congestion and based on the congestion they argue and argue about use there is a sort of coupling methods which essentially you know looks at 2 copies of the Sun Run Walk which are a couple in some way and look at the pro will do that they're far away from each other there are also some more analytical methods like locks so many qualities and what if I look solo inequalities which I know hard to sublease But what we do establish them they give very sharp bones and in this work like the hope is still as a new Want to this list and that is local speculates mention.

[00:07:08]
This is a new ish one in the sense that it's been already used in the literature for or you know the proof of the matter explosion here. But in some sense before our work it wasn't very uplifting also what we're doing in our work is you know making it more applicable so I'm going to explain this in further detail just keep this in the back of your head.

[00:07:32]
Match right expansion so like this done this higher order on a walk on the set of metric basis. Ok so the plan for today is for a link going to be. Hired to run walks by using what is called local spectral expression. And the rough idea is that I'm not going to be looking at this big process I'm going to be looking at some small graphs and I'm going to be showing all these small graphs expand and because of this the higher order run and walk expense.

[00:08:05]
Ok so the main observation that we're going to be making is some local mixing implies a lot of mixing and by local mixing I'm really meaning we're going to be looking at a such a some local use. And these local views are going to be much smaller graphs like the big graph that I care about and they're going to be much easier to argue on hopefully and then we're going to be leveraging that make same to our You global mix.

[00:08:28]
Ok so I want to be explaining what these local graphs are to do that I need to explain you know what a simplicial complex just because. It's just more convenient. You know to introduce this notion to explain where these graphs live or you know I'm going to be giving a definition of these local droughts which are calling grass.

[00:08:51]
Ok related work again you're not the 1st person the 1st people to consider local spectra expansion there is a long body of work that you know I'm expecting no want to digest. But the basic idea is people have studied Ok let's look at local view of crops and if they're expanding then this whole big process expanding.

[00:09:12]
The most relevant for us in this is the work of coffin open home. So it was the best results prior to ours and this was used and the result of an aerial view and always go on. To prove the Fosamax Fos mixing of the fire to run and walk on the set of metric bases Ok so this is remarkable because we didn't know any other method that was useful and office of the make of this very beautiful proof is using some basic facts about high dimensional expansion our local spec like fashion.

[00:09:43]
Because so or our work is going to be the strongest one in the list but interestingly enough we're just going to find that all this is a little speech works Ok and to be able to make a non-trivial statement about the argument all you need to do is argue some basic points of its effects can so without Let me argue about let me introduce you to some political process.

[00:10:06]
So simplistic complex is just a dumb word collection worthless collection of subsets or some universe what does it mean it means if I contain a subset of my collection I also contain all of that subsets Ok so I'm not I'm looking at arbitrary complexes and it's folk I'm just little simple shit complexes.

[00:10:26]
What that basically means is that you can think off the top of a collection that you care about let's call it on it up. On the plus one of the month itself that's going to define x. just to be develop word closure of I go so x. is going to contain all the subsets of sets and all I got so if your comments are you must you can think of Ok to be content sorry you can go on the go to be containing all proper cue cards or paragraphs then he's also going to be content the next are just going to be a partial colorings so I'm going to be referring to like d. in the plus one as a dimension of my simple show complex because you know what the magician seem to get angry if you're you know not making this plus one so just maybe making them happy and the cardinal dimension is going to be just the cardinality of the largest set minus one.

[00:11:16]
Elements in this collection are just going to be called the faces of my simplicial complex and again whenever I have a face of my simplicial complex I'm going to be referring to its cardinality minus one as a dimension of my face. Because I'm also going to be introducing the short term notation x j to you not to the set of Face the Nation which almost any j. faces.

[00:11:39]
Ok so with this notation takes the is just going to be the collection I care about and x. minus one is just going to be the empty set is white minus one plus one is 0 there is only one you know side of Flight 0 and stamp the set.

[00:11:54]
Ok some remarks if you look at a graph a nice way to vertices that is just going to be a pure simplicial complex if I mention one and it is sort of like if it makes you happy or inserting the general to simplicial complex as you can think of people as one in 4 hyper graphs so x. minus one is just going to be damned to set again example is maybe a set of verses x. one is going to be the set of edges x. there's going to be a set of triangles etc etc And if the is going to be the collection of the characters.

[00:12:24]
Ok So far everything clear. Sure I'm yeah I'm also include Yeah. I'm going to be using this is just for convenience because it's an abstract definition. And it's not the most useful. Ok so what I'm going to be looking at as a get a specific kind of higher order Runamuck that I'm going to be studying is the so-called on up walk on x.t. again this is the same thing I introduced in the 1st slide just different terms and it's bolted on approach because I start with a face from mixtape I pick one element out from them I go one step down and then I go up one step.

[00:13:15]
And I pick this essentially uniformly at random from 60 among all the things that contain why I look at the same random walk as before the same on the walk as all the walks that I. You know introduce so far. This is just so we have a. Reference Point Ok so we're going to be using local spectral expression to study this walk so what is it me it means we're going to be looking at some graphs.

[00:13:44]
And these graphs are essentially we are actually going to be living in this complex. Ok so let's also be a j. face so if you once you can take x. to be the collection of proper colorings and also can be a partial coloring and I'm going to be looking at the lat us all the faces an x. which contain off.

[00:14:05]
Ok so I'm not good at drawing the stuff but what I want to think is that at the bottom level at like at the top of the spire midterm that I'm going to have the face. At the top. I'm going to have all the faces an extent that contain all 5 again so if I'm thinking about colorings also is just going to be a partial coloring this layer is just going to contain all the colors that are consistent but often and what I care is this really local view.

[00:14:34]
What's a spoiler. Like some looking at this part of the simplicial complex which contains stuff of the from Alfie and x. y. and I'm going to be you know visualizing a graphic there Ok so what is the word Dick said of the scruff going to be the vortex of the scruff is going to be held to the vertices x. in my universe so that also union x. is a j. plus one face so again we all thought is a partial coloring x. is just going to be a word 6 color pair that is consistent with alpha so you know.

[00:15:13]
If x. is say the vertex you on blue this means that also doesn't assign the color blue to any of the neighbors a few Ok and I have. The sort of edges x. y. such that x. y. Union all file is a j. plus 2 face. And again this is the similar thing is just a paper all right text color pair which are consistent in this coloring setting Ok.

[00:15:46]
Ok so also is just maybe a part of the. Bottom of it. Like the end. Should If so light colored so also is a partial color and. So the top is going to be all the colors that are consistent with alpha so when the scruff is the right is in the scarf is generally off the former u.n. blue or whatever un blue is a consistent you know what extent.

[00:16:16]
To all the coverings it. Is the size of office so if you're in colorings j. plus one is going to be the number of colors that you assign to. Vertices. Ok I'm also going to be adding this great w. Alpha. Which is a socially going to be. Which is going to be corresponding to the number of top level faces that contain Ok so if you ask if I have an x. x. y..

[00:16:49]
The weight I'm going to be assigning quick to it is the number of top level faces that contain an x. y. So again with this calling and ology It just means the number of color inks that can complete this partial coloring of an x.y.. Ok so I mean I guess a reasonable interpretation is that you have this edge here and all the thing above is like pulling it up.

[00:17:12]
So this is what this weights that we offer corresponds to. Let me talk a little bit more about the link graph in some sense this link graph is the local view of all fun this is going to be the local meeting that we are going to be arguing about somebody making it more precise but for now just think of it as like a local vingle that the office lease when he looks out of his indoor at a sunny day.

[00:17:36]
So I want to be writing golf. With some abuse of notation for the run walk back tracks where we make transitions of it's probably proportional to w. of. Ok sometimes the 2nd I have you know you have this run the walk is going to correspond to spectral motion or correlation.

[00:17:54]
Because when you really think about this w. off a quantity. It is just whole correlated are x. and y. condition on Alpha. Ok so hopefully Ok so now let me give me another concrete then let me give you another concrete example suppose I got g.. I said x. to be the collection of all in the plan the sets of.

[00:18:21]
And I'm going to be making this assumption case not to lock the key is not a large book a smaller than an over the top of one where Delta is a max of the ring g. I'm going to be explaining away the that in a minute but let's 1st ask ourselves a question.

[00:18:38]
Of size. What does the link graph just look like. Ok so the vertices are going to be diverted to says I can add to us and still remain in the band set for order I can complete this independence that as you know next to an independence that of skerrick making this assumption especially to trivialize this you know if you're bored at this point do you want some problem to work on you can think about this it's not too hard but you know whatever.

[00:19:09]
So what are these words is this. So I have this independence at us and the only road this is I can't add to as and remain independents that are the voices on the other words expound us which means if they have an edge I can't add up so the Rotex said that I care about is essentially all the other vertices so I look at us and I look at the sorts expounder it take the United States a compliment this is Yes Ok.

[00:19:37]
What about the sort of edges. Well these things are not connected to as anyway. Right if they're living in the us they're going to be an edge precisely one they are not an edge in the original graph right so these things are living in vs salt whenever they are not an edge in the.

[00:20:01]
When we yes there are going to be an edge in my link graph so this graph g.s. is just maybe the complement graph of the Indies Sabra on b s. Ok and I'm going to be. I'm going to be assigning every edge weight again. So x. and y. is just going to be the number of in the sky not contain as an x. y..

[00:20:26]
Because one interesting observation if I have still. My graph is going to be annoyed it why because k. minus 2 plus 2 is 2 and turns out if you have it in the pan instead of slice k. there is only one in the pan instead of that contains even that yourself yet this was just an example to hopefully make things more clear are there any questions.

[00:20:55]
We're going to. Try and we'll see it. Right. Maybe I didn't understand the question. The more we. Clear. Great so let me talk about what was known before. So called home through the falling bond and again this was the bond that was used to salvage the matter of expansion.

[00:21:38]
So you look at all these link graphs and again these are the graphs of our transitions are taken with respect to the way it's defined and if all these things have all you know you bonded by Grandma then this Donald bloke I have is going to have been the 2nd single about it but it's actually really 2nd eigenvalue and this thing is going to be because of some definite If you and I do I doubt but what we have is a bond on as I say the officer on a book.

[00:22:07]
Ok so this is a very good theorem. And by the way this condition is called locals like that if you are simple to complex satisfying This is called a gum a local speckle expander it like it's some generalisation of all spectral expansion on Graph So if you're a graph that is a local spectral expander that is just an expanding earth Ok for that matter it's complex what you know is that if you look at these links they're going to be complete multi-part I profs so their 2nd item of the 0 and indeed this is all that I really want to waste on in the zone to prove the matter and explain conjecture I guess no matter expansion theorem and the bad thing is like if I want to make this smaller than one i.e. the very aggressive.

[00:22:56]
Right I need this to be order one of the discord. And for the places that I would like to apply this theorem then these going to be on the gun rights so I require an explanation one over and square it on all the graphs and when you think about it there aren't that many grass which have like that's expansion so in this sense the all in this is the only scenario where discourse one theorem.

[00:23:23]
Is useful for us in mixing is the. Metroid setting and that setting is already resolved so what you're doing what you want to do is would like to extend to John Edwards general some to complex as general comments or logics So we have this font it's a little more hostile to the eye but what we're doing instead is instead of looking instead of opting for global bond on the expansion of these links we're bombing them layer by layer.

[00:23:55]
And then peer getting this phone which I claim that is a refinement of the coughing up and it might not be obvious but let me. Explain a piece like why it's more reasonable than the coffee shop and I bought. A case of suppose all these link routes are connected so this is the local area that I was talking about then if I have a connect to the graph I know that the 2nd that emotion is going to be bothered by one but it's a strictly smaller than one and then this thing is going to be non-zero so this term is always going to be smaller than one so I'm getting something non-trivial just by making the assumption that my graphs are connected.

[00:24:36]
Because in particularly this means I no longer need to assume the lowest smaller one over I know I don't need to require that done ways or they want to be squared so I can hope to apply this theorem for our settings that are you know not the matter any special case.

[00:24:54]
Ok. And again like this turn looks a little weird this is going to come from an induction. So I'm going to make induction proof by relating this to a random walk and it's the minus one and 2 are on the Mocha next to my stool and every time I make inductions that I want to acquire one of these things.

[00:25:18]
Will get. So. Before I talk about this proof and applications of this result let me give you a problem so you 1st need to be able to compute the cytomel use and this is not easy if you want to write this way to magic so this weight function for something that lives on the lower levels this potentially more is a really hard counting problem and I don't want to do it right you don't want to solve it you don't want to reduce a hard problem to another hard problem you want to just be lazy and get away with it so what do we do we appeal to the stream of Oppenheim this is what we do in our paper there is an alternative way that I guess you might have already heard about from you was going to talk that I'm also going to talk about briefly but we say is this this was something proven by Oppenheimer and he says essentially the expression trickles down what does it mean if you have.

[00:26:16]
It On some level and little the links in the level below are connected means that you can prove a moderate spawn the expansion on the links on the lower level so why is this useful for us this is useful for us because links on level d. minus 2 as I talked about before their own weight so all I need to do is doing some basic spectrograph theory right again why is this because the weights I have are proportional to the top level faces that continue if you're a d. minus 2 level link the minus 2 plus 2 is d. So there's only one thing that contains e. and that's yourself so you have.

[00:26:53]
You have. Always a graph. Ok So using this would prove an interesting results that I'm not going to present a proof of but I think that's. Pretty neat and it's not something you should be wowed away by it's just something you couldn't do before so there's folklore result that I.

[00:27:09]
Found the proof for in full I'm not sure whether they're the ones that they prove that because it's a fairly simple proof. Ok let's look at the simple independence of complex and listen just on a pull on all independent sets of size and over 2 dots. Ok so by using some coupling arguments you can show that this random process is rapidly mixing in particular you can see some pole or college in the plan the sets of size and over 2 dots up again we have a modest.

[00:27:43]
Improvement office where instead of looking at says that of size and over to it all to be looking at sets of size and over. The smallest eigenvalue of the normal at the normalized basis metrics so why is this something interesting. Because this at the premier You can always show that this is bonded by the operator norm of the adjacency metrics and if you have a graph of Maxim d.v. Delta.

[00:28:14]
You have you know that's all bonded by Delta So you as I actually imply this result from this broad interesting thing of this is to halt one we actually have a spec log and so I don't know how to us speculating for this case without using this machinery in fact I don't know any other arguments so maybe it's on me but you know this is something that I'm really excited about and also in the case where we are far away from a bipartite prof this as a parameter is going to be really small Actually you don't even need far away from bipartite if you have a plan or graph or not some degree Delta.

[00:28:52]
Delta is going to be order squared so really you know that it's going to be order scored Delta. So you're essentially doing the best you can do so you know there's a classical result that says any graph of proximity Delta has an independent set of size and over Delta plus one and we're getting really close to that case when we're far from fiber expanders again this is not something you should be blown away by just intrigued.

[00:29:18]
There's all sorts of approach by an early on always got on so what they do is they look at the hard core distribution for independent sets what does this mean this means they assign an independence that us with proportional to t. to you know the cardinality of festivities some per meter.

[00:29:37]
That as I actually. Tells you which direction are we going to be by us for bigger size or smaller sets. And they use correlation to get ideas and you know stronger spatial mixing. The 2nd item always and this is something that makes sense because when you think about the the whites that you're doing in the random walk they're all proportional to conditional probabilities right the conditional probability of picking something based on Ok really have this say you're in like all.

[00:30:10]
Going off right so I'm not sure that made things clearer or confused you more but let me tell you that this thing like this problem the approximate counting problem was considered by whites who you know proved strong spatial mixing for this and then based on this he gave a deterministic audit all the rhythm for estimating this below what is called the unique structural for bomb the degree graphs.

[00:30:37]
And interestingly enough by using our theorem these ideas on there really aren't always going to have shown. So long as your beloved uniqueness trash old you can actually some pull from this distribution by using the down up all. Ok so by using the standard deduction from approximate counting in approximate some playing this actually improves rise as a result to all graphs again giving up on deterministic determinism but I don't think that's such a big deal there are some other applications that we use this.

[00:31:16]
To establish mixing up front in walks a certain complex is astonished by Liam $100.00 young so they build this like times are like construction a fight emotional expanders like some bullshit complexes in which to stand up walk makes as fast. As I show you they have the 2nd one with these bonded by one hearth.

[00:31:35]
They give a proof in the paper based on the composition technique of germ sun to tell them to go to so this takes like them pages and it's just a black box up location it's pretty cool and then you can also do something much inks from gentle graphs or simple intersection of partition that writes.

[00:31:57]
We can do Jim Sinclair we go to inquire about what we can do if there is a matching of size are in a graph not necessarily by part i.v. can some pull agro can sample a matching of size are witchery from this so all this is a different algorithms than germs and Claire So it's interesting to me that we can do something interesting in every doubt during the full Jim Sinclair algorithm Ok.

[00:32:23]
So it's called high dimension expanded from expanders. So they built simplicial complexes in which he's hired to run amok Max Foster and what they prove is that it's all done by j.c. in their complexes but in the bottom what like in the bottom layer you have a very good explanation but in all the subsequent levels you have expansion on health and they try to argue that you know the high road to run and walks now were less makes very fast and this takes some time pages again.

[00:32:53]
Papers are equal construction and I think if you have an easier proof of this is just makes the paper my strong. Ok. So this is what you're proving again in case you forgot just let me make some remarks in the paper of an Early on all these gun they use Come on G.'s order whatever and mine's j. so not order one of squared and further if you look at the top level links are there going to be links of course and size so you can hope for better than cause the expansion so the fact that this product form is there is useful because it indeed some telescoping and days and you get.

[00:33:32]
You've got problems I'm mixing. For our application we actually never might for our application we need to come will want to j. but whatever. Type examples for this formula so in fact the proof that I'm going to be presenting in a few minutes is very similar to the proof of the explanation of Johnson graphs and in fact if you look at the Johnson graph this is going to be so Johnson golf is just hired to run a walk on the complete complex where all the all the Kayleigh meant that subsets are contained in the top face on the top layer Ok again there's going to be induction proof this is where it comes from just keep it at the back of your mind so let me talk about the.

[00:34:19]
Proof a little. So the idea we have is Ok we have this run amuck p.v. down up. Via the stop on 1 pm you want to relate that to the Donna problem walk on. The minus one. Ok so what you do is we draw this bipartite graph on the left hand side I'm going to put all the faces of the minus one on the right hand side I'm going to put all the faces of a mission d. And you know I'm going to put an edge between the left hand side on the right hand side of the left hand side is contained in the right hand side.

[00:34:56]
Then the idea is the stunnel run walk was just you know you're here you go to run them sets you contain and then you go back to a set that you know you contain all the stuff that isn't here so if you look at a square like the squared like the sides do in their goals you're going to have 2 things one of the things you're going to have on the diagonal corresponding to is just needed on the run walk out for x. of the Donna walk.

[00:35:30]
And you're going to have this other on the wall here that's 1st starts on from extreme minus one goes and then comes back so this is another random walk. And you have these interesting formulas and you know this is a bipartite cross so I know that this part is the adjoint map of this part so particularly these 2 things on this inspector.

[00:35:52]
And also their p.c. so I'm going to be using longer 2 instead of seeing the 2 because whatever. So let me remind you what what I just said defines trying to walk the Uptown walk on x. t. minus one and the optimum work is just I start from the minus one face I sample uniform face that contains me and then I go down by deleting around the mountains.

[00:36:17]
Ok so this is the other blocked agonal. And in particular they have the same arguments. Right so i minutes to shrunk the state space of my around and walk. What did I say Ok now I. Still to make the induction prefer a little. More rigorous I need to define our general Donald and general opt out walks so what does it mean I want to push these Dano plugs and up on walks to intermediate bubbles so I can as I'm sure push it all the way to 0 on for easier I'm going to be able to do something smart Ok so I have this on p.j. optimal and exchange and what it means is that you're still going up and going down the similar way but when you're going up your loop you're thinking that essentially all the stuff in Ecstasy that's going to better is pulling you up and you're going up with the proportional to the number of faces that pull you up.

[00:37:22]
And similarly I'm going to find this done up walk on b j o n x c sorry and they're going you're going down normally when you're going down they are going up you're going up proportional to the number of stairs that are pulling the up. Because someone is aeration is.

[00:37:42]
I can do the same bipartite argument as I did before and this is why I need to define in a weird way and say that the down up walk on eggs j. plus one and optimal can exchange they have the same non-zero spectra because you know. This part is the.

[00:38:01]
Adjoint of that part of the talks and. Another interesting thing to note is that these run walls of the same stationary distribution in the station distribution is just going to be proportional to the number of stuff that continue up stairs. Ok. So one more. Ok And this isn't useful for the elections that So one more observe ation I guess this is unfortunate choice of colors but what is the optimum walk on the 0 and $1.00 so optimal on the vertices so what I do is this I start from artex I go to something that I just didn't do.

[00:38:44]
You know purport to be quality proportional to the number of stuff that is pulling me up stairs and then I go down to random a very So I guess like this can't be read but what I'm doing down is I'm really treating one of these things and when that is.

[00:38:59]
Ok so what is this thing. This thing is just a. Function I defined earlier. This is this is the weights on the empty link so what I'm actually doing is just plain old lazy random walk on the Answer link. And you know by. Just using some basic facts about speckle shifts I can say that the 2nd item out of the of the walk is just one plus the 2nd eigenvalue of the m. to link by half Ok so this is essentially going to be a reduction basis.

[00:39:44]
And let me be the 1st started you know. So I know. Spectral property of the 0 up down. I can do this argument about core spectral ity when I went to Rowan like transfer some knowledge I have on p.j. op done to p.j. plus down up. Scientists were spectral ity So all that I need to do is relating the.

[00:40:14]
Done up organ up on the same level. So. Let me make this more formal or you know more understandable so the scruffy one that goes up and down I know the spectrum of this this is just a plain old Li's run walk on the empty. Then what I do is I say this is course spectral to this random or graph p one done up and then what I want to do is I want to relate.

[00:40:46]
This run walk to p one up down by just saying all the way there spectra should be close and then I want to research Pete this argument. You know about a way like once I come here this is going to be me and auctions that so I don't have recently put this argument and eventually push this to get me to give me a bone on p.v. minus found out and then I want to say Ok this is going to be close to the optimum walk on the minus one and this is core spectral to walk on x.t. summed up.

[00:41:20]
Ok so this is the proof idea and if you've seen this before Fiona Johnson graphs Torino are the score of the theory type stuff this is exactly the same both just up. Close and as different. Look at I need to define one more. Run Walk. And this is going to be Denali as it were of the Uptown walk and this means I'm doing the same uptown thing I'm not taking in the loops so I'm saying Ok once you go there you can't return back.

[00:41:57]
So what is this I start from an off the tee. Of Fortune choice of course so when I'm going up I'm going to something next. And then when I'm going down I pick something from. I pick something from all 50 that I have and I'm swapping it from which x. writes I'm doing something non-trivial So if you're thinking pieces hero down up.

[00:42:23]
It's just going to be the Runnable Patricks of a graph. Ok. And again I can write a functional dependency for this in terms of the original run work much Rex Let's not worry about stuff. So what I'm going to be showing is that this metrics and dark matter x. they're close to each other spectra.

[00:42:48]
This is the p. is the order and the thing I'm using about the p. is the order is that whenever this thing is smaller than the out think that I don't know the of this is dominated by the I tell you well you have this so all Ok.

[00:43:03]
So finally after all this talk I can explain all got finally after all this talk I can explain you why we care about this higher to run walks and living better if you could actually read the labels on the pictures but essentially if you're conditioning to subsequent steps of your random walk on having the same thing on having the same awful part so I go from an alpha why I pick some ex ups there's a lot of x. y. and then I'm going to offer you next.

[00:43:34]
Then essentially you're taking a random walk in the step in the graph g. Alpha. Right so if you're mixing golf or you're essentially mixing in the run walk conditioned on containing off. Right so this is one useful observation and the other useful observation is that. We're doing the Donald walk.

[00:44:00]
Here essentially walking on a quick look when you're going up and down this corresponds to taking a walk in the clique would solve loops in this of your text of the scruff So it turns out if you want to write around what matters of this you get exactly the projector to the top of a space of jail of Java but so this is going to be the important part.

[00:44:28]
So let me just because what this means that its earlier what is a is dominated by b. in this piece the order it means that the quadratic form of form over a is dominated by the quadratic from over b.. And if I look at g. all the minus shelf some conditioning boardwalks on containing off so I have this thing and this is just the part of g. also that lives on the I can space perpendicular to one y. because I told you is the projector to space.

[00:45:01]
Ok so. This is just going to be bounded by the 2nd item of the of she also has a projector to the audience space program the 3rd one. And finally. I can right the Stone Exactly. Because I'm missing something in it turns out this is why we call our paper really improve them as a friend walks or hired around a box because previews for only does this and then we say if you do this you're going to be actually getting some significant improvement.

[00:45:36]
So let me give you a proof I just need to introduce one more thing and that is whenever I have a function g. faces and i j minus one face I can define a lot of my function on the space just by saying Ok f. off of x. is just going to be x. So I'm just going to be looking at what this function of C's mistress ticked in on face that one table phone.

[00:46:02]
Ok so we'll let's look at this let's look at the quadratic from over the fence so the idea now was. I can use a preserve as a ration before a lot of expectation so what I'm going to do is I'm going to condition this run walk on essentially intersecting on all 5 between the 2 subsequent steps.

[00:46:23]
So I'm going to have this expectation and then I know that if I condition this room open off I'm going to hit your floor and if I condition this random walk on all the fun going to jail for up so I get this term. So I can do the calculation I did before this is a calculation from last page and no I recall that this thing is bounded by the most famous one let me just plug it in.

[00:46:57]
And finally I did a lot of expectation again. This is what is in my result at the end so I don't want to do the induction perfectly because I don't think it's very instructive and I think this observations are really why we care about local spectra expansion because when we're looking at the local views it is very easy to argue about them like in this framework.

[00:47:26]
Ok so this is called Gollum method by the way just I know like need to make this a little more precise by defining the rights problem distributions so there's a lot of expectation is really something much more than a lot of like speculation that takes a while to prove but this is the basic idea behind it.

[00:47:45]
If you have this this is easy to finalize. Ok so this was all that I haue of some questions so stripped spectral expansion of all 4 animal designs that. It's controlled the contraction in the Elton arm. And you have this alternative What if I look soul in quality and what if I look so lovely quality control is a construction of relative entropy.

[00:48:12]
Can we extend these techniques to prove what if other social inequality on hired runnable side in this is a very interesting question this is already partially done by Mary crying Hengel aneurism I saw they do this for the matter of complex but the proof is like the proof just doesn't generalize for.

[00:48:28]
Other complexes other than the matter of complex so they're using some interesting connections between the human beings 0. Sum the composition of entropy. It's very interesting to me because normally just because you have you know there's doesn't you know you can get a tight I don't even get a tight spot on the what if I look so of course that they managed to do it somehow.

[00:48:52]
The other question is why different types of front and walls that we can use local special information to analyze so one thing I'm very curious about is if you can look at the random walk constructed by German Sinclair to some Paul on your project x. is restricts my run walk to the level of x. and minus one of much ngs or it's not going to be connected so my random walk can't do it us and germs and Clara can do it it's because they're using some more advanced Metropolis rule so if I have some more advanced Metropolis for can I still use my a fancy tools my fancy toys still prove convergence I think is an interesting question.

[00:49:39]
And the other question I have is can we get a more concrete connection between correlation to k. and spatial mixing so I really don't know is grounded to something in their paper. But in some songs what they do was show that something stronger than spatial mixing bowls and there are even basic cases where we don't know whether spatial mixing calls strong spatial mixing sorry so can we do something smart to give a more concrete when action so $100.00 stand for example is the normal way we argue about.

[00:50:13]
The. The eigenvalues on links can we maybe marry the proof of oftentimes theorem with some ideas from correlation again so I think this would be interesting because if you can say it's actually whenever I have strong spatial mixing mixing up there on a walk you know it's going to be huge because you're taking the 2 on seemingly related things and you're marrying them.

[00:50:41]
And the other thing on the other question how does Ken relax need to have expression all eggs so all for example for just run walk example I know about the search for this independence that example that I have. No doubt the complex is connected below all levels and over the plus one.

[00:51:04]
But I don't know how to prove an explanation on all that was 11 or Delta plus one because something very weird to stop an ng and I need to argue about minimum degrees of the links so can I do something on average can I do something by arguing about.

[00:51:22]
Expectation of a local neighborhoods of things because when you think about it even if you're demanding an explanation on every link layer you're demanding too much so maybe we can. Suffice. Thank you for your attention. To suppose. So it was posed but he would. Have to. Right to do.

[00:52:03]
So this method will not work there. But. It was. Right in the. Well heeled. I don't know why this was one of the special cases that we tried to improve it on. We couldn't get anything not sure whether it could have had any because we didn't try hard enough or because it was hard but I believe that like this trying a free case needs to be like somewhat leverage of what I was thinking about.

[00:52:44]
The paper I think we got out where they're putting mixing of flour that I mix for trying to figure out. And I think my similar ideas should be useful for the setting too because in some sense the only setting where decide emotional expression machine should be useful may have some global lack of structure and that manifests itself.

[00:53:07]
Locally So I think trying to free should be a very natural case to look at I just don't know how to analyze a very direct this point.