 Thank you very much, Stefan, for the invitation. So my topic is random operator theory. Who has ever seen a random operator in this audience? OK, those people should now leave the room, because what I'm going to talk about is really the absolute basics of random operator theory, because I envisioned that most of you will be actually probabilists and have never seen a random operator before. OK, so what is random operator theory? Well, from the mathematical point of view, it's a topic in which sits somewhere in between analysis, probability, and mathematical physics. Its motivation mainly comes from physics. So let me start describing to you the physical phenomena which I want to talk about, which is very on-voke among theoretical physicists at the moment, in which connects to stuff which is understood by the physicists, say, for almost 40 years now. But it took a while for mathematicians to catch up. So see, what are we talking about? Let's imagine a disordered quantum system. What is a disordered quantum system? Physically, this might be a semiconductor, which is doped in a disordered way. So you might think of an alloy where you have one sort of atom at an atomic grid, at a lattice. And another sort of atom may be taking also these positions in this lattice, in this atomic lattice, but this is done in a random way. That is a physical alloy. And what we want to physically describe is the transport properties of the electrons in such a solid. As you know, if this would be perfect solid, then quantum mechanics would require you and me to study periodic Schrodinger operators. And what this order now introduces into the game is that periodicity is completely destroyed and you have a random Schrodinger-type operator. Now, what is the physical prediction for those systems? Well, it is what I drew in me somehow on my first slide, namely, I think of sort of an electron gas which is filled up to the Fermi level. Never mind if you understand what this is and just put some excess electrons into the solid just by taking a lead and pumping electrons on that point. And then what we want to see, or that's the physical phenomenon in me which is observed, is that in certain situations in me which are attached to this order being present in the semiconductor, the excess charge cannot move. Where it can move a bit, it can smear out at me, but it cannot wander off to infinity. So it will remain forever in the vicinity in the way you put this lead. Now, this is, if you think about in me somehow such a system in me from the physical point of view, in me this is quite in contradiction in me to what your professor in theoretical physics told you in me what will happen. Namely, if you have a typical physical system which is an interacting electron in the system, in me then equilibrium states, which are the stationary states in me of the system, they spread uniformly over the energy shell, in me that's what's in statistical mechanics known as the argonica hypothesis. And that requires in me somehow that such things and we can actually, it may not happen. So when people in me somehow measure these things or measure absence of transport or as bloch in Munich who has a device of trapping atoms, cold atoms in me in a cavity and shooting light in this cavity and we saw that effectively in me these atoms will actually be subject in me to a random potential. What he observes is that on long time scale, long in the sense of an experiment can never reach an infinite time scale. But a long time scale in me in this experiment once he puts initially in me all atoms in the left half of the cavity, they will, you will see in me this step of having more atoms in the left half of the cavity for basically arbitrarily long times as long as you can run this experiment, okay? And this is not something in me which you were taught in your physics course what will happen in me because what will normally happen if you take a gas in me and just push everything in me to the left half of me of this chamber here, releasing it will of course equilibrate it, okay? So this is the physics phenomena in me now mathematicians and we cannot really grasp at the moment I mean the total sort of breadth of the whole phenomena is in the sense of in me that what we can really prove is that these phenomena exist in non-interacting physical systems in me and that was already predicted in me by old works in me of Anderson in the 50s and that's the phenomenon of Anderson localization. So once you switch off in me interaction and we won't cause in me exactly this type of behavior Anderson localization, okay? So even if they are physicists in the audience what you should bear in mind is in me that what I'm talking about in me is the real foundations the mathematical foundations of a phenomenon which you now quite frequently talk about in the physics labs, which is this MBA in me and that's at the moment a challenge in me to prove this in me really in rigorous terms. Okay, something for the younger generation. Now what I want to do in me in my talk is on my three talks in me in total in me is in part one, which is this lecture. I want to really go back in me to Anderson's old work and explain to you why indeed in me we have for non-interacting systems this phenomenon of localization so absence of transport and all that. Now this phenomenon and we goes with a wealth of other exciting phenomena. You could mention the quantum Hall effect. But since I thought in me this is much more of a probability audience and I took another aspect of this Anderson localization for non-interacting systems in me which one can actually show that this goes along with which is so-called Poisson eigenvalue statistics and that will be the talk tomorrow and on Wednesday in me I will go a bit in the direction of showing to you what might or might not survive or what we can at least show in certain systems in me which then mimic interactions and the best framework for that and we are quantum spin chains. Okay, so that's a bit the plan of these lectures. So let me go now to describing really the most simple random operator and which you can think of which is really Anderson's good old model of a an operator on some kind of a graph. So in the following G might be a graph. Actually I'm a bit sloppy here. So G would be the vertices of me of the graph and I'm just ignoring the edges in my notation and the Hilbert space in me which I want to work with is the space of square sum of the sequences of the vertices of this graph. On the Hilbert space I'm going to install a random operator in me which consists of two parts. Now the first part in me is entirely non-random and from a physical point of view models hopping of electrons in this solid. You can think of the graph being for example ZD so having in me some kind of a crystalline structure and I want to take in me some other days the simplest hopping in me which would be nearest neighbor hopping so I'm just taking in me the adjacency operator on that graph so I can, it acts on a psi, psi is an element in this A2 space. Psi is a function and we can think of a function which assigns every lattice point here a complex value in such a way in me that the modulus squared of these complex values are summable in me over my graph and A of psi is really just the sum over the other vertex X say here over the neighboring lattice sides. So in this CD here for D equal to two as is drawn here if this is X, the sum of Y would range over the four vertices being just adjacent to this vertex X. Okay, so that's a non-random object and you'll be probably familiar with that in your course on maybe functional analysis or graph theory. So suppose, let me talk about this later. Okay, so to add some randomness one can pursue in me sort of many ideas but the simplest one in me would be to just think of the randomness being created in me from some kind of random potential. Now, what is a potential? A potential in me somehow on this A2 space is just a multiplication operator and I'm multiplying by values V of X attached in me to the vertices in these values and we are now taking random in the simplest process on my graph and it's just IID random variables on this graph. So for simplicity in me things can be boosted up and we too much more general distributions and we were just for simplicity of this talk let me always suppose that the probability distribution of one of those variables is taken to be absolutely continuous with respect to Lebesgue measure and that its density which I'm going to call rho is bounded, okay? And also in me somehow since I don't want to make in me too much fuss in me with technicalities I'm going to take the shortcut of stating my result in me always in the case in me with the support of this probability distribution is compact. All of that in me maybe generalized but I want to keep in me things simpler. Now, in order to tune up randomness I mean since I want to keep this probability distribution fixed I introduce another parameter in which is just a real parameter lambda without loss of generality always greater than zero or equal to zero sometimes which for fixed probability distribution in me then scales in me the disorder strength. So this lambda in me is there of me to say we will maybe in me this term dominates and we all this term dominates this term the being dominant in me for lambda large I mean this term dominant in me for lambda small. Okay, what is the basic question in me in this game? So now you can easily show in me that the operator in me which I've installed in me is a bounded self-adjoint operator. So if this is a bounded self-adjoint operator and be the simplest thing which we can do a spectral theory. Okay, so what I'm going to ask about in me first is in me what is the spectrum of this operator? Now what would you guess? Here comes the first question to you. So if lambda is equal to zero and my graph is ZD the regular lattice what would you guess in me would be the spectrum of this operator or what have you learned? Now see in me then we're exactly in this situation which physicists in me describe in me as an ideal solid in me which is a translation and variant operators invariant in me to all lattice shifts in me if G is really ZD. Okay, so but in this case all eigen functions are what? The good old in me block waves, e to the ikx where k in me ranges over the Brion zone so the reciprocal lattice in me of ZD, okay? Now you can work things out and let me show this in me on the next slide, you can work things out in me as showing in me that the usual block floquet analysis from functional analysis works. Who has never heard of block floquet analysis? Aha, probably it's in the audience. Okay, so maybe I should, I mean I have these exercises so maybe we should do this in the exercise session and the basic idea you know is in me so you have generalized eigen functions psi of x which look as e to the ik dot x where k is somehow in the Brion zone so the reciprocal lattice in me of Z to the D would be something like e minus pi pi and the torus e to the D. Okay, if you plug this in in me you can see in me that these things in me will actually reproduce in me and the eigen value and we will have something to do with the cosine. Okay, so for example we take D equal to one, right? So what is it if you plug it in and you get so e to the ik plus k of x plus one plus e to the ik of x minus one which is equal to e to the ikx times something like me two times in the cosinus of k. Okay, so that's an eigen function. Granted it's not in the Hilbert space in me so you have to mumble something about Fourier analysis in me here in order to make this rigorous in me but that shows to you that indeed in me somehow when lambda is equal to zero here we are the spectrum in me of this guy is just the interval in which is given in me by two times cosine k where k ranges in me over minus pi to pi in case D equal to one and that's in me this band from minus two to two. Okay, so the whole interval here will be filled up by spectral values in case lambda is equal to zero. Okay, now from your functional analysis course in me you might or might not and we have learned the distinction that if one studies spectra of self-adjoint operators then one can ask for a refined question namely remember in me there's the Lebesgue theorem in me for measures you can decompose every measure according to an absolutely continuous part with respect to Lebesgue something which is singular continuous and something which is pure point. Okay, corresponding to me to this decomposition is a decomposition of the whole spectrum of a self-adjoint operator so every spectrum of a self-adjoint operator H can be decomposed into an absolutely continuous part a pure point part in the singular continuous part. Okay, now as it turns out in case lambda is equal to zero and this is the first thing which we need to understand then this spectrum of this operator would be purely absolutely continuous. Now absolutely continuous spectra we singled and we buy non-square integrable or continuous spectra are signaled and we buy non-square integrable eigenfunction and that's what we have here this is a non-square integrable piece. Okay, so down here we have this band of absolutely continuous spectrum and now the question from this random operator perspective the first question which you can ask is now what happens if I switch on lambda? Okay, and the striking thing what will happen in me is that well just like in statistical mechanics things are actually dimension dependent it's not only dependent on the strength of the disorder but it's also dimension dependent. Okay, so there's a bit of theory in which I'm not going to go into and I want you and me to just because for time reasons otherwise I cannot talk about exciting stuff there's a bit of theory in which I want you to swallow and I can give you a reference in the very end which is the following see in principle you might think oh this is a random operator and we saw all of the spectrum it should be random I mean this should depend on the realisation now however in case you're on a lattice that's not true so you have shifts on the lattice and things are not translation invariant but they are translation invariant on average what does that mean? Well, see if I take a realisation of the random potential some values on this lattice CD and I shift everything and this would be a valid realisation of the random potential okay now if I shift everything and this is this operator and we will then be unitary equivalent to the original random operator since it's unitary equivalent the spectra will agree okay so essentially it's from this perspective and we're not utterly surprising that the spectrum as a set and also the components in the Lebesgue decomposition they are actually non-random and that is a consequence before general you can take this IID random potential but you could also just switch between an agotic process on CD and a good thing here we will ensure that the spectra and all of the components of the spectra are non-random objects okay so what we will prove we will not prove the whole story don't worry what we will prove is part of the following picture maybe if I now switch on lambda which let's get to this value of lambda in supposedly rho in me is now just say in me the equidistribution between negative a half and a half then so the first question is in me wears the spectrum as a set well see that requires a proof in me but let me do some hand waving here that really indicates in me how the real proof goes so see in me is your probabilist in me you certainly know in me that in this on this lattice CD there will be arbitrarily large regions in me where the random potential now is just very close to being negative one half to being a constant any constant in me somehow in this interval now what happens in me if I take an operator and add a constant where it just shifts the spectrum right so you're going to create if I subtract in me minus one half in me I'm just going to shift in me this interval in me to minus negative a half up there but all these states in me which I then can reach in me with lambda times negative one half in me they will be in the spectrum why is that in me well you know in me I can just so-called why sequences that is approximate eigenfunctions in me which I accommodated in this large regions over the potentialist very close in me to being that constant value and therefore in me this value in me is realized in me in the spectrum okay so I can tell the story in me for any value in me therefore in me we have now in me sort of understood in me why the spectrum in me is indeed nothing is in me but the spectrum of this adjacency matrix which is this interval here just augmented in me by lambda times a half in minus lambda times a half on two sides and everything in between all spectral values in me will be realized okay so these lines here in me are the boundaries of the almost true spectrum then the next question is so okay that's an easy exercise actually in me that you can prove in me without doing anything that's basically in me sort of constructing close sort of approximate eigenstates in me to realizing that this is true now here comes the hard question namely compute in me the spectral decomposition so tell me in me what is the AC part of that spectrum what is the pure point part of the spectrum and what is the singular continuous part of that spectrum that is now not so easy at all and the prediction in me by physicists in me is the following as I said in me the exciting thing is in me this all depends on the dimension and in dimension one and two it is extremely striking in me that independent how large the disorder is the continuous part of the spectrum will just go away so in other words if I just move to lambda being arbitrarily tiny but positive here all the spectrum is pure point for d equal to one and d equal to two now this is really a quantum effect because think about it you know a classical particle which moves on a slightly roughened line because it moves in a potential in me which is a bit sort of rough then if you just give it enough energy in me then the particle can escape to infinity because it just rumbles over all of this roughness quantum mechanics says now that's not true anymore for a quantum particle this quantum particle will be trapped so if you take an arbitrary sort of soft in me but still random surface a quantum particle in one dimension will be trapped okay there's only pure point spectrum and there's only bound states I explain in me this sort of what this has to do with bound states in me later but that's the striking thing we should surprise you okay this happens to be true in me also in two dimensions in starting with three dimensions the picture changes and what physicists predict is that there is actually a so called mobility edge for small this order lambda when you take the spectrum at the edges of this interval the only pure point spectrum no transport will see a reposal level statistics but there will be a bubble of AC states transport in so called random matrix statistics in me for this problem that's the big picture in me which physicists paint okay now the purpose of this talk is to show you in me that once I go off enough here that all the states are localized and that's what we will see in me somehow in these two lectures that's the complete proof in which I will give okay but that is the challenge and I should say as I will remark in the very end is we have absolutely no idea about how to grasp this inner point very good in me there's a very good understanding in me how to grasp to grasp pure point spectrum absence of transport possible level statistics but this really really escapes us okay that's one of the reasons why I'm not saying anything about this okay now so now let's go in me to the math so as I said in me the task will be to show in me that all these things here occur at least it's sufficiently high disorder in arbitrary dimensions now how do we go about that okay so here's your rough reminder in me on spectral theory so remember all the spectral information about a self-adjoint operator H is encoded in the resolvent which is the inverse in me of this operator of course I can only take the inverse away from the spectrum the spectrum of a self-adjoint operator is a subset of the real line and we therefore I'm being a bit cautious here and we're just taking things and we sort of off the real line and since I'm working on a small L2 space the particular functions which are associated to lattice sites X which have the property in me that they are one if Y is equal to X in zero is these are L2 functions okay so these would be the evaluation functionals and that's my notation and before it now they form a basis of this little L2 space and we went X runs through all of the vertices and we saw all information about this self-adjoint operator H is contained in the matrix elements of the resolvent in terms of the spaces and that is referred to as the greens function now what is spectral theory where spectral theory is the theory in me about the measures associated through this steatius relation to the operator H and an arbitrary selection of vectors here X and Y so all information about the spectrum in me is contained in the spectral measure when X runs through the lattice sites okay is that clear so for example with the spectrum of H which would be the union of the supports of the spectral measures associated with X and X that is with the spectrum as a set and then you can if I ask about mu X X that's a real valued Vorel measure I can do my Lebesque decomposition whether this is absolutely continuous with respect to Lebesque singular continuous or pure point and depending on whether X has the property that this produces a pure point measure singular continuous measure in the X this vector in me then spans part of me of the or then creates me in a certain way absolutely continuous eigenstates your singular continuous eigenstates or pure point eigenstates okay so this what we're going to study is the Green's function and as you will see in me somehow all information in me can be about this random operator or non-random operator but the safe adjourn operator and we can be recaptured and be from this function now since this is the spectrum measures we are just complex valued Vorel measures an important quantity in me which I want to look at in this lecture in me is the so-called eigenfunction correlator also known in me to mathematicians as to the total variation measure of these spectral measures now this would be abbreviated by qx and Vorel said I the last equality in me is to give you an intuition I mean also in me that was the quantity written down by physicists first of what this guy is okay so suppose in me somehow h is a matrix right so the matrix and we just consist of a safe adjourn matrix there are some eigenvalues which make up the spectrum and there are some eigenfunctions let me call it the normalized eigenvectors psi of E okay so what would be the first of all the first exercise so if h is a matrix E1 of TEN then let me suppose somehow they have simple eigenvalues J in the so-called normalized eigenvectors normalized as always and we have a Hilbert space so there's a scalar product and a norm and I normalize it with respect to this norm they would be called psi of EJ so what would actually be in me the measure of mu in this case is the first exercise well I would sum over all eigenvalues in my Borel set I and then I have a weight and then this weight would be just equal to let me see if I have done it right psi E of X psi E of Y bar where bar denotes complex conjugation that would be the spectral measure of the simple matrix here okay what is the total variation measure where the total variation measure of this discrete measure is just what I wrote down here okay you just take as weights the absolute values of the eigenfunction at X and Y and that's where this name eigenfunction correlator comes from this is such an informative quantity because in quantum mechanics we want to study quantum mechanical time evolution quantum mechanical time evolution is governed by the Schrodinger equation the Schrodinger equation for those of you who have never seen that that's IDT that's a linear equation and before a vector psi and that's given by H of psi of t okay or in other words the time evolved state or the time evolved vector from just some vector in my Hilbert space which at time zero psi naught is just the unitary operator generated in me by the safer joint operator H which applied to psi naught okay this is a unitary time evolution and so it preserves the norm in scalar products in me in my Hilbert space that's sort of one of the defining features in me of the Schrodinger equation now so suppose as my initial vector I start something in me which is start my particle in me at a lattice point y and suppose in me somehow energetically I restrict in me my time evolution to an energy window i so what is this in me this is the spectral projection of the safer joint operator H onto the Borel-Zi now you could take in me the Schrodinger equation and be with that particular initial state then it would be time evolved to this state and let me sorry something which I haven't said is since I have a physics background and since this is mathematical physics the scalar product is linear in the second coordinate and we not the first coordinate so it's linear in that beast and we are not that beast okay so if I start in me somehow with that vector take my time evolution and then ask what is the probability of finding in me the quantum particle at point x in my graph what would be this probability where the probability the probabilistic interpretation in quantum mechanics and we would be in me that psi0 of x squared would be the probability to find me the quantum state described by psi0 at x okay now you see in me somehow why we won't normalize wave functions in me because if this is supposed to be a probability and when you sum up over x in the graph it better be 1 okay so that's the probability that's really the quantum probability of finding um particle which is modeled by this wave function psi0 at x in the graph okay now now absence of transport we would be then the question of me that if I start here at y my lattice that's my y and I consider a far away x here that even if I take the time evolution sort of for an arbitrarily long time that the modulus square of this beast will be small if x is far away from y that is absence of transport that is in me somehow the quantity in me which I want to control now why is this inequality um true see right see so what is this this is this object here in me can be so let me take in me the left hand side and re-express it in terms of the spectral measure so what is it well that's the integral over the spectral measure associated with x and y integrated over e uh I uh is the energy interval to which I restrict everything and then I take e to the minus it e that is in me what the spectral theorem tells you and now in me this inequality in me is just pulling the absolute value inside in me the integral right it's just the trivial uh uh triangle inequality me that's what we are doing here so controlling the eigenfunction correlator allows you in me to control in me the the transport in me that is in me what this what this box in me tries to um tries to tell you now I've started in me telling you about your point spec trying all that jazz in me in so how do we go back in me to if we now control in me the eigenfunction correlator in therefore in the transport in me through the simple inequality how do we control the spectral information well um here is a black box which you might or might have not learned in me in your functional analysis course in me which is a theorem in me which actually goes back only to the 70s it's named rage theorem in me after who el amra in John Resco and ends we've contributed in me various parts of this and it tells you that uh quite physical thing namely um that the information about the spectrum is encoded in fact in the transport so there's a one to one correspondence in between transport information in me and spectrum and that goes as follows so here's the rage theorem uh in its simplest form it tells it's an abstract theorem and beyond on here with spaces take a safer joint operator as our random operator has nothing to do with randomness and you take a sequence of compact operators and we would converge strongly to the identity okay anybody in me who doesn't know what strong convergence is I mean don't be ashamed and be you know this is functional analysis so uh it basically means in me that for psi the edge the limit of A tends to infinity of here in this case the limit is one of Al that goes to zero so we're measuring the convergence in the hybrid space of these vectors when you apply the operators to them okay so it tells you there's a beautiful one to one correspondence and transport information and that goes as follows namely you measure uh the norm of the time evolved vector psi when applied Al to it and you require uh that in cesaro average when you first take the cesaro average so first take Al to infinity and then take take T to infinity and then take Al to infinity that this quantity goes to zero and the vectors in me which will satisfy this property in me they are the vectors in me which contribute to the continuous spectrum of the self-adjoint operator so they span the orthogonal subspace in me of uh continuous states okay so the Hilbert space in me is an orthogonal sum of pure point states in the continuous states you may or may not uh subspan in me by absolutely continuous states and singular continuous states okay that's how this works now the point spectrum in me they have the property in me that when you time evolve and ask in me about the remainder uniformly in time maybe this goes to zero when you blow up in me Al now what's the physical intuition behind this so you can think of these Al as if you're on my graph so think of a CD so let me now put more structure on this graph in me namely uh this is an infinite graph say and let me take Al as the projections and we turn the balls of radius L okay now so just a multiplication operator in me which is one if axis inside and L and zero outside I mean that's the operator which I want to consider when L tends to infinity these projections clearly converge to one as L tends to infinity because you eventually in me get the whole graph and what is this statement in me then in concrete well it says in me that a state in me from the continuous subspace has the has the property in me that it's quantum probability so this probability psi t of x being in some set that is by definition so see so I said the quantum probability of being at x and it's just psi of x squared and the probability of finding a particle in B we would be then the sum over x in B so you take this time evolved vector and you ask is the particle inside the box now when upon says our averaging and taking the box we even arbitrarily large and it's not there and we then you're sure it's it's a continuous state that is very physical in me because continuous states are the states and we would contribute to transport so once in me you you have such a state in me which is near here then if you wait long enough and then it would just disappear in me from this room that's what this says and we at least and we upon says our averaging in contrast when you take pure point spectral states they correspond to bound states in me namely when you ask in me what this quantity here is in me this is proportional in me to it's sort of the square square root of the proper the quantum probability of not finding the particle in the in the ball of radius L at least after time T in me and it says in me that even in me if I look uniformly in T when I make the box and be large enough then this goes to zero in me so in other words if I have such a state in me which is a time zero here then if I wait long enough in me it will still remain in a vicinity in me of this room that is in me what this this rage theory entails you and that should now in me somehow make a bit more explicit in me what I said before in me namely that it is to us of interest whether states here are only bound states and therefore there is no transport or states and we are only continuous states and therefore there is transport it's kind of a one to one correspondence now if you look closely in me what physicists believe is in these operators there is no singular continuous spectrum so that's why there is not C in me but AC let me go back to this okay now so the first lecture we will be as I said in me somehow proving an upper bound on the eigenfunction correlator and I will do this in me by actually first proving an upper bound on the Green's function and for this I will just follow Anderson's original arguments now if you there are all kinds of problems with this paper but there is a beautiful idea here in me which you as probabilists in me will actually appreciate now see in me if what we're aiming at is the regime in me when lambda is tremendously large so as a first in me guess and you could think in me one way to understand in me this resolvent of this operator is to say oh when lambda is large in me let me forget about a so let me treat a as a perturbation right this is in fact I mean not such a brilliant idea in general we but for large disorder it works now how do we do perturbation theory in functional analysis and we we apply in me the really only equation in me which is suited for that which is the resolvent equation which since I've included a minus a in H that's how it reads it's up with a plus sign so you could say I went to first approximation of me the perturbed resolvent is the unperturbed one which is just these and then there's the first order me perturbation now of course first order perturbation doesn't give you anything because there's this unknown quantity here and we so let us just expand until we somehow everything I mean just just expand and we write down the full north and orman series iterate and be this resolving equation so what will you get well in fact I mean what you get is something which will probably show up in other talks and I want to do this in me sort of sort of careful so let's take the first equation I mean let's now somehow put matrix elements on them this is something which you should really understand so what's the first term where the first term in me is the x y matrix element of a multiplication operator that we know what it is because if I take the inverse of a multiplication operator it's still a multiplication operator so it's diagonal in this given basis of states and it multiplies by v of x minus c okay so what else do we have I mean where we have plus okay so what is a so let me see if I come here with an x I mean there's a multiplication operator I mean it will throw out v of x minus c and then we have delta x a 1 over h minus c delta y so what is a a by definition I mean it was the sum over w with a property I mean that there are neighbors to x okay the matrix element the x y matrix element of a and it was normalized to 1 so what we end up here is a full of the perturbed operator okay so this is what I get when I take matrix element on the first equation and if I iterate this thing of course what I will get is what well I would then somehow accumulate an expansion where I start in x then eventually I will end up in y and at each step I'm just going to make somehow steps of size 1 and I accumulate factors of 1 over lambda times the potential at that side where I am minus e that's what I will end up with I can go back and when I just expand everything out oh there's a minus sign and I'm wrong on this slide and I eventually erase this so kill this minus sign because I got confused when preparing this so you get all the matrix elements in general if you put another operator a here that's what it is but since these are adjacency matrices we get only walks of step 1 and I accumulate for each walk gamma I accumulate factors of the term 1 over lambda times there's a lambda missing here minus e okay now what's the intuition so first of all the first thing which you may ask is this expansion convergent well that's unfortunately no in general however we have this free parameter z and if z has the property that the imaginary part of z is bigger than the magnitude of those guys a here of the operator norm of a then you can easily see somehow that this is a convergent sum by the way one thing which I also for those of you have never seen that there's always a a priori bound for a safer join operator which bounds the norm of h minus z and also in the case where h is just a multiplication operator in terms of 1 over the modulus of the imaginary part before h is a joined okay that's how you see that this is a convergent expansion at least if we take the imaginary part and we large enough let's suppose this for the moment we're anyway repeating only Anderson's argument it's a hand waving now what would you say I mean now you have all kinds of expansions and be from x to y now the intuition is basically the v of x that's now a random process and as long as the degree of this graph is smaller and we then lambda times v of x minus z this is in fact of a convergent expansion because see the entropy and we would show accumulating in this expansion as you always accumulating degree many terms here but as long as we somehow in modulus and this kills the degree and this is a convergent expansion which in fact is more than convergent it produces factors which are smaller than 1 so as long as the degree is smaller than the modulus of the inverses here you know you get just decay in each step and that's why you know somehow what will come out of that is an exponential decay of this Green's function now okay so that's how Andersen proved exponential decay of the Green's function there's of course something wrong with that where there's the small print that this expansion is a bit not convergent here at least somehow for z's where we're interested in the Green's function that's next to the real axis that the Green's function decays exponentially of the real axis maybe I don't need Andersen for that that's just abstract functional analysis so that's the first problem then if we ignore the convergence of this expansion and the next problem if we take now z to be real or close to real is that of course not every v of x will stay away from e so there will be resonant spots that will be in this expansion when you accumulate now here's a rescue so okay so the first thing which as a probability you can ask is how bad if you take z real how bad is the blow up on a spot where things now explode you know how bad it is the probability that this 1 over the modulus of lambda vx minus some constant in v is bigger than t that's the same as saying lambda vx minus alpha stays smaller than 1 over t and since I assumed that my probability distribution was absolutely continuous with the bounded density it's proportional to 1 over t of course it's also proportional to 1 over lambda okay I'm going to tell you this just already in advance this is a week L1 bound as you know in particular in tails if you take these potentially singular denominators to a fractional power to some power s smaller than 1 then things are actually integrable so the singularities which appear in these expansions that and in particular things go down with the disorder strength there's always a 1 over lambda which makes these probabilities small now but we have to deal with the fact that these expansions which Anderson writes down at first we are not convergent in the regime z close to the real axis where we want to look at and there's a technique in which you as probably might have guessed which in your language we would just be the statement that I mean let me now loop erase this expansion so instead of when we sum over all walks which connect x to y let me loop erase things in sum over self-abiding walks now this again is the same misprint so you just ignore these factors minus 1 which I killed eventually so what would be the idea of this loop erasure where was my yes put it here let me not use let me not use let me not use normal resolvent equation but actually let me kill in this operator starting from x all matrix elements which connect x to the outside world to the rest of the lattice so in two dimensions let me kill these four terms now if x is not equal to y then the corresponding resolvent equation we would read as follows the perturbation is not anymore in A but the perturbation is now in these orange bonds so it's only part of A in which I'm expanding it's only somehow these four terms in which I'm erasing in the first step so what would be it well let me talk about loop erasing this you could have come back at me somehow to x in loop erasing this we would be meaning that you wait until you last exit x in terms of the functional analysis statement is that you take the resolvent equation erase this here and out pops the function from x to x the first step is not anymore 1 over v of x but it's the full green's function and then you connect to a hopping and neighboring x let me call this v here and then from v we continue with the rest of the operator and then everything which lives in the complement of x so just take the operator on the complement of x so g just subtract x from it call this green's function as such and that's your loop erased in v version once you iterate this what you will accumulate then it's not factors of 1 over v of x here but what you will accumulate is factors of full resolvent but which always work in the complement of the path up to time k and that is called the feedback expansion now what about its convergence well its convergence is wonderful because as you know if we go now to a finite graph there is just a finite number of terms in this sum it can only be a finite number of self-awarding graph and walks it can be on a finite graph so on a finite graph I can even take z arbitrary close to the real axis and do this just a finite sum now the argument in which I have to make in order to prove the gd case exponentially is that these factors here which I accumulate will be typically very small small in comparison to what where the entropy is still the same the entropy in each step is the degree which we are getting from this expansion so here comes the proof in principle the full proof that for large disorder the greens function decays exponentially it is this slide so let's take the one step resolvent equation which results from erasing those orange hoplings now let's average this greens function or the modulus of this greens function with respect to this order that's this capital E here and for reasons which will be clear in a minute I don't want to take the greens function the average of the modulus of the greens function whereby I rather want to take a fractional moment fractional moment means s smaller than 1 now apply triangles in equality to that and use the fact that I've set things in v up such that the modulus of a is equal to 1 what you will arrive at in v is a sum over the diagonal greens function times in v a problem which looked like the problem before in v you have to still walk v to y in the complement of x now this is now an expected value of the product wouldn't it be for this product and would this expected value factorize one could argue as follows let me take in the expected value of the greens function of this diagonal term which pops up here ok so now I claim that when lambda is large in v this case is 1 over lambda to the s and I show you this in v somehow in a moment now I claim even further even when I condition on everything aside in v from the random variable at x where for brevity of this slide I abbreviate this conditional expectation of IE of x that's just this integral here I have IIT random variables then this integral is bounded by 1 over lambda to the s take this for granted and in this expected value we now just integrate over v of x now see let's get things up in this fienbergish way the second factor here that plays cool under this average because it does not depend on v of x that's the result in the complement so indeed it's not an expected value of a product which I have to compute it's just an expected value of a single factor and this is bounded by 1 over lambda to the s that's what I inserted and what I get in v is I get v of sum so there's a certain entropy in this expansion I get an expected value of a gadget in v which looks as the thing which I originally looked at but I got now this factor of 1 over lambda and now iterate this thing iterate it how many times iterate it in the distance of x to y times in each step I get well degree many terms of the graph times the factor of 1 over lambda to the s if the degree times this factor here is smaller than 1 iteration just gets you an exponential decay right I mean that's what you get so okay so what I show to you is an argument actually a rigorous argument why when lambda is large enough the greens function decay is exponentially and you know what the decay rate is the decay rate is given in terms of lambda in the degree of the graph in this explicit form okay so this relied on observation two observations namely the first observation is that this factor here in v was even when I only integrate on v of x bounded by c s times divided by lambda to the s and if you think about it and be sort of a bit closer it also relied on the fact that these quantities here we are a priori bounded by a constant because when I iterate eventually I'll have to throw away the last term how do I throw away the last term and be well I throw it away because I know it's bounded I don't want to show to you how at the moment why these things are bounded but you know when x is equal to y I will show to you why they are bounded and that is this this lemma here now how do you what's the magic behind this lemma so here is something in which if you haven't seen this before that is actually the heart of I would say 30% of papers on problems on graphs and a lot of other things now see what we are left when we want to prove this inequality basically you have to ask yourself how does this omega of x enter in this Green's function so how does it depend on it now that is the question which in abstract ways can be framed and be as follows the H is an operator which is some H0 plus lambda vx times a rank 1 projection onto the vector delta x I am going to write this rank 1 projection in this physicist way and basically what I am asking is let me call this p so basically the question which I am asking is how does the resolvent of H look in the subspace p because this is nothing else but delta x 1 over h minus d delta x p so that is the question which I am asking now here is a general theory in which I think it is worth learning about which is very old it is called it answers the question that when you take a self-adjoint operator in some hybrid space you have an orthogonal projection can be rank 1 or not rank 1 it doesn't matter and you add to this self-adjoint operator think of this as H0 A an operator which just lives in that subspace p that is what this identity here says and then the resolvent on that subspace of the perturbed operator is nothing else but the resolvent of the unperturbed operator on that subspace modified me by an inverse and that is now an inverse only on that subspace pH of an operator k which depends on z which happens to be just the projected version of the resolvent now this is a general fact of life it is a very useful identity and one thing which I want you to do is think about how one would prove this at least for matrices and now once we prove this we can answer how gxx depends why because let us just take p of x p and what I suggested here then this operator a is really this creature here and it says that the full resolvent once I project it from both sides on this one-dimensional subspace that is nothing else but the operator k times the inverse of 1 plus a, what is a in this case well a is nothing else but lambda v of xp times k times p so in other words gxx is of that form with a quantity alpha which we are going to look at closer can actually be identified so this alpha of z this is nothing else but 1 over minus the xx matrix element of h0 in this case here but once you know that this is true of course you know that once you integrate over v of x this integral is bounded and that is what I showed to you in this inequality remember we have a week at one bound so every fractional moment is bounded okay deteriorates and we once you go and v to s equal to 1 but that doesn't matter because we can stay at any fractional moment okay so I would hand out these slides and we think about how to prove this it's an easy exercise and that's just a very small corollary how much time do I have and I think I'm going infinitely slow okay so what I showed to you is the argument why the fractional moment of the Green's function decays exponentially even if z is arbitrarily close to the real axis because remember in this whole argument here the bound which you get is independent of z of the complex energy okay so we have such an inequality independent of z and we end the remainder of what we have to show in order to show absence of transport and absence of continuous spectrum is in fact and we have to go to this eigenfunction correlator so the next step in the proof of in fact this theorem here is to convert the bound on the Green's function to a bound on the eigenfunction correlator and if we would succeed and we would then have proven that at large disorder there is no transport there is no continuous spectrum so they can only be if there is spectrum and there is only pure point spectrum now this result in me as it stands and it was for d equal to 1 proven by Consouya in fact actually also we are looking at the eigenfunction correlator for arbitrary d it was proven in this form and we first by Eisenmann using exactly not exactly but more or less and we somehow with the method in which I talked about now I should say and we somehow this method is known and we under the name fractional moment method one of its beauties is actually its simplicity as you see and we can really go through the full proof and if you're only interested in Green's functions then the full proof is really just more or less one slide there are other techniques in me to prove localization and we see localization is basically what are you what are you trying to do I mean well localization in me is this phenomena in me that when you think about in me this path expansions of the Green's functions one shocking thing in me to a probabilist is of course in me the quantities which are involved there we are not non-negative and these are now complex quantities in me which you have to sum up but what you have to show is in me that since there's enough in me somehow randomness in this path expansions the blow ups which are there they are they are there in me because we are in the we are searching looking at the Green's function in the spectrum you know there will be resonances in me in these in these systems and these are physical in me because the phenomenon which you try to defeat is that you could have an eigenstate which lives here and then by quantum tunneling in me you can move over there so these resonances in me which experience in these expansions they are completely physical and you have to see in me that because because of this self-avoidiness in me of the path the environment is always refreshed and that's why these resonances in me really don't play in me such a tremendous role in me at least for large disorder and that's the phenomenon of localization in me in a mathematical proof controlling these resonances in me was done in me not by looking at an integrated version of the Green's function but by probabilities that these blow-ups are severe or not so severe in a multi-scale analysis in me and that was the original proof in me by Fouillet and Spencer who could actually prove the decay of the Green's function but couldn't go in me the last step to really prove me the absence of transport okay good so the remainder of the probably of this lecture of the tomorrow I will spend now in me on showing how to go in me from the Green's function to the Eigen function correlator that's an important step in order to control in me really the absence of transport now correlator is a very fundamental quantity it's the total variation measure of the spectral measure so it's actually good to pause hearing me and collect me some of its properties you know I was at points and we sort of cheating and telling you this expansion with a Fienberg expansion that's self-adjoining walks this is all convergence we don't have to worry because we can go to a finite volume but ultimately we're interested in the infinite volume and we saw one of the things which you should understand is if I take my Hamiltonian just cut off the graph to a finite box say what's the relation of the Eigen function correlator on the finite box to the Eigen function correlator in the infinite system and here's an unfortunate fact and we this quantity is lower semi-continuit in this cut-off if you prove bounds for finite volume they're inherited by the infinite volume limit that's just a general fact of life that when you take a sequence of operators which converges in the strong resolvent sense their total variation measures there at least and we some are following this lower semi-continuity for open sets and that's enough strong resolvent convergence I don't know if I should say something about this basically this is the reconversion of the associated spectrum measures that is what makes this proof when we work now what's the relation of the Eigen function correlator to the Green's function there's a lot to say in fact it's a singular limit namely if we're on a finite graph I can always resort to finite graphs the Eigen function correlator of a bounded interval is in fact nothing is but the Green's function is fractional moment integrated over that interval take somehow a singular now this will blow up why would this blow up let me do I have some blackboards still probably not let me try it here somehow why would this blow up if I'm on a finite graph H is a matrix so what is GXY at any energy E this is nothing but the sum of the Eigen values EJ of 1 over EJ minus E and then there's a certain weight and the weight is just the normalized Eigen function at X and it's complex conjugated Y okay now once so now if E doesn't hit one of those Eigen values this is a well defined quantity it's a finite sum over non-singular terms but once I integrate over E basically what you're integrating here when you take the modulus is a 1 over X singularity that is not such a good idea however if you plug in a fractional moment and then suddenly 1 over X singularity becomes integral right so once S is smaller than 1 this gadget here wants to integrate over E this is finite as long as S is smaller than 1 now what is claimed here is with that when you take the limit S tends to 1 what you will effectively get out here is the sum of the moduli over these two terms why is that clear we will mentally decompose this interval here here's an Eigen value that's E1 that's E2 decompose this interval in terms of somehow I ends or I1 I2 which stay which are sort of centered in the equidistant in the next Eigen value now from this integration over I1 so in this sum all terms want to integrate over I1 we are non-singular when Ej is not E1 so in the limit S tends to 1 this will give 0 this part of the integration now once you pick out one of those singular terms what you have to check can check this is that once I integrate just one of those factors the normalization is done in such a way that I really get a conical delta that I really get 1 as long as M is equal to I1 and that's how you pick out these terms which are equal to the Eigenfunction correlator so the Eigenfunction is not an alien object it's a singular limit unfortunately if we now take expected values we could think of somehow just finishing the proof but that doesn't work because once you look at what we've proved here there is this annoying quantity C of S and C of S was quite explicit this is sort of a bound on these integrals here which you cannot really improve when they blow up with S tending to 1 now you might say but I have a factor 1 over 1 minus S in front yes you do have this but unfortunately also mu of S then goes to 0 and then you will lose your exponential decay so directly using Mv this simple observation Mv doesn't get us anywhere and that's why we have to be a bit smarter Mv than that Mv and turn another crank and investigate a bit closer the relations of Green's function and Eigenfunctions and I think Mv I will do this Mv tomorrow so the random wall expansion actually the fact that image of Z is greater than it's not a necessary condition no it's actually convert very close to the real aspect yes you can actually sure so his comment is about that I was a bit too timid here that is actually true you can say more about this but still Mv this doesn't really solve your problem because so in some sense Mv somehow that's a red herring indeed Mv the convergence can be Mv taken under control however Mv what's really wrong with this expansion is and you see this Mv somehow in this picture want to take non-self adjoining non-self avoiding walks you know in this in this land is Mv these resonance spots where lambda V of X minus the energy where you want to look at is actually much much larger than the degree of the graph okay and once in Mv taken non-self avoiding expansion you can avoid Mv somehow going back to these resonance spots over and over again and that's just a not very good idea so what you should at least do Mv you should avoid all those blow up points in this expansion in order to see exponential decay excuse me yes sorry much much smaller Mv than that because Mv that is what's going to kill you the argument by this decay is exponentially once you run into these resonance spots and if you're in an infinite system there will be an infinite number of those guys and if you don't take care of your expansion in a better way than just running into these resonance spots over and over again you're doomed so that's why Mv at least a soft form of avoiding resonances is necessary before any of those ideas to work and the feedback expansion is somehow the most dramatic version of that you'll never come back to a site which you visited before