## Geometric Functional Analysis & Probability Seminar

#### To suggest a talk (yours or somebody else's) contact Ronen Eldan, Itai Benjamini, Gady Kozma or Gideon Schechtman.

To join the mailing list, contact Amir Gonen (amir.gonen@weizmann.ac.il). This seminar also has a google calendar entry: Weizmann - GFAP Seminar  (Geometric Functional Analysis and Probability)

### Upcoming talks

March 8th, 2018

Erwin Bolthausen (UZH)

On the high temperature phase in mean-field spin glasses

We present a new way to derive the replica symmetric solution for the free energy in mean-field spin glasses. Only the Sherrington-Kirpatrick case has been worked out in details, but the method also works in other cases, for instance for the perceptron (work in progress), and probably also for the Hopfield net. The method is closely related to the TAP equations (for Thouless-Anderson-Palmer). It does not give any new results, presently, but it gives a new viewpoint, and it looks to be quite promising. As the TAP equations are widely discussed in the physics literature, also at low temperature, it is hoped that the method could be extended to this case, too. But this is open, and probably very difficult.

March 15th, 2018

Elie Aidekon (Paris)

Points of infinite multiplicity of a planar Brownian motion.

Points of infinite multiplicity are particular points which the Brownian motion visits infinitely often. Following a work of Bass, Burdzy and Khoshnevisan, we construct and study a measure carried by these points. Joint work with Yueyun Hu and Zhan Shi.

March 22nd, 2018

Faculty retreat - no seminar

March 29th, 2018

Yinon Spinka (TAU) + Mark Rudelson (UMich)

TBA

April 26th, 2018

Anirban Basak (WIS)

TBA

May 17th, 2018

Ron Peled (TAU)

TBA

May 24th, 2018

Emanuel Milman (Technion)

TBA

May 31st, 2018

Bhaswar Bhattacharya (UPenn)

TBA

June 7th, 2018

TBA

### List of previous talks sorted backwards

Jan 18th, 2018

Oren Louidor (Technion) + Alexander Glazman (TAU)

Oren Louidor: Dynamical freezing in a spin-glass with logarithmic correlations.

We consider a continuous time random walk on the 2D torus, governed by the exponential of the discrete Gaussian free field acting as potential. This process can be viewed as Glauber dynamics for a spin-glass system with logarithmic correlations. Taking temperature to be below the freezing point, we then study this process both at pre-equilibrium and in-equilibrium time scales. In the former case, we show that the system exhibits aging and recover the arcsine law as asymptotics for a natural two point temporal correlation function. In the latter case, we show that the dynamics admits a functional scaling limit, with the limit given by a variant of Kolmogorov's K-process, driven by the limiting extremal process of the field, or alternatively, by a super-critical Liouville Brownian motion. Joint work with A. Cortines, J. Gold and A. Svejda.

Alexander Glazman: Level lines of a random Lipschitz function

We consider the uniform distribution on Lipschitz functions on the triangular lattice, i.e. all integer-valued functions which differ by 0 or 1 on any two adjacent vertices. We show that with a positive probability such a function exhibits macroscopic level lines. Instead of working directly with Lipschitz functions we map this model to the loop $O(2)$ model with parameter $x=1$. The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. A main tool in the proof is a positive association (FKG) property that was recently shown to hold when $n \ge 1$ and $0 Jan 4th, 2018 Sebastien Bubeck (Microsoft) + Percy Deift (NYU) Sebastien Bubeck: k-server via multiscale entropic regularization I will start by describing how mirror descent is a natural strategy for online decision making, specifically in online learning and metrical task systems. To motivate the k-server problem I will also briefly recall what we know and what we don't know for structured state/action spaces in these models. Using the basic mirror descent calculations I will show how to easily obtain a log(k)-competitive algorithm for k-paging. I will then introduce our new parametrization of fractional k-server on a tree, and explain how to analyze the movement cost of entropy-regularized mirror descent on this parametrization. This leads to a depth*log(k)-competitive (fractional) algorithm for general trees, and log^2(k) for HSTs. I will also briefly mention dynamic embeddings to go beyond the standard log(n) loss in the reduction from general metrics to HSTs. Joint work with Michael B. Cohen, James R. Lee, Yin Tat Lee, and Aleksander Madry. Percy Deift: Universality in numerical analysis with some examples of cryptographic algorithms. We show that a wide variety of numerical algorithms with random data exhibit universality. Most of the results are computational, but in some important cases universality is established rigorously. We also discuss universality for some cryptographic algorithms. Joint work with C. Pfrany, G. Menon, S. Olver, T. Trogdan and S. Miller. Dec 28th, 2017 Amir Dembo (Stanford) + Yuval Peres (Microsoft) Amir Dembo: Large deviations theory for chemical reaction networks. The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Andrea Agazzi and Jean-Pierre Eckman, we utilize Lyapunov stability theory to bypass this challenge and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Yuval Peres: Trace reconstruction for the deletion channel In the trace reconstruction problem, an unknown string$x$of$n$bits is observed through the deletion channel, which deletes each bit with some constant probability q, yielding a contracted string. How many independent outputs (traces) of the deletion channel are needed to reconstruct$x$with high probability? The best lower bound known is linear in$n$. Until 2016, the best upper bound was exponential in the square root of$n$. We improve the square root to a cube root using statistics of individual output bits and some inequalities for Littlewood polynomials on the unit circle. This bound is sharp for reconstruction algorithms that only use this statistical information. (Similar results were obtained independently and concurrently by De, O’Donnell and Servedio). If the string$x$is random, we can show a subpolynomial number of traces suffices by comparison to a random walk. (Joint works with Fedor Nazarov, STOC 2017, with Alex Zhai, FOCS 2017 and with Nina Holden & Robin Pemantle, preprint (2017).) Dec 21st, 2017 Nadav Yesha (King's College London) CLT for small scale mass distribution of toral Laplace eigenfunctions In this talk we discuss the fine scale$L^2$-mass distribution of toral Laplace eigenfunctions with respect to random position. For the 2-dimensional torus, under certain flatness assumptions on the Fourier coefficients of the eigenfunctions and generic restrictions on energy levels, both the asymptotic shape of the variance and the limiting Gaussian law are established, in the optimal Planck-scale regime. We also discuss the 3-dimensional case, where the asymptotic behaviour of the variance is analysed in a more restrictive scenario. This is joint work with Igor Wigman. Dec 7th, 2017 Matan Harel (TAU) Discontinuity of the phase transition for the planar random-cluster and Potts models with$q > 4$The random-cluster model is a dependent percolation model where the weight of a configuration is proportional to q to the power of the number of connected components. It is highly related to the ferromagnetic q-Potts model, where every vertex is assigned one of q colors, and monochromatic neighbors are encouraged. Through non-rigorous means, Baxter showed that the phase transition is first-order whenever$q > 4$- i.e. there are multiple Gibbs measures at criticality. We provide a rigorous proof of this claim. Like Baxter, our proof uses the correspondence between the above models and the six-vertex model, which we analyze using the Bethe ansatz and transfer matrix techniques. We also prove Baxter's formula for the correlation length of the models at criticality. This is joint work with Hugo Duminil-Copin, Maxime Gangebin, Ioan Manolescu, and Vincent Tassion. Nov 30th, 2017 Fanny Augeri (WIS) Large deviations principles for random matrices In this talk, I will try to present some techniques to handle the problem of large deviations of the spectrum of random matrices. I will focus on the case of macroscopic statistics of the spectrum of Hermitian matrices - in particular Wigner matrices - as the empirical distribution of the eigenvalues, the largest eigenvalue or the traces of powers. In a first part, I will be concerned with the so-called objective method''. Coined by David Aldous, this method consists in introducing, given a sequence of random objects, like random finite graphs, a new infinite random object from which one can deduce asymptotic properties of the original sequence. In the context of random matrices, this method has been mainly advertised by Balint Virag, and proven effective in showing universality results for the so-called beta-ensembles. Regarding large deviations of random matrices, this objective method'' amounts to embed our sequence of matrices with growing size into an appropriate space on which one is able to understand the large deviations, and carry out a contraction principle. I will review several large deviations principles obtained by this method, given by interpretations of random matrices as either dense or sparse graphs, and point out the limits of this strategy. In a second part, I will present a different approach which is inspired from Ledoux's proof of the large deviations of Wiener chaoses. I will give a large deviations principles for the traces of Gaussian Wigner matrices using this strategy. Similarly as for Wiener chaoses, where the deviations are obtained by translations in the direction of the reproducing kernel, the large deviations of the traces of Gaussian Wigner matrices are due to additive perturbations of the underlying matrix. If time permits, I will explain how this approach can be generalized to large deviations governed by the same phenomenon. In particular, this approach enables us to partially recover some large deviations results for a family of Wigner matrices which exhibit a heavy-tail phenomenon'', meaning that the deviations of their spectrum are due to the deviations of a negligible proportion of the entries. Nov 23rd, 2017 Naomi Feldheim (WIS) Persistence of Gaussian Stationary Processes Consider a real Gaussian stationary process, either on$Z$or on$R$. What is the probability that it remains positive on$[0,N]$for large$N$? The relation between this probability, known as the persistence probability, and the covariance kernel of the process has been investigated since the 1950s with motivations stemming from probability, engineering and mathematical physics. Nonetheless, until recently, good estimates were known only for particular cases, or when the covariance kernel is either non-negative or summable. In the first hour of the talk we will discuss new spectral methods which greatly simplify the analysis of persistence. We will then describe its qualitative behavior in a very general setting. In the second hour, we will describe (very) recent progress. In particular we will show the proof of the spectral gap conjecture'', which states: if the spectral measure vanishes on an interval containing 0 then the persistence is less then$e^{-cN^2}$, and this bound is tight if the measure is non-singular and compactly supported. Time permitting, we will also discuss tiny persistence'' phenomena (of the order of$e^{-e^{cN}}$). Based on joint works with Ohad Feldheim, Benjamin Jaye, Fedor Nazarov and Shahaf Nitzan. Nov 9th, 2017 Ilya Goldsheid (Queen Many University) Real and complex eigenvalues of the non-self-adjoint Anderson model June 29th, 2017 Amir Dembo (Stanford) The criticality of a randomly-driven front Consider independent continuous-time random walks on the integers to the right of a front R(t). Starting at R(0)=0, whenever a particle attempts to jump into the front, the latter instantaneously advances k steps to the right, absorbing all particles along its path. Sly (2016) resolves the question of Kesten and Sidoravicius (2008), by showing that for k=1 the front R(t) advances linearly once the particle density exceeds 1, but little is known about the large t asymptotic of R(t) at critical density 1. In a joint work with L-C Tsai, for the variant model with k taken as the minimal random integer such that exactly k particles are absorbed by the move of R(t), we obtain both scaling exponent and the random scaling limit for the front at the critical density 1. Our result unveils a rarely seen phenomenon where the macroscopic scaling exponent is sensitive to the initial local fluctuations (with the scaling limit oscillating between instantaneous super and sub-critical phases). June 15th, 2017 Pyotr Nayar (Technion) Gaussian mixtures with applications to entropy inequalities and convex geometry We say that a symmetric random variable X is a Gaussian mixture if X has the same distribution as YG, where G is a standard Gaussian random variable, and Y is a positive random variable independent of G. In the first part of the talk we use this simple notion to study the Shannon entropy of sums of independent random variables. In the second part we investigate, using Gaussian mixtures, certain topics related to the geometry of$B_p^n$balls, including optimal Khinchine-type inequalities and Schur-type comparison for volumes of section and projections of these sets. In the third part we discuss extensions of Gaussian correlation inequality to the case of p-stable laws and uniform measure on the Euclidean sphere. Based on a joint work with Alexandros Eskenazis and Tomasz Tkocz. June 8th, 2017 Nishant Chandgotia (Tel Aviv University) Irrational rotations, random affine transformations and the central limit theorem It is a well-known result from Hermann Weyl that if alpha is an irrational number in [0,1) then the number of visits of successive multiples of alpha modulo one in an interval contained in [0,1) is proportional to the size of the interval. In this talk we will revisit this problem, now looking at finer joint asymptotics of visits to several intervals with rational end points. We observe that the visit distribution can be modelled using random affine transformations; in the case when the irrational is quadratic we obtain a central limit theorem as well. Not much background in probability will be assumed. This is in joint work with Jon Aaronson and Michael Bromberg. March 9th, 2017 Sasha Shamov Conditional determinantal processes are determinantal A determinantal point process governed by a locally trace class Hermitian contraction kernel on a measure space$E$remains determinantal when conditioned on its configuration on an arbitrary measurable subset$B \subset E$. Moreover, the conditional kernel can be chosen canonically in a way that is "local" in a non-commutative sense, i.e. invariant under "restriction" to closed subspaces$L^2(B) \subset P \subset L^2(E)$. Using the properties of the canonical conditional kernel we establish a conjecture of Lyons and Peres: if$K$is a projection then almost surely all functions in its image can be recovered by sampling at the points of the process. Feb 9th, 2017 Alexander Fish (Sydney) The values of quadratic forms on difference sets, measure rigidity and equidistribution. Given a quadratic form Q in d variables over the integers, e.g. Q(x,y,z) = xy - z^2, and a set of positive density E in Z^d, we investigate what kind of structure can be found in the set Q(E-E). We will see that if d >= 3, and Q is indefinite, then the measure rigidity, due to Bourgain-Furman-Lindenstrauss-Mozes or Benoist-Quint, of the action of the group of the symmetries of Q implies that there exists k >=1 such that k^2*Q(Z^d) is a subset of Q(E-E). We will give an alternative proof of the theorem for the case Q(x,y,z) = xy - z^2 that uses more classical equidistribution results of Vinogradov, and Weyl, as well as a more recent result by Frantzikinakis-Kra. The latter proof extends the theorem to other polynomials having a much smaller group of symmetries. Based on joint works with M. Bjorklund (Chalmers), and K. Bulinski (Sydney). Jan 19th, 2017 Jay Rosen Tightness for the Cover Time of$S^{2}$Let M be a smooth, compact, connected two-dimensional, Riemannian manifold without boundary, and let$ C_{\epsilon}$be the amount of time needed for the Brownian motion to come within (Riemannian) distance$\epsilon$of all points in M. The first order asymptotics of$ C_{\epsilon}$as$\epsilon$goes to 0 are known. We show that for the two dimensional sphere$\sqrt{C_{\epsilon}}-2\sqrt{2}\left( \log \epsilon^{-1}- \frac{1}{4}\log\log \epsilon^{-1}\right)$is tight. Joint work with David Belius and Ofer Zeitouni. Jan 12th, 2017 Ran Tessler (ETH) + Assaf Naor (Princeton) Ran Tessler (11:00): A sharp threshold for Hamiltonian spheres in a random 2-complex We define the notion of Hamiltonian sphere - a 2-complex homeomorphic to a sphere which uses all vertices. We prove an explicit sharp threshold for the appearance of Hamiltonian spheres in the Linial-Meshulam model for random 2-complexes. The proof combines combinatorial, probabilistic and geometric arguments. Based on a joint work with Zur luria. Assaf Naor (12:00): A new vertical-versus-horizontal isoperimetric inequality on the Heisenberg group, with applications to metric geometry and approximation algorithms. In this talk we will show that for every measurable subset of the Heisenberg group of dimension at least 5, an appropriately defined notion of its "vertical perimeter" is at most a constant multiple of its horizontal (Heisenberg) perimeter. We will explain how this new isoperimetric-type inequality solves open questions in analysis (an endpoint estimate for a certain singular integral on$W^{1,1}$), metric geometry (sharp nonembeddability into$L_1\$) and approximation algorithms (asymptotic evaluation of the performance of the Goemans-Linial algorithm for the Sparsest Cut problem). Joint work with Robert Young.

Jan 5th, 2017

Amir Dembo

Title: Walking within growing domains: recurrence versus transience

When is simple random walk on growing in time d-dimensional domains recurrent? For domain growth which is independent of the walk, we review recent progress and related universality conjectures about a sharp recurrence versus transience criterion in terms of the growth rate. We compare this with the question of recurrence/transience for time varying conductance models, where Gaussian heat kernel estimates and evolving sets play an important role. We also briefly contrast such expected universality with examples of the rich behavior encountered when monotone interaction enforces the growth as a result of visits by the walk to the current domain's boundary. This talk is based on joint works with Ruojun Huang, Ben Morris, Yuval Peres, Vladas Sidoravicius and Tianyi Zheng.

For previous years see: 2010-2016, 2007-2010, 2005-2007 (maintained by Boaz Tsaban) 2000-2005 (maintained by Gideon Schechtman)