Seminar on Stochastic Processes

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Autumn Semester 2019

Date / Time

Speaker

Title

Location

25 September 2019
17:15-18:15

Augusto Teixeira
IMPA

Event Details

Seminar on Stochastic Processes

Title	Random walk on the simple symmetric exclusion process
Speaker, Affiliation	Augusto Teixeira, IMPA
Date, Time	25 September 2019, 17:15-18:15
Location	HG G 43
Abstract	In this talk we will discuss some of the history of random walks on dynamical random environments and we will present a recent result where the environment is given by a simple symmetric exclusion process. For this model, we are able to prove a law of large numbers for the displacement of the walk (for all but two densities of the underlying particle system) as well as a central limit theorem throughout its ballistic regimes. The main technique that we employ is a renormalization scheme that brings its inspiration from percolation theory.

Random walk on the simple symmetric exclusion processread_more

HG G 43

2 October 2019
17:15-18:15

Dominik Schröder
ITS

Event Details

Seminar on Stochastic Processes

Title	Hermitian Cusp Universality and Non-Hermitian Edge Universality
Speaker, Affiliation	Dominik Schröder, ITS
Date, Time	2 October 2019, 17:15-18:15
Location	HG G 43
Abstract	In the last decade, the Wigner-Dyson-Mehta (WDM) universality conjecture has been proven for very general Hermitian random matrix ensembles in the bulk and at the edge of the self consistent density of states (scDOS). The scDOS of Wigner-type matrices can, in addition, also feature cubic root cusp singularities for which we prove universality in the real and complex symmetry class through an optimal local law and the fast relaxation to equilibrium of Dyson Brownian motion. This result completes the last remaining case of the WDM conjecture for Wigner-type random matrices. About universality for non-Hermitian matrices much less is known. We prove that for non-Hermitian i.i.d. matrices the local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements are Gaussian. The proof relies on the optimal local law in the cusp regime of Hermitian random matrices, and a supersymmetric estimate on the least singular value of shifted Ginibre ensembles. This estimate on the least singular value improves the classical smoothing bound from [Sankar, Spielman, Teng (2006)] in the transitional edge regime.

Hermitian Cusp Universality and Non-Hermitian Edge Universalityread_more

HG G 43

9 October 2019
17:15-18:15

Thomas Lehéricy
Universität Zürich

Event Details

Seminar on Stochastic Processes

Title	Uniform mixing time and bottlenecks in random quadrangulations
Speaker, Affiliation	Thomas Lehéricy, Universität Zürich
Date, Time	9 October 2019, 17:15-18:15
Location	HG G 43
Abstract	We prove a lower bound on the size of bottlenecks in uniform quadrangulations, valid at all scales simultaneously. We use it to establish upper bounds on the uniform mixing time of the lazy random walk on uniform quadrangulations, as well as on their dual. The proofs involve an explicit computation of the Laplace transform of the number of faces in truncated hulls of the uniform infinite plane quadrangulation.

Uniform mixing time and bottlenecks in random quadrangulationsread_more

HG G 43

16 October 2019
17:15-18:15

Tobias Hurth
Université de Neuchâtel

Event Details

Seminar on Stochastic Processes

Title	A factorization formula for the partition function in the parabolic Anderson model
Speaker, Affiliation	Tobias Hurth, Université de Neuchâtel
Date, Time	16 October 2019, 17:15-18:15
Location	HG G 43
Abstract	We consider a continuous-time simple symmetric random walk on the integer lattice Z^d in dimension greater than or equal to 3. The random walk is subject to a random potential induced by independent two-sided Brownian motions linked with the sites in Z^d. In the high-temperature regime, for the partition function that corresponds to starting from site x at time s and ending on site y at time t, we will state and motivate a factorization formula of the type obtained for example by Sinai (1995) and Kifer (1997) for different polymer models. We shall explain that the error term in the formula is uniformly small not just in the diffusive regime \|x-y\| < (t-s)^{1/2}, but up to \|x - y\| < (t-s)^{1-epsilon} for epsilon arbitrarily small. We will then outline how the factorization formula can be used to show that the global stationary solution to the semidiscrete stochastic heat equation attracts solutions whose initial data grows subexponentially. The talk is based on a project with Kostya Khanin and Beatriz Navarro Lameda. Fedor Nazarov helped us with the proof of the factorization formula.

A factorization formula for the partition function in the parabolic Anderson modelread_more

HG G 43

30 October 2019
17:15-18:15

Yueyun Hu
Université Paris 13

Event Details

Seminar on Stochastic Processes

Title	The Derrida-Retaux conjecture on recursive models
Speaker, Affiliation	Yueyun Hu, Université Paris 13
Date, Time	30 October 2019, 17:15-18:15
Location	HG G 43
Abstract	This talk is based on a joint work with Xinxing Chen, Victor Dagard, Bernard Derrida, Mikhail Lifshits, and Zhan Shi. We study a max-type recursive model which was introduced by Derrida and Retaux (2014) as a simplified hierarchical renormalization model to understand the depinning transition of a line in presence of strong disorder. It is expected to have many universality properties at or near criticality, though few of these predicted properties have been rigorously proved so far. In the nearly supercritical regime we prove that under a suitable integrability assumption on the initial distribution, the free energy vanishes at the transition with an essential singularity with exponent 1/2. This gives a weaker answer to a conjecture of Derrida and Retaux (2014). Other behaviours are obtained when the integrability condition is not satisfied.

The Derrida-Retaux conjecture on recursive modelsread_more

HG G 43

6 November 2019
17:15-18:15

Afonso Bandeira
ETH Zürich

Event Details

Seminar on Stochastic Processes

Title	Hidden rare events, statistical-to-computational gaps, and hardness in the Sherrington-Kirkpatrick Hamiltonian
Speaker, Affiliation	Afonso Bandeira, ETH Zürich
Date, Time	6 November 2019, 17:15-18:15
Location	HG G 43

Hidden rare events, statistical-to-computational gaps, and hardness in the Sherrington-Kirkpatrick Hamiltonian

HG G 43

13 November 2019
17:15-18:15

Franco Severo
IHES

Event Details

Seminar on Stochastic Processes

Title	Coincidence of critical parameters for percolation of Gaussian Free Field level-sets
Speaker, Affiliation	Franco Severo, IHES
Date, Time	13 November 2019, 17:15-18:15
Location	HG G 43
Abstract	We consider level-sets of the Gaussian Free Field on Z^d, for d>2, above a given height parameter h \in R. As h varies, this defines a canonical site percolation model with slow (polynomial) decay of correlations. We prove that three natural critical parameters associated to this model (h_{*}, h_{} and \bar{h}) respectively describing a strongly non-percolative regime, the emergence of an infinite cluster, and a strongly percolative regime, actually coincide. Combined with previous results, this equality has many implications regarding the geometry of GFF level-sets, both in the subcritical and supercritical regimes. The core of the proof is a new interpolation scheme that integrates out the long-range dependency of the GFF, thus allowing to import results from finite-range percolation models. In order to implement this strategy, we make extensive use of renormalization techniques. Based on a joint work with Hugo Duminil-Copin, Subhajit Goswami and Pierre-François Rodriguez.

Coincidence of critical parameters for percolation of Gaussian Free Field level-setsread_more

HG G 43

20 November 2019
17:15-18:15

Kilian Raschel
Université de Tours

Event Details

Seminar on Stochastic Processes

Title	Confined random processes and Galois theory of difference equations
Speaker, Affiliation	Kilian Raschel, Université de Tours
Date, Time	20 November 2019, 17:15-18:15
Location	HG G 43
Abstract	Some models of random walks or Brownian motions in cones can be studied via functional equations that satisfy their Laplace transforms or generating functions. This approach applies to various situations, allowing to study first passage times, extinction probabilities, numbers of paths, stationary probabilities for reflected Brownian motion, etc. It is rather easy to reformulate these functional equations in terms of q-difference equations, for example f(q*s) - f(s) = g(s), where f is the unknown function (typically the generating function), while g and q are known and depend on the model. Tools from the theory of difference equations are then perfectly adapted, in particular to characterize the algebraic nature of the solution, or even to compute it. In this talk we will present several examples: we will begin by the enumeration of quadrant walks, for which recent works by Dreyfus, Hardouin, Roques and Singer characterize the differential transcendence of the generating functions. We will also study planar reflected Brownian motion and present a joint work with Bousquet-Mélou, Elvey Price, Franceschi and Hardouin, giving a complete characterization of the Laplace transforms of the stationary distributions. We will conclude by showing some open problems.

Confined random processes and Galois theory of difference equationsread_more

HG G 43

27 November 2019
17:15-18:15

Francis Comets
Université Paris-Diderot

Event Details

Seminar on Stochastic Processes

Title	Random walk avoiding its convex hull: finite memory case
Speaker, Affiliation	Francis Comets, Université Paris-Diderot
Date, Time	27 November 2019, 17:15-18:15
Location	HG G 43
Abstract	Fix integers d≥2 and k≥d−1. Consider a random walk X(0),X(1),… in R^d in which, given X(0),X(1),…,X(n) (n≥k), the next step X(n+1) is uniformly distributed on the unit ball centred at X(n), but conditioned that the line segment from X(n) to X(n+1) intersects the convex hull of {0,X(n−k),…,X(n)} only at X(n). For k=∞ this is a version of the model introduced by Angel et al., which is conjectured to be ballistic, i.e., to have a limiting speed and a limiting direction. We establish ballisticity for the finite-k model, and comment on some open problems. In the case where d=2 and k=1, we obtain the limiting speed explicitly: it is 8/(9π^2). The talk is based on a joint work with Mikhail Menshikov and Andrew Wade.

Random walk avoiding its convex hull: finite memory caseread_more

HG G 43

4 December 2019
17:15-18:15

Franco Flandoli
Università di Pisa

Event Details

Seminar on Stochastic Processes

Title	Effect of transport noise on PDEs
Speaker, Affiliation	Franco Flandoli, Università di Pisa
Date, Time	4 December 2019, 17:15-18:15
Location	HG G 43
Abstract	Linear PDEs of transport type are regularized by the addition of an extra transport term of stochastic type; this is a phenomenon discovered around 2010 and consolidated by different techniques and on different examples. However, the effect on nonlinear PDEs is much less clear. Two results for point vortex solutions and point charge solutions of 2D Euler equations and 1D Vlasov-Poisson equations respectively, indicate that a rich noise has to be considered, opposite to the linear case where a simple space-independent noise suffices to regularize. Some confirmations that such rich noise may have regularizing properties on nonlinear models came for Leray alpha model and dyadic models of turbulence. We now have new insight into the case 3D Navier-Stokes equations, that will be explained in the talk. The results mentioned above, the classical and the new ones, have been obtained by several authors including M. Gubinelli, E. Priola, D. Barbato, L. Galeati, D. Luo and myself.

Effect of transport noise on PDEsread_more

HG G 43

11 December 2019
17:15-18:15

Christophe Garban
Université de Lyon

Event Details

Seminar on Stochastic Processes

Title	Inverted orbits of exclusion processes, diffuse-extensive- amenability and (non-?)amenability of the interval exchanges
Speaker, Affiliation	Christophe Garban, Université de Lyon
Date, Time	11 December 2019, 17:15-18:15
Location	HG G 43
Abstract	After a brief introduction on the notion of amenability for groups, I will focus on the group of interval exchanges (the IET group) which is believed to be amenable. One of the (many) equivalent criteria to show that a group is amenable is Kesten's criterion on the return probabilities of random walks. In the case of G=IET, a recent work by Juschenko, Matte Bon, Monod and De La Salle provides a new criterion which is also of probabilistic nature. This new criterion involves the size of the inverted orbit of a certain random walk on the wobbling group W(Z^d) of permutations of Z^d. The aim of this talk will be to introduce natural models of random walks on permutations of Z^d for which this criterion can be analyzed. The talk will not require any prerequisites.

Inverted orbits of exclusion processes, diffuse-extensive- amenability and (non-?)amenability of the interval exchangesread_more

HG G 43

18 December 2019
17:15-18:15

Romain Tessera
Université Paris Orsay

Event Details

Seminar on Stochastic Processes

Title	Quantitative measure equivalence between finitely generated groups
Speaker, Affiliation	Romain Tessera, Université Paris Orsay
Date, Time	18 December 2019, 17:15-18:15
Location	HG G 43
Abstract	We initiate a quantitative study of measure equivalence (and orbit equivalence) between finitely generated groups that extends the classical setting of Lp measure equivalence. Orstein Weiss' theorem says that all amenable groups are orbit equivalent. Here we investigate quantitative versions of this result. We also study the stability of hyperbolicity under Lp measure equivalence.

Quantitative measure equivalence between finitely generated groupsread_more (CANCELLED)

HG G 43

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