Seminar on Stochastic Processes

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19 February 2020
17:15-18:15

Marianna Russkikh
Department of Mathematics, MIT

Event Details

Title	Dimers and embeddings
Speaker, Affiliation	Marianna Russkikh, Department of Mathematics, MIT
Date, Time	19 February 2020, 17:15-18:15
Location	Y27 H 12
Abstract	One of the main questions in the context of the universality and conformal invariance of a critical 2D lattice model is to find an embedding which geometrically encodes the weights of the model and that admits “nice” discretizations of Laplace and Cauchy-Riemann operators. We establish a correspondence between dimer models on a bipartite graph and circle patterns with the combinatorics of that graph. We describe how to construct a 't-embedding' (or a circle pattern) of a dimer planar graph using its Kasteleyn weights, and develop a relevant theory of discrete holomorphic functions on t-embeddings; this theory unifies Kenyon's holomorphic functions on T-graphs and s-holomorphic functions coming from the Ising model. We discuss a concept of `perfect t-embeddings’ of weighted bipartite planar graphs. We believe that these embeddings always exist and that they are good candidates to recover the complex structure of big bipartite planar graphs carrying a dimer model. Based on: joint work with R. Kenyon, W. Lam, S. Ramassamy; and joint work with D. Chelkak, B. Laslier.

Dimers and embeddingsread_more

Y27 H 12

26 February 2020
17:15-19:00

Dr. Vitali Wachtel
Universität Augsburg

Event Details

Title	First-passage times over moving boundaries for random walks with non-identically distributed increments
Speaker, Affiliation	Dr. Vitali Wachtel, Universität Augsburg
Date, Time	26 February 2020, 17:15-19:00
Location	Y27 H 12
Abstract	We consider a random walk $S_n$ with independent but not necessarily identical distributed increments. Assuming that increments satisfy the Lindeberg condition, we investigate the tail behaviour of the stopping time $T_g=\min\{n:x+S_n\leq g_n\}$ for a large class of boundaries $\{g_n\}$. We also prove limit theorems for $S_n$ conditioned on$\{T_g>n\}$. At the end of the talk we shall also consider the situation when the Lindeberg condition is not fulfilled, but the central limit theorem is still valid.

First-passage times over moving boundaries for random walks with non-identically distributed incrementsread_more

Y27 H 12

4 March 2020
17:15-19:00

Ofer Zeitouni
University of Minnesota & Weizmann Institute

Event Details

Title	Random perturbations of non-Normal matrices
Speaker, Affiliation	Ofer Zeitouni, University of Minnesota & Weizmann Institute
Date, Time	4 March 2020, 17:15-19:00
Location	Y27 H 12
Abstract	Non-normal matrices can exhibit a high sensitivity to perturbations, as the case of a matrix with a single block in its Jordan decomposition shows. Yet, when randomly perturbed, the spectrum of such matrices exhibits a remarkable stability. For example, the spectrum of randomly perturbed non-Hermitian Toeplitz matrices concentrates on certain spectral curves, that are typically outside the spectrum of the unperturbed operator. Further, the outliers process (away from the spectral curve) turns out to have interesting characteristics. I will discuss these developments. Based on joint works with Anirban Basak, Elliot Paquette, and Martin Vogel.

Random perturbations of non-Normal matricesread_more (CANCELLED)

Y27 H 12

25 March 2020
17:15-18:15

Cyril Marzouk
Ecole Polytechnique, Palaiseau

Event Details

Title	On scaling limits of random planar maps
Speaker, Affiliation	Cyril Marzouk, Ecole Polytechnique, Palaiseau
Date, Time	25 March 2020, 17:15-18:15
Location	Y27 H 12
Abstract	A planar map is a surface homeomorphic to a 2-sphere obtained by (topologically) gluing polygons together. In 2011 Grégory Miermont proved that a uniform random gluing of quadrangles converges in distribution as the number of quadrangles tends to infinity, once properly rescaled, to a continuum random object called the Brownian map, in the same spirit that the rescaled simple random walk converges to the Brownian motion; this result was simultaneous obtained by Jean-François Le Gall who also established the convergence to the same limit of uniform random gluing of triangles, or of any fixed polygon (with even number of sides…). After recalling the basics of this theory, I will discuss some recent extension of this result, in the spirit of Donsker’s Theorem, when more generally one takes a uniform random gluing of an arbitrary list of polygons: the Brownian map arises in the limit as soon as there is no "macroscopic" polygon.

On scaling limits of random planar mapsread_more (CANCELLED)

Y27 H 12

27 May 2020
17:15-18:15

Yves Le Jan
Université Paris-Sud

Event Details

Title	Title T.B.A.
Speaker, Affiliation	Yves Le Jan, Université Paris-Sud
Date, Time	27 May 2020, 17:15-18:15
Location	Y27 H 12

Title T.B.A. (CANCELLED)

Y27 H 12

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