Seminar on Stochastic Processes

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Autumn Semester 2021

Date / Time

Speaker

Title

Location

22 September 2021
17:15-18:15

Dr. Barbara Dembin
ETH Zurich, Switzerland

Event Details

Seminar on Stochastic Processes

Title	Large deviation principle for the streams and the maximal flow in first passage percolation
Speaker, Affiliation	Dr. Barbara Dembin, ETH Zurich, Switzerland
Date, Time	22 September 2021, 17:15-18:15
Location	HG G 19.1
Abstract	We consider the standard first passage percolation model in the rescaled lattice Z^d/n for d>= 2: with each edge e we associate a random capacity c(e)>= 0 such that the family (c(e))_e is independent and identically distributed with a common law G. We interpret this capacity as a rate of flow, i.e., it corresponds to the maximal amount of water that can cross the edge per unit of time. We consider a bounded connected domain Ω in R^d and two disjoint subsets of the boundary of Ω representing respectively the source and the sink, i.e., where the water can enter in Ω and escape from Ω. We are interested in the maximal flow, i.e., the maximal amount of water that can enters through Ω per unit of time. A stream is a function on the edges that describes how the water circulates in Ω. In this talk, we will present a large deviation principle for streams and deduce by contraction principle an upper large deviation principle for maximal flow in Ω. This is a joint work with Marie Théret.

Large deviation principle for the streams and the maximal flow in first passage percolationread_more

HG G 19.1

29 September 2021
17:15-18:15

Dr. Linxiao Chen
ETH Zurich, Switzerland

Event Details

Seminar on Stochastic Processes

Title	Analytic combinatorics in several variables and application to multivariate local limit theorems
Speaker, Affiliation	Dr. Linxiao Chen, ETH Zurich, Switzerland
Date, Time	29 September 2021, 17:15-18:15
Location	HG G 19.1
Abstract	In this talk I will present a recipe for reading the asymptotics of a multi-dimensional infinite array of numbers from its generating function. In the bivariate case, this means reading the asymptotics of a_{m,n} as m,n → ∞ and m/n^θ → s (where θ>0 is fixed and s>0 is a variable) from the function A(x,y)= Σ_{m,n≥0} a_{m,n} x^m y^n. I will explain why such asymptotics are useful for studying probabilistic models, in particular for establishing uniform local limit theorems with exotic limit distributions, with some examples from random map models. Previously, similar recipes were available only in the case where θ=1 and the function A is rational. In contrast, our method works for any θ>0 and algebraic function A, but under some additional conditions on the singularity structure of A. If time permits, I will explain why the old and new methods treat disjoint cases, and how they might be combined together in the future.

Analytic combinatorics in several variables and application to multivariate local limit theoremsread_more

HG G 19.1

6 October 2021
17:15-18:15

Prof. Dr. Jiří Černý
Universität Basel

Event Details

Seminar on Stochastic Processes

Title	Giant component in the supercritical level-set percolation of GFF on regular expanders
Speaker, Affiliation	Prof. Dr. Jiří Černý, Universität Basel
Date, Time	6 October 2021, 17:15-18:15
Location	HG G 19.1
Abstract	In recent works with A. Abächerli, we showed that the level set percolation of the zero-mean Gaussian free field on a certain class of regular expander graphs exhibits a phase transition. A slight drawback of this result is our description of the supercritical phase where we only could show that the largest connected component is "mesoscopic". In the talk, I will first describe the problem, and then explain how to remove this drawback and how to show that the supercritical level set has an essentially unique giant component.

Giant component in the supercritical level-set percolation of GFF on regular expandersread_more

HG G 19.1

20 October 2021
17:15-18:15

Dr. Wei Qian
University of Paris-Saclay

Event Details

Seminar on Stochastic Processes

Title	Geodesics in the Brownian map: Strong confluence and geometric structure
Speaker, Affiliation	Dr. Wei Qian, University of Paris-Saclay
Date, Time	20 October 2021, 17:15-18:15
Location	HG G 19.1
Abstract	I will talk about a joint work with Jason Miller where we establish results on all geodesics in the Brownian map, including those between exceptional points. First, we prove a strong and quantitative form of the confluence of geodesics phenomenon which states that any pair of geodesics which are sufficiently close in the Hausdorff distance must coincide with each other except near their endpoints. Then, we show that the intersection of any two geodesics minus their endpoints is connected, the number of geodesics which emanate from a single point and are disjoint except at their starting point is at most 5, and the maximal number of geodesics which connect any pair of points is 9. For each k=1,…,9, we obtain the Hausdorff dimension of the pairs of points connected by exactly k geodesics. For k=7,8,9, such pairs have dimension zero and are countably infinite. Further, we classify the (finite number of) possible configurations of geodesics between any pair of points, up to homeomorphism, and give a dimension upper bound for the set of endpoints in each case. Finally, we show that every geodesic can be approximated arbitrarily well and in a strong sense by a geodesic connecting typical points. In particular, this gives an affirmative answer to a conjecture of Angel, Kolesnik, and Miermont that the geodesic frame, the union of all of the geodesics in the Brownian map minus their endpoints, has dimension one, the dimension of a single geodesic.

Geodesics in the Brownian map: Strong confluence and geometric structureread_more

HG G 19.1

27 October 2021
17:15-18:15

Dr. Cécile Mailler
University of Bath

Event Details

Seminar on Stochastic Processes

Title	The ants walk: finding geodesics in graphs using reinforcement learning
Speaker, Affiliation	Dr. Cécile Mailler, University of Bath
Date, Time	27 October 2021, 17:15-18:15
Location	HG G 19.1
Abstract	How does a colony of ants find the shortest path between its nest and a source of food without any means of communication other than the pheromones each ant leave behind itself? In this joint work with Daniel Kious (Bath) and Bruno Schapira (Marseille), we introduce a new probabilistic model for this phenomenon. In this model, the nest and the source of food are two marked nodes in a finite graph. Ants perform successive random walks from the nest to the food, and the distribution of the n-th walk depends on the trajectories of the (n-1) previous walks through some linear reinforcement mechanism. Using stochastic approximation methods, couplings with Pólya urns, and the electric conductances method for random walks on graphs (which I will explain on some simple examples), we prove that, depending on the exact reinforcement rule, the ants may or may not always find the shortest path(s) between their nest and the source food.

The ants walk: finding geodesics in graphs using reinforcement learningread_more

HG G 19.1

3 November 2021
17:15-18:15

Dr. Armand Riera
Universität Zürich

Event Details

Seminar on Stochastic Processes

Title	Spine representations of non-compact models and isoperimetric inequalities in Brownian geometry
Speaker, Affiliation	Dr. Armand Riera, Universität Zürich
Date, Time	3 November 2021, 17:15-18:15
Location	HG G 19.1
Abstract	We will provide a unified construction of the three main models of non-compact Brownian geometry, namely the Brownian plane, the infinite volume Brownian disk and the Brownian half-plane. We will explain how these three models can be encoded by a random infinite tree with non-negative labels under different - degenerated - conditionings. This new construction allows to understand the precise form of geodesics towards the boundary, to obtain a multiplicity of explicit computations and to investigate relations between these models. This part of the talk is based on a joint work with Jean-François Le Gall. We will then use this construction to establish sharp bounds on the probability of having a short cycle separating a large ball - centered in the root - from infinity in the Brownian plane. Finally, if time allows, we will combine all these results to determine the isoperimetric profil of the Brownian plane.

Spine representations of non-compact models and isoperimetric inequalities in Brownian geometryread_more

HG G 19.1

10 November 2021
17:15-18:15

Prof. Dr. Juhan Aru
EPFL

Event Details

Seminar on Stochastic Processes

Title	A characterisation of the continuum Gaussian free field in d ≥ 2 dimensions.
Speaker, Affiliation	Prof. Dr. Juhan Aru, EPFL
Date, Time	10 November 2021, 17:15-18:15
Location	HG G 19.1
Abstract	We will look at the statement and the proof of the title, try to see what might be surprising about it and what is not, what might be useful about it and what is not.

A characterisation of the continuum Gaussian free field in d ≥ 2 dimensions.read_more

HG G 19.1

17 November 2021
17:15-18:15

Tsviqa Lakrec
Universität Zürich

Event Details

Seminar on Stochastic Processes

Title	Scenery Reconstruction for a Random Walk on Random Scenery with Adversarial Error Insertion
Speaker, Affiliation	Tsviqa Lakrec, Universität Zürich
Date, Time	17 November 2021, 17:15-18:15
Location	HG G 19.1
Abstract	Consider a simple random walk on the integers, which are colored at random. Look at the sequence of the first N steps taken and colors of the visited integers. From it, you can deduce the coloring of approximately √N integers. Suppose an adversary may change δ N entries in that sequence. What can be deduced now? This is the adversarial scenery reconstruction problem. We discuss the history of scenery reconstruction, the connection with ergodic theory, and present the following new theorem on adversarial scenery reconstruction: for any θ<0.5,p>0, there are N₀, δ₀ such that if N>N₀ and δ < δ₀ then with probability >1-p we can reconstruct the coloring of >N^θ integers.

Scenery Reconstruction for a Random Walk on Random Scenery with Adversarial Error Insertionread_more

HG G 19.1

24 November 2021
17:15-18:15

Prof. Dr. Afonso Bandeira
ETH Zurich, Switzerland

Event Details

Seminar on Stochastic Processes

Title	Noncommutative Matrix Concentration Inequalities
Speaker, Affiliation	Prof. Dr. Afonso Bandeira, ETH Zurich, Switzerland
Date, Time	24 November 2021, 17:15-18:15
Location	HG G 19.1
Abstract	Matrix Concentration inequalities for the spectrum of random matrices, such as Matrix Bernstein, have played an important role in many areas of pure and applied mathematics. These inequalities are intimately related to the celebrated noncommutative Khintchine inequality of Lust-Piquard and Pisier. In this talk we leverage ideas from Free Probability to remove the dimensional dependence in the noncommutative Khintchine in a range of instances, yielding sharp bounds in many settings of interest. As a byproduct we develop non-assymptotic matrix concentration inequalities that capture non-commutativity (or, to be more precise, "freeness"). Joint work with March Boedihardjo and Ramon van Handel, more information at arXiv:2108.06312 [math.PR].

Noncommutative Matrix Concentration Inequalitiesread_more

HG G 19.1

8 December 2021
17:15-18:15

PD Dr. Lucile Laulin
Université de Bordeaux

Event Details

Seminar on Stochastic Processes

Title	The elephant random walk and some related processes
Speaker, Affiliation	PD Dr. Lucile Laulin, Université de Bordeaux
Date, Time	8 December 2021, 17:15-18:15
Location	HG G 19.1
Abstract	The elephant random walk (ERW) is a fascinating discrete-time random walk on integers which was introduced in the early 2000s by two physicists in order to investigate how long-range memory affects the behavior of the random walk. In this talk, I will present how martingale theory can be used to obtain results on the asymptotic behavior of the ERW and some of its related processes (center of mass, reinforced version…). I will also explain how it is possible to estimate the memory parameter of the ERW.

The elephant random walk and some related processesread_more

HG G 19.1

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