Seminar on Stochastic Processes

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Autumn Semester 2014

Date / Time

Speaker

Title

Location

17 September 2014
17:15-18:15

Clément Hongler
EPF Lausanne

Event Details

Seminar on Stochastic Processes

Title	Ising model: making probabilistic sense of some conformal field theory ideas
Speaker, Affiliation	Clément Hongler, EPF Lausanne
Date, Time	17 September 2014, 17:15-18:15
Location	HG G 43
Abstract	Scaling limits of critical planar lattice models are conjectured to be described by Conformal Field Theory (CFT). The most classical example is the Ising model, which is expected to be described by a minimal model of CFT. We will discuss some progress in this direction: the proof of the conjectured formulae for the correlation functions using discrete complex analytic spinors and ways to make sense of the very objects and methods of CFT. Based on joined works with (mostly) Stéphane Benoist, Dmitry Chelkak, Konstantin Izyurov, Kalle Kytöla and Stanislav Smirnov.

Ising model: making probabilistic sense of some conformal field theory ideasread_more

HG G 43

1 October 2014
17:15-18:15

Friedrich Götze
Universität Bielefeld

Event Details

Seminar on Stochastic Processes

Title	Spectral distribution of random matrices and free probability
Speaker, Affiliation	Friedrich Götze, Universität Bielefeld
Date, Time	1 October 2014, 17:15-18:15
Location	HG G 43
Abstract	We investigate the asymptotic spectral distribution of functions, in particular products, of classes of non-hermitian random matrices and discuss its relation to free Probability. Furthermore, we discuss two approaches to asymptotic expansions of Voiculescu's entropy for sums of free random variables.

Spectral distribution of random matrices and free probabilityread_more

HG G 43

8 October 2014
17:15-18:15

Igor Kortchemski
Universität Zürich

Event Details

Seminar on Stochastic Processes

Title	Random stable looptrees and percolation on random planar maps
Speaker, Affiliation	Igor Kortchemski, Universität Zürich
Date, Time	8 October 2014, 17:15-18:15
Location	HG G 43
Abstract	In this talk, we will introduce a class of random compact metric spaces L_alpha indexed by alpha in (1,2) and which we call looptrees. They are made of a collection of random loops glued together along a random tree structure, can informally be viewed as dual graphs of alpha-stable Lévy trees and are coded by a spectrally positive alpha-stable Lévy process. We will see that stable looptrees are universal scaling limits, for the Gromov-Hausdorff topology, of various combinatorial models. In particular, the stable looptree of parameter 3/2 is closely related to the scaling limits of cluster boundaries in critical site-percolation on large random triangulations. Based on joint works with Nicolas Curien.

Random stable looptrees and percolation on random planar mapsread_more

HG G 43

15 October 2014
17:15-18:15

Vincent Tassion
Université de Genève

Event Details

Seminar on Stochastic Processes

Title	A universal behavior in Divide-and-Color percolation
Speaker, Affiliation	Vincent Tassion, Université de Genève
Date, Time	15 October 2014, 17:15-18:15
Location	HG G 43
Abstract	We study Divide-and-Color percolation on a planar lattice G, defined as follows. First, sample a Bernoulli bond percolation at a subcritical parameter p < pc(G). This yields a random partition of V into finite clusters; we say that two clusters at distance 1 from each other are adjacent. In a second step, assign one color to each cluster independently of the others. The color is chosen to be black with probability r and white with probability 1-r. For fixed p < pc(G), we observe an infinite path of adjacent black clusters, as soon as the parameter r exceeds a critical density rc(p). We prove that on the square lattice rc(p) converges to 1/2 when p tends to pc(G). This result solves a conjecture due to Beffara and Camia and is related to the universality of critical planar percolation.

A universal behavior in Divide-and-Color percolationread_more

HG G 43

22 October 2014
17:15-18:15

Nikolaos Zygouras
University of Warwick

Event Details

Seminar on Stochastic Processes

Title	Scaling limits of disorder relevant and marginally relevant systems
Speaker, Affiliation	Nikolaos Zygouras, University of Warwick
Date, Time	22 October 2014, 17:15-18:15
Location	HG G 43
Abstract	We consider statistical mechanics models defined on a lattice, in which disorder acts as an external “random field”. Such models are called disorder relevant, if arbitrarily weak disorder changes the qualitative properties of the model. We show that, when the homogeneous model has a non-trivial continuum limit, disorder relevance manifests itself via the convergence of the disordered model to a disordered continuum limit, whose partition function is given in terms of a Wiener chaos expansion. Such are the cases of the 1+1 directed polymer (DPRM), the disorder pinning model with renewal exponent $\alpha >1/2$ and the 2d critical random field Ising model. Changing some parameters of the models, e.g. the dimension $d$ in the $1+d$ DPRM or the exponent $\alpha$ in the case of the pinning model may produce models, which are insensitive to disorder (disorder irrelevance). The point where this transition happens ($d=2$ for DPRM or $\alpha=1/2$ for pinning) is called marginal and it is the most subtle to analyse. In particular, the Wiener chaos expansion characterising the continuum limit of the corresponding disorder relevant model fails to make sense. However, a scaling limit, which is universal among the different marginally relevant models, still exists. Based on Joint work with F. Caravenna and R. Sun.

Scaling limits of disorder relevant and marginally relevant systemsread_more

HG G 43

29 October 2014
17:15-18:15

Patrik Ferrari
Universität Bonn

Event Details

Seminar on Stochastic Processes

Title	Free energy fluctuations for directed polymers in 1+1 dimension
Speaker, Affiliation	Patrik Ferrari, Universität Bonn
Date, Time	29 October 2014, 17:15-18:15
Location	HG G 43
Abstract	The Kardar-Parisi-Zhang (KPZ) universality class includes directed polymers in random media in 1+1 dimension. According to the universality conjecture, for any finite temperature, the fluctuations of the free energy (e.g. for point-to-point) directed polymers is expected to be distributed as the (GUE) Tracy-Widom distribution in the limit of large system size. This distribution arose first in the context of random matrices. Detailed results as the fluctuation laws for models in the KPZ were, until recently, available only for "zero temperature models". We consider two models at positive temperature, a semi-discrete and the continuum directed polymer models, and determine the law of the free energy fluctuations.

Free energy fluctuations for directed polymers in 1+1 dimensionread_more

HG G 43

5 November 2014
17:15-18:15

Jamal Najim
Université de Marne la Vallée

Event Details

Seminar on Stochastic Processes

Title	Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges
Speaker, Affiliation	Jamal Najim, Université de Marne la Vallée
Date, Time	5 November 2014, 17:15-18:15
Location	HG G 43
Abstract	We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. For this matrix model, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions, we will show that the extremal eigenvalue which converge almost surely towards the edges of the support fluctuate according to the Tracy-Widom law at the scale $N^2/3$. Moreover, given several generic positive edges, we establish that the associated extremal eigenvalue fluctuations are asymptotically independent. Finally, when the leftmost edge is the origin, we prove that the smallest eigenvalue fluctuates according to the hard-edge Tracy-Widom law at the scale $N^2$ (Bessel kernel). As an application, an asymptotic study of the condition number of large correlated Wishart matrices is provided. Joint work with Walid Hachem (Telecom Paristech and CNRS, France) and Adrien Hardy (KTH - Royal Institute of Technology, Sweden).

Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edgesread_more

HG G 43

* 12 November 2014
15:15-16:15

Daisuke Shiraishi
University of Kyoto

Event Details

Seminar on Stochastic Processes

Title	Growth exponent for loop-erased random walk in three dimensions
Speaker, Affiliation	Daisuke Shiraishi, University of Kyoto
Date, Time	12 November 2014, 15:15-16:15
Location	HG G 19.2
Abstract	Let S be the simple random walk on Z^d started at the origin. We think of its path up to time n, say S[0,n], as a random graph. Three objects we discuss are the loop erasure of S[0,n], the graph distance and the effective resistance between the origin and S(n) on S[0,n]. All of them are generated by the path S[0,n] and its asymptotic behavior depends on the dimension. In the talk, I will explain what's known (or unknown) and what I proved about them.

Growth exponent for loop-erased random walk in three dimensionsread_more

HG G 19.2

19 November 2014
17:15-18:15

Charles Bordenave
Université de Toulouse

Event Details

Seminar on Stochastic Processes

Title	Non-backtracking spectrum of random graphs
Speaker, Affiliation	Charles Bordenave, Université de Toulouse
Date, Time	19 November 2014, 17:15-18:15
Location	HG G 43
Abstract	The non-backtracking matrix of a graph is a non-symmetric matrix on the oriented edge of a graph which has interesting algebraic properties and appears notably in connection with the Ihara zeta function and in some generalizations of Ramanujan graphs. It has also been used recently in the context of community detection. In this talk, we will study the largest eigenvalues of this matrix for the Erdos-Renyi graph G(n,c/n) and for simple inhomogeneous random graphs (stochastic block model). This is a joint work with Marc Lelarge and Laurent Massoulié.

Non-backtracking spectrum of random graphsread_more

HG G 43

3 December 2014
17:15-18:15

Christophe Garban
ENS Lyon

Event Details

Seminar on Stochastic Processes

Title	KPZ formula derived from Liouville heat kernel
Speaker, Affiliation	Christophe Garban, ENS Lyon
Date, Time	3 December 2014, 17:15-18:15
Location	HG G 43
Abstract	In this talk, I will give a new way to obtain the Knizhnik--Polyakov--Zamolodchikov (KPZ) formula of Liouville quantum gravity, based on the heat kernel of Liouville Brownian motion. This derivation of the KPZ formula was first suggested by F. David and M. Bauer in order to get a geometrically more intrinsic way of measuring the dimension of sets in Liouville quantum gravity. I will start this talk by a short introduction on Liouville quantum gravity and the KPZ relation. Joint work with N. Berestycki, R. Rhodes and V. Vargas.

KPZ formula derived from Liouville heat kernelread_more

HG G 43

10 December 2014
17:15-18:15

Jérémie Bettinelli
Institut Elie Cartan, Nancy

Event Details

Seminar on Stochastic Processes

Title	Geodesics in Brownian surfaces
Speaker, Affiliation	Jérémie Bettinelli, Institut Elie Cartan, Nancy
Date, Time	10 December 2014, 17:15-18:15
Location	Y27 H 25
Abstract	In this talk, we introduce a class of random metric spaces called Brownian surfaces, which generalize the famous Brownian map to the case of topologies more complicated than that of the sphere. More precisely, these random surfaces arise as the scaling limit of random maps on a given surface with a boundary. We will review the known results about these rather wild random metric spaces and we will particularly focus on the geodesics starting from a uniformly chosen random point. This allow to characterize some subsets of interest in terms of geodesics and, in particular, in terms of pairs of geodesics aiming at the same point and whose concatenation forms a loop not homotopic to 0. Our results generalize in particular the properties shown by Le Gall on geodesics in the Brownian map, although our approach is completely different.

Geodesics in Brownian surfacesread_more

Y27 H 25

17 December 2014
17:15-18:15

Benjamin Sudakov
ETH Zürich

Event Details

Seminar on Stochastic Processes

Title	Hamiltonicity problem in random graphs
Speaker, Affiliation	Benjamin Sudakov, ETH Zürich
Date, Time	17 December 2014, 17:15-18:15
Location	HG G 43
Abstract	A Hamilton cycle in a graph G is a cycle visiting each vertex of G exactly once, and a graph is called Hamiltonian if it contains a Hamilton cycle. Hamiltonicity, named after Sir Hamilton who studied it in the 1850s, is one of the most important and extensively studied concept in graph theory. The fundamental problem whether various random graph models contain a Hamilton cycle traces back to the original pioneering works of Erdos–Renyi introducing random graphs more than 50 years ago. In this talk we describe the techniques applied in the study of Hamiltonicity of random graphs, survey some classical results and present some new ones as well (focusing on the recent work with Krivelevich and Lubetzky).

Hamiltonicity problem in random graphsread_more

HG G 43

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Organizers: Jean Bertoin, Erwin Bolthausen, Ashkan Nikeghbali, Pierre Nolin, Martin Schweizer, Benjamin Shlein, Alain-Sol Sznitman

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