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Seminar on stochastic processes

Members of the probability group are involved in co-organizing remote specialized seminars that take place on Tuesdays and Thursdays:

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Autumn Semester 2017

Date / Time Speaker Title Location
27 September 2017
17:00-18:00 		Vincent Beffara
Université de Grenoble
Event Details

Seminar on Stochastic Processes

Title Percolation in random analytic functions
Speaker, Affiliation Vincent Beffara, Université de Grenoble
Date, Time 27 September 2017, 17:00-18:00
Location HG G 43
Percolation in random analytic functions
HG G 43
4 October 2017
17:00-18:00 		Nicolas Perkowski
Humboldt Universität Berlin
Event Details

Seminar on Stochastic Processes

Title Particle approximation and martingale formulation of the stochastic Burgers equation
Speaker, Affiliation Nicolas Perkowski, Humboldt Universität Berlin
Date, Time 4 October 2017, 17:00-18:00
Location HG G 43
Abstract I will present a martingale formulation for the stationary stochastic Burgers equation, a singular stochastic PDE that is conjectured to be “weakly universal” in the sense that a wide range of weakly asymmetric conservative systems converges to this equation, and I will discuss how to apply the martingale formulation in order to prove this weak universality. Special emphasis will be given to the exclusion process in contact with reservoirs, which converges under the right assumptions to Burgers equation with Dirichlet boundary conditions, an equation for which the correct mathematical formulation is surprisingly subtle.
Particle approximation and martingale formulation of the stochastic Burgers equationread_more
HG G 43
11 October 2017
17:00-18:00 		Ewain Gwynne
MIT
Event Details

Seminar on Stochastic Processes

Title The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity
Speaker, Affiliation Ewain Gwynne, MIT
Date, Time 11 October 2017, 17:00-18:00
Location HG G 43
Abstract We discuss the first proof that the discrete conformal embeddings of certain random planar maps approximate their continuum counterparts. In particular, we show that the Tutte embeddings (a.k.a. harmonic or barycentric embeddings) of the mated-CRT maps maps converge to $\gamma$-Liouville quantum gravity (LQG). Mated-CRT maps are discretized matings of correlated continuum random trees, and $\gamma$ ranges from 0 to 2 as one varies the correlation parameter. We also show that the associated space-filling path on the embedded map converges to space-filling SLE (in the annealed sense) and that the embedded random walk converges to Brownian motion (in the quenched sense). Our proof proceeds by way of a quenched scaling limit result for random walk in a certain inhomogeneous random environment. The mated-CRT map provides a coarse-grained approximation of other random planar maps which can be bijectively encoded by pairs of discrete random trees---e.g., the UIPT, spanning-tree weighted maps, and bipolar-oriented maps---so our results suggest a possible approach for proving that embeddings of these planar maps also converge to LQG. Based on joint work with Jason Miller and Scott Sheffield https://arxiv.org/abs/1705.11161.
The Tutte embedding of the mated-CRT map converges to Liouville quantum gravityread_more
HG G 43
18 October 2017
17:00-18:00 		Antonio Auffinger
Northwestern University
Event Details

Seminar on Stochastic Processes

Title The SK model is FRSB (full replica symmetry breaking) at zero temperature
Speaker, Affiliation Antonio Auffinger, Northwestern University
Date, Time 18 October 2017, 17:00-18:00
Location HG G 43
Abstract In the early ‘80s, Giorgio Parisi wrote a series of ground breaking papers where he introduced the notion of replica symmetry breaking. His powerful insight allowed him to predict a solution for the SK model by breaking the symmetry of replicas infinitely many times. In this talk, we will prove Parisi's prediction at zero temperature for the mixed p-spin model, a generalization of the SK model. We will show that at zero temperature the functional order parameter is full-step replica symmetry breaking (FRSB). We will also describe the importance of this result for the description of the energy landscape. Based on works with Wei-Kuo Chen (U. of Minnesota) and Qiang Zeng (Northwestern U.).
The SK model is FRSB (full replica symmetry breaking) at zero temperatureread_more
HG G 43
25 October 2017
17:00-18:00 		Ellen Powell
ETH Zürich
Event Details

Seminar on Stochastic Processes

Title An invariance principle for branching diffusions in bounded domains
Speaker, Affiliation Ellen Powell, ETH Zürich
Date, Time 25 October 2017, 17:00-18:00
Location HG G 43
Abstract I will discuss branching diffusions in a bounded domain D of R^d, in which particles are killed upon hitting the boundary. It is known that such a process undergoes a phase transition when the branching rate exceeds a critical value: a multiple of the first eigenvalue of the diffusion. The main focus of this talk will be the genealogical tree associated with the critical process, when it is conditioned to survive. I will prove that this converges to Aldous' Continuum Random Tree under appropriate rescaling.
An invariance principle for branching diffusions in bounded domainsread_more
HG G 43
1 November 2017
17:00-18:00 		Sasha Sodin
Queen Mary University of London
Event Details

Seminar on Stochastic Processes

Title Non-Hermitian random Schroedinger operators
Speaker, Affiliation Sasha Sodin, Queen Mary University of London
Date, Time 1 November 2017, 17:00-18:00
Location HG G 43
Abstract Non-Hermitian random Schroedinger operators were put forth in the mid 1990-s by Hatano and Nelson, and mathematically studied by Goldsheid and Khoruzhenko. We shall discuss some results obtained in a recent joint work with I. Goldsheid.
Non-Hermitian random Schroedinger operatorsread_more
HG G 43
8 November 2017
17:00-18:00 		Alberto Chiarini
ETH Zürich
Event Details

Seminar on Stochastic Processes

Title Invariance principle for the degenerate dynamic random conductance model
Speaker, Affiliation Alberto Chiarini, ETH Zürich
Date, Time 8 November 2017, 17:00-18:00
Location HG G 43
Abstract After the brilliant result of Papanicolau and Varadhan (1979) in the case of bounded stationary and ergodic environments, there has been a recent upsurge in the research of quenched homogenization in random media. In particular, to identify the optimal conditions that a general stationary and ergodic environment must satisfy in order to obtain the convergence to a non-degenerate Brownian motion, is still an open problem. In this talk, we study a continuous-time random walk on $mathbb{Z}^d$ in an environment of dynamic random conductances. We assume that the law of the conductances is ergodic and stationary with respect to space-time shifts. We prove a quenched invariance principle for the random walk under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by the celebrated Moser’s iteration scheme. This is joint work with S. Andres, J-D. Deuschel and M. Slowik.
Invariance principle for the degenerate dynamic random conductance modelread_more
HG G 43
15 November 2017
17:00-18:00 		Noemi Kurt
TU Berlin
Event Details

Seminar on Stochastic Processes

Title An individual-based model for the Lenski experiment, and the deceleration of the relative fitness
Speaker, Affiliation Noemi Kurt, TU Berlin
Date, Time 15 November 2017, 17:00-18:00
Location HG G 43
Abstract The Lenski experiment investigates the long-term evolution of bacterial populations. Its design allows the direct comparison of the reproductive fitness of an evolved strain with its founder ancestor. It was observed by Wiser et al. (2013) that the mean fitness over time increases sublinearly, a behaviour which in the biological literature is commonly attributed to effects like clonal interference or epistasis. In this talk we present an individual-based probabilistic model that captures essential features of the design of the Lenski experiment. We assume that each beneficial mutation increases the individual reproduction rate by a fixed amount, which corresponds a priori to the absence of epistasis. Using an approximation by near-critical Galton-Watson processes, we prove that under some assumptions on the model parameters which exclude clonal interference, the relative fitness process derived from the microscopic model converges, after suitable rescaling, in the large population limit to a power law function. This is joint work with Adrián González Casanova, Anton Wakolbinger, and Linglong Yuan.
An individual-based model for the Lenski experiment, and the deceleration of the relative fitnessread_more (CANCELLED)
HG G 43
22 November 2017
17:00-18:00 		Xin Sun
Columbia University
Event Details

Seminar on Stochastic Processes

Title On the conformal structure of random planar maps
Speaker, Affiliation Xin Sun, Columbia University
Date, Time 22 November 2017, 17:00-18:00
Location HG G 43
Abstract Liouville quantum gravity is a random surface which is conjectured to describe the scaling limit of conformally embedded random planar maps. We present some progress on a program towards proving this conjecture in the case of uniform triangulations, using a percolation-based embedding which we call the Cardy embedding. Based on joint works with Bernardi, Garban, Gwynne, Holden, Lawler, Li, Miller, Sepulveda, and Sheffield.
On the conformal structure of random planar mapsread_more
HG G 43
29 November 2017
17:00-18:00 		Alexander Drewitz
Universität Köln
Event Details

Seminar on Stochastic Processes

Title Sign clusters of the Gaussian free field percolate on $\mathbb Z^d$, $d \ge 3$
Speaker, Affiliation Alexander Drewitz, Universität Köln
Date, Time 29 November 2017, 17:00-18:00
Location HG G 43
Abstract We consider level set percolation for the Gaussian free field on the Euclidean lattice in dimensions larger than or equal to three. It had previously been shown by Bricmont, Lebowitz, and Maes that the critical level is non-negative in any dimension and finite in dimension three. Rodriguez and Sznitman have extended this result by proving that it is finite in all dimensions, and positive in all large enough dimensions. We show that the critical parameter is positive in any dimension larger than or equal to three. In particular, this entails the percolation of sign clusters of the Gaussian free field. This talk is based on joint work with A. Prévost (Köln) and P.-F. Rodriguez (Los Angeles).
Sign clusters of the Gaussian free field percolate on $\mathbb Z^d$, $d \ge 3$read_more
HG G 43

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