Seminar on Stochastic Processes

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Spring Semester 2018

Date / Time

Speaker

Title

Location

21 February 2018
17:15-19:00

Jian Ding
Wharton University of Pennsylvania

Event Details

Seminar on Stochastic Processes

Title	Random walks among Bernoulli obstacles
Speaker, Affiliation	Jian Ding, Wharton University of Pennsylvania
Date, Time	21 February 2018, 17:15-19:00
Location	Y27 H 25
Abstract	We consider random walks on Z^d (for d\geq 2) with random Bernoulli obstacles, where the random walk will be killed upon hitting an obstacle. We will present some recent results on the behavior of the random walk conditioned on survival for a large time. This is based on joint works with Changji Xu, and a work with Ryoki Fukushima, Rongfeng Sun and Changji Xu.

Random walks among Bernoulli obstaclesread_more

Y27 H 25

28 February 2018
17:15-18:15

Aser Cortines Peixoto
Universität Zürich

Event Details

Seminar on Stochastic Processes

Title	Dynamical Freezing in a Spin Glass System with Logarithmic Correlations
Speaker, Affiliation	Aser Cortines Peixoto, Universität Zürich
Date, Time	28 February 2018, 17:15-18:15
Location	Y27 H 12
Abstract	We consider a continuous time random walk on the two-dimensional discrete torus, whose motion is governed by the discrete Gaussian free field on the corresponding box acting as a potential. More precisely, at any vertex the walk waits an exponentially distributed time with mean given by the exponential of the field and then jumps to one of its neighbours, chosen uniformly at random. We prove that throughout the low-temperature regime and at in-equilibrium time scales, the process admits a scaling limit as a spatial K-process driven by a random trapping landscape, which is explicitly related to the limiting extremal process of the field. This demonstrates dynamical freezing in a spin glass system with logarithmically correlated energy levels.

Dynamical Freezing in a Spin Glass System with Logarithmic Correlationsread_more

Y27 H 12

7 March 2018
17:15-18:15

Guillaume Rémy
ENS Paris

Event Details

Seminar on Stochastic Processes

Title	The Fyodorov-Bouchaud formula and Liouville conformal field theory
Speaker, Affiliation	Guillaume Rémy, ENS Paris
Date, Time	7 March 2018, 17:15-18:15
Location	Y27 H 12
Abstract	Starting from the restriction of a 2d Gaussian free field (GFF) to the unit circle one can define a Gaussian multiplicative chaos (GMC) measure whose density is formally given by the exponential of the GFF. In 2008 Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of this GMC. In this talk we will give a rigorous proof of this formula. Our method is inspired by the technology developed by Kupiainen, Rhodes and Vargas to derive the DOZZ formula in the context of Liouville conformal field theory on the Riemann sphere. In our case the key observation is that the negative moments of the total mass of GMC on the circle determine its law and are equal to one-point correlation functions of Liouville theory in the unit disk. Finally we will discuss applications in random matrix theory, asymptotics of the maximum of the GFF, and tail expansions of GMC.

The Fyodorov-Bouchaud formula and Liouville conformal field theoryread_more

Y27 H 12

14 March 2018
17:15-18:15

Noemi Kurt
Technische Universität 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, Technische Universität Berlin
Date, Time	14 March 2018, 17:15-18:15
Location	Y27 H 12
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

Y27 H 12

21 March 2018
17:15-18:15

Loïc Richier
Ecole Polytechnique, Palaiseau

Event Details

Seminar on Stochastic Processes

Title	Some properties of discrete stable maps
Speaker, Affiliation	Loïc Richier, Ecole Polytechnique, Palaiseau
Date, Time	21 March 2018, 17:15-18:15
Location	Y27 H 12
Abstract	The purpose of this talk is to discuss some properties of random planar maps such that the degree of a typical face falls within the domain of attraction of a stable distribution with parameter $\alpha\in(1,2)$. These maps, that we call discrete stable, have attracted a lot of attention. In 2011, Le Gall and Miermont proved that discrete stable maps admit subsequential scaling limits, suggesting the existence of a stable counterpart to the Brownian map. At the same time, Borot, Bouttier and Guitter established a connection between discrete stable maps and the O(n) loop models on planar maps. In the first part of the talk, we will investigate the scaling limits of large faces of discrete stable maps (or, equivalently, large loops in the O(n) model). The motivation comes from a conjecture stating that these large faces are self-intersecting in the so-called dense regime $\alpha\in(1,3/2)$, and self-avoiding in the so-called dilute regime $\alpha\in(3/2,2)$. In the second part, we will deal with the bond percolation model on discrete stable maps in the dilute regime. We will discuss a duality property showing that at criticality, the open percolation cluster of the origin is itself a discrete stable map in the dense regime, with explicit parameter. This result is inspired by recent work of Miller, Sheffield and Werner, who established such a duality property in Conformal Loop Ensembles. To conclude, we will mention other results concerning the bond percolation model, such as the scaling limits of percolation clusters and the sharpness of the phase transition. Partially joint work with Nicolas Curien and Igor Kortchemski.

Some properties of discrete stable mapsread_more

Y27 H 12

28 March 2018
17:15-18:15

Nathanaël Enriquez
Université Paris-Sud

Event Details

Seminar on Stochastic Processes

Title	Two equivalent versions of the Riemann Hypothesis
Speaker, Affiliation	Nathanaël Enriquez, Université Paris-Sud
Date, Time	28 March 2018, 17:15-18:15
Location	Y27 H 12
Abstract	I will present in this talk two equivalent versions of the Riemann Hypothesis. The first one concerns the probability for two iid integers with geometric distribution to be co-prime. In this respect, it can be viewed as an analog of von Koch's result about the probability of an integer to be prime. The second one, which apparently has no connection with the first one, concerns the asymptotic number of convex polygonal lines with integer vertices, joining the origin to the point of coordinates $(n,n)$. This work was done with J. Bureaux.

Two equivalent versions of the Riemann Hypothesisread_more

Y27 H 12

11 April 2018
17:15-18:15

Laszlo Erdös
IST Austria

Event Details

Seminar on Stochastic Processes

Title	Spectral rigidity for addition of random matrices
Speaker, Affiliation	Laszlo Erdös, IST Austria
Date, Time	11 April 2018, 17:15-18:15
Location	Y27 H 12
Abstract	A classical theorem of Voiculescu asserts that the eigenvalue of density of the sum of two hermitian matrices in a random relative basis is given by the free convolution. We show that this result holds down to the smallest local scale including the spectral edges. Our technique extends to certain non-hermitian situations as well and provides a local version of the single ring theorem on the optimal scale as well. The talk is based upon joint works with Kevin Schnelli and Zhigang Bao.

Spectral rigidity for addition of random matricesread_more

Y27 H 12

18 April 2018
17:15-18:15

Benny Sudakov
ETH Zürich

Event Details

Seminar on Stochastic Processes

Title	Asymptotics in bond percolation on expanders
Speaker, Affiliation	Benny Sudakov, ETH Zürich
Date, Time	18 April 2018, 17:15-18:15
Location	Y27 H 12
Abstract	The evolution of the largest connected component has been studied intensely in a variety of random graph processes, starting with celebrated work of Erdös-Renyi, who in 1960 proved that a random subgraph of an $n$-vertex complete graph undergoes a phase transition at edge probability $1/n$ when, asymptotically almost surely, a linear-sized ("giant") component appears. In this talk we consider edge percolation on a family of high-girth $d$-regular expanders. Alon-Benjamini-Stacey in 2004 established that a critical probability for the appearance of a giant component in this case is $p_c=1/(d-1)$. Our main result recovers the sharp asymptotics, similar to the classical Erdos-Renyi result, of the size and degree distribution of the vertices in the giant component at any $p>p_c$. On the other hand we show that, unlike the situation in the classical random graph case, the second largest component in edge percolation on a regular expander, even with an arbitrarily large girth, can have size at least $n^a$ for any fixed $a<1$. Several related results will be discussed as well. Joint work with Krivelevich and Lubetzky.

Asymptotics in bond percolation on expandersread_more

Y27 H 12

25 April 2018
17:15-18:15

Noam Berger
TU München

Event Details

Seminar on Stochastic Processes

Title	Monotonicity for random walk among random conductances with a drift
Speaker, Affiliation	Noam Berger, TU München
Date, Time	25 April 2018, 17:15-18:15
Location	Y27 H 12
Abstract	We investigate the speed of a random walk among i.i.d. uniformly elliptic conductances in the presence of a drift, and ask whether the speed is a monotone function of the drift. In cases with large disorder, it is not hard to produce examples where the speed is not monotone. Our main result is that in low disorder the speed is indeed monotone. Based on joint work with Nina Gantert and Jan Nagel.

Monotonicity for random walk among random conductances with a driftread_more

Y27 H 12

2 May 2018
17:15-18:15

Tom Hutchcroft
University of Cambridge

Event Details

Seminar on Stochastic Processes

Title	Percolation on hyperbolic graphs
Speaker, Affiliation	Tom Hutchcroft, University of Cambridge
Date, Time	2 May 2018, 17:15-18:15
Location	Y27 H 12
Abstract	In Bernoulli bond percolation, we independently decide to delete or retain edges of a graph with retention probability p. On most infinite graphs, percolation undergoes a phase transition in the sense that there exists a critical parameter 0 < p_c < 1 such that below p_c there is no infinite connected component, and above p_c there is some infinite connected component. Benjamini and Schramm (1996) conjectured that on any nonamenable transitive graph, percolation also undergoes a second phase transition from non-uniqueness to uniqueness of the infinite cluster: That is, there exists 0 < p_c < p_u \leq 1 such if p_c < p < p_u then there are infinitely many infinite clusters, while if p > p_u there is a unique infinite cluster. In this talk, I will describe a proof of this conjecture under the additional assumption that the graph in question is Gromov hyperbolic. The proof will also establish that percolation on any Gromov hyperbolic graph has mean-field critical exponents.

Percolation on hyperbolic graphsread_more

Y27 H 12

9 May 2018
17:15-18:15

Rob Morris
IMPA, Brazil

Event Details

Seminar on Stochastic Processes

Title	The sharp threshold for making squares
Speaker, Affiliation	Rob Morris, IMPA, Brazil
Date, Time	9 May 2018, 17:15-18:15
Location	Y27 H 12
Abstract	Many of the fastest known algorithms for factoring large integers rely on finding subsequences of randomly generated sequences of integers whose product is a perfect square. Motivated by this, at the 1994 ICM Pomerance posed the problem of determining the threshold of the event that a random sequence of $N$ integers, each chosen uniformly from the set $\{1,\dots,x\}$, contains a subsequence, the product of whose elements is a perfect square. In 1996, Pomerance gave good bounds on this threshold and also conjectured that it is sharp. A few years ago, in major breakthrough, Croot, Granville, Pemantle and Tetali significantly improved these bounds, and stated a conjecture as to the location of this sharp threshold. In the talk we will discuss a proof of their conjecture, which combines techniques from number theory and probabilistic combinatorics. In particular, at the heart of the proof lies a self-correcting random process of non-uniform hypergraphs. Joint work with Paul Balister and Béla Bollobás.

The sharp threshold for making squaresread_more

Y27 H 12

23 May 2018
17:15-18:15

David Belius
Universität Zürich

Event Details

Seminar on Stochastic Processes

Title	TAP-Plefka variational principle for mean field spin glasses
Speaker, Affiliation	David Belius, Universität Zürich
Date, Time	23 May 2018, 17:15-18:15
Location	Y27 H 12
Abstract	In this talk I will recall the Thouless-Anderson-Palmer (TAP) approach to the Sherrington-Kirkpatrick model from one of the earliest papers on this model, and describe how it can be reinterpreted as a variational principle in the spirit of the Gibbs variational principle and the Bragg-Williams approximation. Furthermore I will present a rigorous proof of this TAP-Plefka variational principle in the case of the spherical Sherrington-Kirkpatrick model.

TAP-Plefka variational principle for mean field spin glassesread_more

Y27 H 12

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