Seminar on Stochastic Processes

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Spring Semester 2022

Date / Time

Speaker

Title

Location

23 February 2022
17:15-18:15

Dr. Shuta Nakajima
Universität Basel

Event Details

Seminar on Stochastic Processes

Title	A variational formula for large deviations in First-passage percolation under tail estimates
Speaker, Affiliation	Dr. Shuta Nakajima, Universität Basel
Date, Time	23 February 2022, 17:15-18:15
Location	Y27 H12
Abstract	We consider first passage percolation with identical and independent weight distributions. In this paper, we study the upper tail large deviations under a tail assumption on the distribution. We prove that the corresponding rate function is described by the so-called discrete p-Capacity, and we study its asymptotic. The talk is based on joint work with Clement Cosco.

A variational formula for large deviations in First-passage percolation under tail estimatesread_more

Y27 H12

2 March 2022
17:15-18:15

Dr. Max Fathi
UFR de Mathématiques, Université de Paris

Event Details

Seminar on Stochastic Processes

Title	Stability of the spectral gap under a curvature-dimension condition
Speaker, Affiliation	Dr. Max Fathi, UFR de Mathématiques, Université de Paris
Date, Time	2 March 2022, 17:15-18:15
Location	Y27 H12
Abstract	A theorem of Lichnerowicz (1958) states that the spectral gap (or sharp Poincare constant) of a smooth n-dimensional Riemannian manifold with curvature bounded from below by n-1 is bounded by n, which is the spectral gap of the unit n-sphere. This bound has since been extended to metric-measure spaces satisfying a curvature-dimension condition. In this talk, I will present a result on stability of the bound: if a space has almost minimal spectral gap, then the pushforward of the volume measure by a normalized eigenfunction is close to a Beta distribution with parameter n/2, with a sharp estimate on the L1 optimal transport distance. Joint work with Ivan Gentil and Jordan Serres.

Stability of the spectral gap under a curvature-dimension conditionread_more

Y27 H12

16 March 2022
17:15-18:15

Dr. Giuseppe Genovese
University of Zurich

Event Details

Seminar on Stochastic Processes

Title	Reconstruction of a random pattern with restricted Boltzmann machines
Speaker, Affiliation	Dr. Giuseppe Genovese, University of Zurich
Date, Time	16 March 2022, 17:15-18:15
Location	Y27 H12
Abstract	A Restricted Boltzmann machine (RBM) is a Gibbs probability distribution of great theoretical and methodological relevance in machine learning. It is defined on a bipartite graph, with one layer (so-called visible) usually made of binary variables encoding the data, and a second ancillary layer (so-called hidden). One says that the RBM retrieves a pattern, if any algorithmic search initialised in proximity of it will not end up so far (or alternatively, the patterns and the local minima of the energy are close enough). I will present some recent results showing that the ability of a RBM to retrieve a random pattern depends on the choice of the distribution of the hidden layer. Indeed efficient retrieval is possible for distributions with a strict sub-Gaussian decay, while strict super-Gaussian tails give poor performance. The case of Gaussian tail (of which the Hopfield model is a special case) is critical and separates these two regimes.

Reconstruction of a random pattern with restricted Boltzmann machinesread_more

Y27 H12

23 March 2022
17:15-18:15

Dr. Jacopo Borga
Stanford University

Event Details

Seminar on Stochastic Processes

Title	The skew Brownian permuton: a new universal limit for random constrained permutations and its connections with Liouville quantum gravity
Speaker, Affiliation	Dr. Jacopo Borga, Stanford University
Date, Time	23 March 2022, 17:15-18:15
Location	Y27 H12
Abstract	Consider a large random permutation satisfying some constraints or biased according to some statistics. What does it look like? In this seminar we make sense of this question introducing the notion of permuton. Permuton convergence has been established for several models of random permutations in various works: we give an overview of some of these results, mainly focusing on the case of pattern-avoiding permutations. The main goal of the talk is to present a new family of universal limiting permutons, called skew Brownian permuton. This family includes (as particular cases) some already studied limiting permutons, such as the biased Brownian separable permuton and the Baxter permuton. We also show that some natural families of random constrained permutations converge to some new instances of the skew Brownian permuton. The construction of these new limiting objects will lead us to investigate an intriguing connection with some perturbed versions of the Tanaka SDE and the SDEs encoding skew Brownian motions. We finally explain how it is possible to construct these new limiting permutons directly from a Liouville quantum gravity decorated with two SLE curves. Building on the latter connection, we compute the density of the intensity measure of the Baxter permuton.

The skew Brownian permuton: a new universal limit for random constrained permutations and its connections with Liouville quantum gravityread_more

Y27 H12

30 March 2022
17:15-18:15

Dr. Peter Nejjar
IAM, Universität Bonn

Event Details

Seminar on Stochastic Processes

Title	Cutoff profile of ASEP on the segment
Speaker, Affiliation	Dr. Peter Nejjar, IAM, Universität Bonn
Date, Time	30 March 2022, 17:15-18:15
Location	Y27 H12
Abstract	We consider the asymmetric simple exclusion proces (ASEP), an interacting particle system that belongs to the so-called Kardar-Parisi-Zhang (KPZ) class of random growth models. Here we study how ASEP on the segment mixes to equilibrium. In particular, we obtain the cutoff window and the cutoff profile of ASEP, giving a precise description of how ASEP converges to equilibrium. Based on joint work with Alexey Bufetov.

Cutoff profile of ASEP on the segmentread_more

Y27 H12

6 April 2022
17:15-18:15

Dr. Gady Kozma
Weizmann Institute of Science

Event Details

Seminar on Stochastic Processes

Title	Harmonic functions with gradient converging to zero at infinity
Speaker, Affiliation	Dr. Gady Kozma, Weizmann Institute of Science
Date, Time	6 April 2022, 17:15-18:15
Location	Y27 H12
Abstract	For which finitely generated groups does there exist a non-constant (discrete) harmonic function whose gradient converges to zero at infinity? We will see a number of examples, including a connection to an open problem on two dimensional simple random walk. Joint work with Gidi Amir and Maria Gerasimova.

Harmonic functions with gradient converging to zero at infinityread_more

Y27 H12

13 April 2022
17:15-18:15

Hugo Vanneuville
Institut Fourier, Université Grenoble-Alpes

Event Details

Seminar on Stochastic Processes

Title	An unbounded nodal surface for 3D analytical Gaussian functions
Speaker, Affiliation	Hugo Vanneuville, Institut Fourier, Université Grenoble-Alpes
Date, Time	13 April 2022, 17:15-18:15
Location	Y27 H12
Abstract	Let f be the 3D Bargmann-Fock field, which is a Gaussian random analytic function from R^3 to R. We prove that a.s. there is an unbounded component in the nodal surface {f=0}. Since 0 is the percolation critical level for f restricted to the plane, this result is equivalent to the fact that the critical level strictly increases between 2D and 3D. This is a classical result for Bernoulli percolation. However, the classical proofs strongly rely on finite energy properties, which are not available for analytical functions. We propose a different approach, that is based on the continuous nature of the model.

An unbounded nodal surface for 3D analytical Gaussian functionsread_more

Y27 H12

27 April 2022
17:15-18:15

Dr. Michel Nassif
CERMICS, École des Ponts Paris Tech

Event Details

Seminar on Stochastic Processes

Title	Additive functionals of large Bienaymé-Galton-Watson trees
Speaker, Affiliation	Dr. Michel Nassif, CERMICS, École des Ponts Paris Tech
Date, Time	27 April 2022, 17:15-18:15
Location	Y27 H12
Abstract	We study the scaling limits of general additive functionals on size-conditioned Bienaymé-Galton-Watson trees. Many usual indices can be written in that form, e.g. the total path length, the Wiener index, and Shao and Sokal’s B1 index. We assume that the offspring distribution is critical with finite or infinite variance. We express the limit as a functional of the stable Lévy tree and study some of its properties. We also describe a phase transition when the toll function depends only on the size and the height of the tree.

Additive functionals of large Bienaymé-Galton-Watson treesread_more

Y27 H12

4 May 2022
17:15-18:15

Dr. William Da Silva
Universität Wien

Event Details

Seminar on Stochastic Processes

Title	Multitype growth-fragmentation and planar excursions
Speaker, Affiliation	Dr. William Da Silva, Universität Wien
Date, Time	4 May 2022, 17:15-18:15
Location	Y27 H12
Abstract	Growth-fragmentation processes are branching processes which model the evolution of a cloud of atoms which may grow and dislocate as time evolves. In a pioneering work, Bertoin, Budd, Curien and Kortchemski describe the branching structure of these particle systems, as well as a particular family obtained in the scaling limit from a Markov peeling process of large random planar maps. We first construct, on a half-planar excursion whose real part is a stable process, a signed version of some of the growth-fragmentation processes revealed by Bertoin, Budd, Curien and Kortchemski. This will prompt us to define a framework for growth-fragmentation with signs, or more generally with types, for which we provide a spinal description. This talk is partly based on joint works with Élie Aïdékon (Fudan University) and Juan Carlos Pardo (CIMAT).

Multitype growth-fragmentation and planar excursionsread_more

Y27 H12

11 May 2022
17:15-18:15

Prof. Dr. Ioan Manolescu
Universität Freiburg, Switzerland

Event Details

Seminar on Stochastic Processes

Title	Rotational invariance in planar FK-percolation
Speaker, Affiliation	Prof. Dr. Ioan Manolescu, Universität Freiburg, Switzerland
Date, Time	11 May 2022, 17:15-18:15
Location	Y27 H12
Abstract	We prove the asymptotic rotational invariance of the critical FK-percolation model on the square lattice with any cluster-weight between 1 and 4. These models are expected to exhibit conformally invariant scaling limits that depend on the cluster weight, thus covering a continuum of universality classes. The rotation invariance of the scaling limit is a strong indication of the wider conformal invariance, and may indeed serve as a stepping stone to the latter. Our result is obtained via a universality theorem for FK-percolation on certain isoradial lattices. This in turn is proved via the star-triangle (or Yang-Baxter) transformation, which may be used to gradually change the square lattice into any of these isoradial lattices, while preserving certain features of the model. It was previously proved that throughout this transformation, the large scale geometry of the model is distorted by at most a limited amount. In the present work we argue that the distortion becomes insignificant as the scale increases. This hinges on the interplay between the inhomogeneity of isoradial models and their embeddings, which compensate each other at large scales. As a byproduct, we obtain the asymptotic rotational invariance also for models related to FK-percolation, such as the Potts and six-vertex ones. Moreover, the approach described here is fairly generic and may be adapted to other systems which possess a Yang-Baxter transformation. Based on joint work with Hugo Duminil-Copin, Karol Kajetan Kozlowski, Dmitry Krachun and Mendes Oulamara.

Rotational invariance in planar FK-percolationread_more

Y27 H12

18 May 2022
17:15-18:15

Prof. Dr. Pierre-François Rodriguez
Imperial College

Event Details

Seminar on Stochastic Processes

Title	Scaling limit of the two-dimensional discrete Gaussian model at high temperatures
Speaker, Affiliation	Prof. Dr. Pierre-François Rodriguez, Imperial College
Date, Time	18 May 2022, 17:15-18:15
Location	Y27 H12
Abstract	The discrete Gaussian model is the Gaussian free field conditioned to be integer-valued. As shown rigorously in celebrated work of Fröhlich and Spencer, the conditioning causes a Berezinskii-Kosterlitz-Thouless type phase transition in two dimensions: the interface exhibits a localized low-temperature and a delocalized (rough) high-temperature phase. I will report on recent work with R. Bauerschmidt and J. Park identifying the macroscopic scaling limit of this field in the rough phase, at sufficiently high temperature.

Scaling limit of the two-dimensional discrete Gaussian model at high temperaturesread_more

Y27 H12

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