Seminar on Stochastic Processes

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Autumn Semester 2016

Date / Time

Speaker

Title

Location

21 September 2016
17:15-18:15

Adrien Kassel
ETH Zürich

Event Details

Seminar on Stochastic Processes

Title	Covariant Symanzik identities
Speaker, Affiliation	Adrien Kassel, ETH Zürich
Date, Time	21 September 2016, 17:15-18:15
Location	HG G 43
Abstract	The goal of this talk is to discuss some stochastic processes associated to a discrete vector bundle of arbitrary rank endowed with a unitary connection (the geometric setup for lattice gauge theory). Specifically, I will describe how the distributions of holonomies of random paths and some covariant Markov random fields are naturally related. This extends classical isomorphism theorems for local time (Dynkin, Eisenbaum, Le Jan, Sznitman) which originated from the ideas of Symanzik in constructive quantum field theory. Joint work with Thierry Lévy (Univ. Paris 6).

Covariant Symanzik identitiesread_more

HG G 43

28 September 2016
17:15-18:15

Pierre Monmarché
Université de Neuchâtel

Event Details

Seminar on Stochastic Processes

Title	Mean Field kinetic particles and the Vlasov-Fokker-Planck equation
Speaker, Affiliation	Pierre Monmarché, Université de Neuchâtel
Date, Time	28 September 2016, 17:15-18:15
Location	HG G 43
Abstract	Consider N kinetic particles in an external potential with a mean field interaction (i.e. each particle is equally influenced by all the others at once). As N go to infinity, two given particles are less and less correlated (this is the propagation of chaos phenomenon), and the density probability of a given particle converges to the solution of a non-linear PDE, the Vlasov-Fokker-Planck (VFP) equation. When the external potential is convex, we prove the rate of convergence to equilibrium in large time for the whole cloud of particles does not, in fact, depend on N, which yields: first, a rate of convergence for the VFP equation, and second, uniform in time propagation of chaos estimates.

Mean Field kinetic particles and the Vlasov-Fokker-Planck equationread_more

HG G 43

5 October 2016
17:15-18:15

Igor Kortchemski
École Polytechnique, Palaiseau

Event Details

Seminar on Stochastic Processes

Title	Lévy processes and random planar structures
Speaker, Affiliation	Igor Kortchemski, École Polytechnique, Palaiseau
Date, Time	5 October 2016, 17:15-18:15
Location	HG G 43
Abstract	We will discuss several models of random discrete planar models (trees, non-crossing configurations, maps) whose asymptotic behavior, as their size grows, is closely related to Lévy processes.

Lévy processes and random planar structuresread_more

HG G 43

12 October 2016
17:15-18:15

Christian Webb
University of Aalto

Event Details

Seminar on Stochastic Processes

Title	Multiplicative chaos in number theory and random matrix theory
Speaker, Affiliation	Christian Webb, University of Aalto
Date, Time	12 October 2016, 17:15-18:15
Location	HG G 43

Multiplicative chaos in number theory and random matrix theory

HG G 43

19 October 2016
17:15-18:15

Bénédicte Haas
Université Paris Nord

Event Details

Seminar on Stochastic Processes

Title	Random trees constructed by aggregation
Speaker, Affiliation	Bénédicte Haas, Université Paris Nord
Date, Time	19 October 2016, 17:15-18:15
Location	HG G 43
Abstract	We study a general procedure that builds random continuous trees by gluing recursively a new branch on a uniform point of the pre-existing tree. This encompasses the famous "line-breaking" construction of the Brownian tree of Aldous. Our aim is to see how the sequence of lengths of branches influences some geometric properties of the limiting tree, such as compactness, height and Hausdorff dimension. This is partly based on a joint work with Nicolas Curien (Université Paris-Sud). At the end of the talk I will also mention some extensions recently obtained by Delphin Sénizergues (Université Paris-Nord) of these results to the random gluing of (random) pointed metric spaces.

Random trees constructed by aggregationread_more

HG G 43

26 October 2016
17:15-18:15

Pierre-Loic Méliot
Université Orsay

Event Details

Seminar on Stochastic Processes

Title	Fluctuations for graphon and permuton models
Speaker, Affiliation	Pierre-Loic Méliot, Université Orsay
Date, Time	26 October 2016, 17:15-18:15
Location	HG G 43

Fluctuations for graphon and permuton models

HG G 43

2 November 2016
17:15-18:15

Hugo Duminil-Copin
Université de Genève and IHES

Event Details

Seminar on Stochastic Processes

Title	Sharpness results via randomized algorithms
Speaker, Affiliation	Hugo Duminil-Copin, Université de Genève and IHES
Date, Time	2 November 2016, 17:15-18:15
Location	HG G 43
Abstract	In this talk, we will provide a new (short) proof of exponential decay/mean-field lower bound for Bernoulli percolation based on randomized algorithms. This proof does not rely on the domain Markov property or the BK inequality. In particular, it extends to continuum percolation models such as Boolean and Voronoi percolation in arbitrary dimension, thus providing the first proof of sharpness of the phase transition for these models.

Sharpness results via randomized algorithmsread_more

HG G 43

9 November 2016
17:15-18:15

Alexander Moll
IHES

Event Details

Seminar on Stochastic Processes

Title	Random partitions and the correspondence principle
Speaker, Affiliation	Alexander Moll, IHES
Date, Time	9 November 2016, 17:15-18:15
Location	HG G 43
Abstract	The classical Hopf-Burgers equation v_t + v v_x = 0 with periodic boundary conditions is a completely integrable system for v: T -> R on the unit circle T. Its hierarchy of commuting conservation laws can be quantized, and the resulting quantum Hamiltonians are simultaneously diagonalized on Schur polynomials. The decomposition of a Glauber coherent state around a classical configuration v into eigenfunctions defines a Schur measure on partitions with specializations determined by v. In the semi-classical limit, we prove a concentration of profiles of Young diagrams around the push-forward along v: T-> R of the uniform measure on T, recovering Okounkov (2003) and in particular Vershik-Kerov (1977) for v(x) = 2 cos x. Moreover, the global fluctuations converge to a Gaussian process, the push-forward along v: T -> R of H^{1/2} noise on T. These results are as predicted by the quantum-classical correspondence principle, although no large deviation principle is currently known for arbitrary Schur measures. Our proofs exploit the integrability of the model via spectral theory of Toeplitz operators as described by Nazarov-Sklyanin (2013) and extend to Jack measures and the periodic Benjamin-Ono system. The talk will begin with the asymptotics of a single Poisson random variable at high intensity and the correspondence principle for a single harmonic oscillator.

Random partitions and the correspondence principleread_more

HG G 43

16 November 2016
17:15-18:15

Yan Fyodorov
King's College London

Event Details

Seminar on Stochastic Processes

Title	How many stable equilibria will a large complex system have?
Speaker, Affiliation	Yan Fyodorov, King's College London
Date, Time	16 November 2016, 17:15-18:15
Location	HG G 43
Abstract	We aim to provide the quantitative answer to the classical question posed by Robert May (1972) "Will a Large Complex System be Stable?". To this end we analyse a generic autonomous nonlinear system of N>>1 randomly coupled ODE's describing degrees of freedom relaxing with the common rate \mu>0. We show that with decreasing rate \mu such systems experience an abrupt transition at some critical value \mu=\mu_C from a trivial phase portrait with a single stable equilibrium into a topologically non-trivial 'absolute instability' regime for \mu_B<\mu<\mu_C where equilibria are exponentially abundant, but typically all of them are unstable. Finally, at even smaller relaxation rate \mu<\mu_B stable equilibria become exponentially abundant, but their fraction to totality of all equilibria remains exponentially small. The revealed picture goes much beyond the May's linear analysis and is expected to be of relevance in the applications of complex systems to ecology, population biology, neural network theory and other areas. The presentation will be based on joint works with Gerard Ben Arous and Boris Khoruzhenko.

How many stable equilibria will a large complex system have?read_more

HG G 43

23 November 2016
17:15-18:15

Maxime Gagnebin
Université de Genève

Event Details

Seminar on Stochastic Processes

Title	Decay of correlations in the XY model
Speaker, Affiliation	Maxime Gagnebin, Université de Genève
Date, Time	23 November 2016, 17:15-18:15
Location	HG G 43
Abstract	We will review the technic used by McBrian and Spencer to obtain an upper bound on the decayof correlations in the XY model. We will then see how it can be modified to prove the same bound in a general class of O(N)-symmetric models, with long range and non-smooth interaction. If time permits, we will see how this is related to the delocalization of a height function and the techniques used in that setting. Based on a joint work with Yvan Velenik.

Decay of correlations in the XY modelread_more

HG G 43

30 November 2016
17:15-18:15

Brent Werness
ETH Zürich

Event Details

Seminar on Stochastic Processes

Title	Constructions of the Gaussian free field and fast simulation of Schramm-Loewner evolutions
Speaker, Affiliation	Brent Werness, ETH Zürich
Date, Time	30 November 2016, 17:15-18:15
Location	HG G 43
Abstract	The Schramm–Loewner evolutions (SLE) are a family of stochastic processes which describe the scaling limits of curves which occur in two-dimensional critical statistical physics models. SLEs have had found great success in this task, greatly enhancing our understanding of the geometry of these curves. Despite this, it is rather dicult to produce large, high-fidelity simulations of the process, with the standard simulation method discretizing the construction of SLE through the Loewner ODE providing a quadratic time algorithm in the length of the curve. Work of Sheffield and Miller has provided an alternate description of SLE with the curve interpreted a flow line of the vector field obtained by exponentiating a Gaussian free field. In this talk, I will describe a new method of approximately sampling a Gaussian free field, and show how this allows us to more efficiently simulate an SLE curve resulting in an algorithm linear in the length of the curve.

Constructions of the Gaussian free field and fast simulation of Schramm-Loewner evolutionsread_more

HG G 43

7 December 2016
17:15-18:15

Cyril Labbé
Université Paris Dauphine

Event Details

Seminar on Stochastic Processes

Title	Cutoff phenomenon for the asymmetric simple exclusion process and the biased card shuffling
Speaker, Affiliation	Cyril Labbé, Université Paris Dauphine
Date, Time	7 December 2016, 17:15-18:15
Location	HG G 43
Abstract	I will present an asymptotic result for the mixing times associated with two Markov chains: the asymmetric simple exclusion process and the biased card shuffling. This result asserts that the distance to equilibrium of the chain, starting from the « worst » initial condition, suddenly drops to 0 at a deterministic time (asymptotically in the size of the system): this is what is usually called the cutoff phenomenon. This is a joint work with Hubert Lacoin.

Cutoff phenomenon for the asymmetric simple exclusion process and the biased card shufflingread_more

HG G 43

14 December 2016
17:15-18:15

Jean-Christophe Mourrat
ENS-Lyon

Event Details

Seminar on Stochastic Processes

Title	Quantitative results in stochastic homogenization
Speaker, Affiliation	Jean-Christophe Mourrat, ENS-Lyon
Date, Time	14 December 2016, 17:15-18:15
Location	HG G 43
Abstract	We will discuss solutions of certain elliptic PDE's with random coefficients. The qualitative theory of homogenization ensures that these solutions become close, in a large scale limit, to solutions of PDE's with constant, "homogenized" coefficients. This is the PDE version of a central limit theorem for reversible random walks in random environments. I will describe a new method allowing to make this convergence quantitative, assuming that the random coefficients are sufficiently mixing. The approach is based on progressively coarsening the equation by a "linearization around the homogenized limit". Joint work with S. Armstrong and T. Kuusi.

Quantitative results in stochastic homogenizationread_more

HG G 43

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