Título: Cutoff for the Glauber dynamics of the discrete Gaussian free field
Palestrante: Reza Gheissari, UC Berkeley
Data: 22/09/2021
Horário: 13:00h
Local: Transmissão online
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Resumo: The Gaussian free field (GFF) is a canonical model of random surfaces in probability theory, generalizing the Brownian bridge to higher dimensions. It arises naturally as the stationary solution to the stochastic heat equation with additive noise (SHE), and together the SHE and GFF are expected to be the universal scaling limit of the dynamics and equilibrium of many random surface models arising in lattice statistical physics. We study the mixing time (time to converge to stationarity, when started out of equilibrium) for the central pre-limiting object, the discrete Gaussian free field (DGFF) evolving under the Glauber dynamics. In joint work with S. Ganguly, we establish that for every dimension d larger than one , on a box of side-length n in Zd, the Glauber dynamics for the DGFF exhibits cutoff at time (d/\pi^2) n^2 \log n with an O(n^2) window. Our proof relies on an "exact" representation of the DGFF dynamics in terms of random walk trajectories with space-dependent jump times, which we expect to be of independent interest.
ID da reunião: 958 0581 3232
Título: Hydrodynamic limit of an exclusion process with vorticity
Palestrante: Davide Gabrielli (Università dell'Aquila)
Data: 20/09/2021
Horário: 15:00hrs às 16:00hrs
Local: Transmissão online
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Resumo: We construct an exclusion process with Bernoulli product invariant measure and having, in the diffusive hydrodynamic scaling, a non symmetric diffusion matrix, that can be explicitly computed. The antisymmetric part does not affect the evolution of the density but it is relevant for the evolution of the current. In particular because of that, the Fick's law is violated in the diffusive limit. Switching on a weak external field we obtain a symmetric mobility matrix that is related just to the symmetric part of the diffusion matrix by the Einstein relation. We show that this fact is typical within a class of generalized gradient models. We consider for simplicity the model in dimension $d=2$, but a similar behavior can be also obtained in higher dimensions. Joint work with L. De Carlo and P. Goncalves
Todas os seminários são ministrados em inglês.
Os vídeos dos seminários passados estão disponíveis nos links abaixo:
Para o segundo semestre, alguns dias depois dos seminários, às gravações ficaram disponíveis AQUI.
Título: Dyson models with random boundary conditions
Palestrante: Aernout van Enter (Groningen University)
Data: 06/09/2021
Horário: 15:00h
Local: Transmissão online.
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Resumo: I discuss the low-temperature behaviour of Dyson models (polynomially decaying long-range Ising models in one dimension) in the presence of random boundary conditions. As for typical random (i.i.d.) boundary conditions Chaotic Size Dependence occurs, that is, the pointwise thermodynamic limit of the finite-volume Gibbs states for increasing volumes does not exist, but the sequence of states moves between various possible limit points, as a consequence it makes sense to study distributional limits, the so-called "metastates" which are measures on the possible limiting Gibbs measures.
The Dyson model is known to have a phase transition for decay parameters α between 1 and 2. We show that the metastate changes character at α =3/2. It is dispersed in both cases, but it changes between being supported on two pure Gibbs measures when α is less than 3/2 to being supported on mixtures thereof when α is larger than 3/2.
Joint work with Eric Endo and Arnaud Le Ny
All the talks are held in English.
The videos of the online seminars are available:
2020
2021-1
For the second semester, a few days after each meeting the video should be available at HERE.
Título: Approximations of the covariance operators of solutions of fractional elliptic SPDEs driven by Gaussian white noise
Palestrante: Alexandre de Bustamante Simas (UFPB & Kaust)
Data: 13/09/2021
Horário: 15:00h
Local: Transmissão online.
Confira AQUI o link para transmissão.
Resumo: In this talk we will briefly present the model we are interested in, which is a fractional elliptic stochastic partial differential equation driven by Gaussian white noise. There is in the literature a standard way to approximate the covariance operator of the solution of such equations, the so-called rational approximation (Bolin and Kirchner, 2020), however this approach uses the solution to build such an approximation. By considering directly the covariance operator, we are able to provide a more computationally efficient approximation. We compute the rate of this approximation in terms of the Hilbert-Schmidt norm. Furthermore, we also obtain, rigorously, the rate of approximation of the so-called lumped mass method. This method is widely used by practitioners and is essential to make it computationally feasible to fit some models in spatial statistics. We obtain the rate of approximation of the lumped mass method in terms of the operator's norm as well as, under some additional restrictions, the Hilbert-Schmidt norm. Finally, we present the usage of these approximations in maximum likelihood estimation. Joint work with David Bolin and Zhen Xiong.
All the talks are held in English.
The videos of the online seminars are available:
2020
2021-1
For the second semester, a few days after each meeting the video should be available at HERE.
Título: The Widom-Rowlinson model: metastability, mesoscopic and microscopic fluctuations for the critical droplet
Palestrante: Elena Pulvirenti (Delft University of Technology)
Data: 30/08/2021
Horário: 15:00h
Local: Transmissão online.
Confira AQUI o link para transmissão.
Resumo: We introduce the equilibrium Widom-Rowlinson model on a two-dimensional finite torus in which the energy of a particle configuration is attractive and determined by the union of small discs centered at the positions of the particles. We then discuss the metastable behaviour of a dynamic version of the WR model. This means that the particle configuration is viewed as a continuous time Markov process where particles are randomly created and annihilated as if the outside of the torus were an infinite reservoir with a given chemical potential. In particular, we start with the empty torus and are interested in the first time when the torus is fully covered by discs in the regime at low temperature and when the chemical potential is supercritical. In order to achieve the transition from empty to full, the system needs to create a sufficiently large droplet, called critical droplet, which triggers the crossover. We compute the distribution of the crossover time and identify the size and the shape of the critical droplet. The analysis relies on a mesoscopic and microscopic description of the surface of the critical droplet. It turns out that the critical droplet is close to a disc of a certain deterministic radius, with a boundary that is random and consists of a large number of small discs that stick out by a small distance. We will show how the analysis of surface fluctuations in the WR model allows us to derive the leading order term of the condensation time and also the correction order term. This is a joint work with Frank den Hollander (Leiden), Sabine Jansen (Munich) and Roman Kotecky (Prague & Warwick).
All the talks are held in English.
The videos of the online seminars are available:
2020
2021-1
For the second semester, a few days after each meeting the video should be available at HERE.
