Título: Exact solution of an integrable particle system.
Autor: Cristian Giardina, University of Modena
Data: 17/03/2021
Horário: 14:00h
Local: Transmissão online.
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ID da reunião: 958 0581 3232
Resumo:
We consider the family of boundary-driven models introduced in [FGK] and show they can be solved exactly, i.e. the correlations functions and the non-equilibrium steady-state have a closed-form expression. The solution relies on probabilistic arguments and techniques inspired by integrable systems. As in the context of bulk-driven systems (scaling to KPZ), it is obtained in two steps: i) the introduction of a dual process; ii) the solution of the dual dynamics by Bethe ansatz. For boundary-driven systems, a general by-product of duality is the existence of a direct mapping (a conjugation) between the generator of the non-equilibrium process and the generator of the associated reversible equilibrium process. Macroscopically, this mapping was observed years ago by Tailleur, Kurchan and Lecomte in the context of the Macroscopic Fluctuation Theory.
[FGK] R. Frassek, C. Giardinà, J. Kurchan, Non-compact quantum spin chains as integrable stochastic particle processes, Journal of Statistical Physics 180, 366-397 (2020).
Título: Thermodynamic Formalism in Neuronal Dynamics and Spike Train Statistics
Palestrante: Rodrigo Cofré (Universidad de Valparaiso)
Data: 15/03/2021
Horário: 15:00h
Local: Transmissão online
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Resumo: The Thermodynamic Formalism provides a rigorous mathematical framework for studying the quantitative and qualitative aspects of dynamical systems. At its core, there is a variational principle that corresponds, in its simplest form, to the maximum entropy principle. It has been used as a statistical inference procedure to represent the collective behavior of complex systems by specific probability measures (Gibbs measures). This framework has found applications in different domains of science. In particular, it has been fruitful and influential in neurosciences. In this talk, I will discuss and briefly review how Thermodynamic Formalism can be exploited in the field of theoretical neuroscience, as a conceptual and operational tool, to link the dynamics of interacting neurons and the statistics of action potentials from either experimental data or mathematical models. I will end this talk by commenting on perspectives and open problems in theoretical neuroscience that could be addressed within this formalism.
Título: "Weak convergence for the scaled cover time of the rooted binary tree"
Palestrante: Santiago Saglietti
Data: 03/03/2021
Horário: 11h às 12h
Local: Transmissão Online
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Resumo: We consider a continuous time random walk on the rooted binary tree of depth n with all transition rates equal to one and study its cover time, namely the time until all vertices of the tree have been visited. We prove that, normalized by 2^{n+1}n and then centered by (log2)n-log n, the cover time admits a weak limit as the depth of the tree tends to infinity. The limiting distribution is identified as that of a randomly shifted Gumbel random variable with rate one, where the shift is given by the unique solution to a specific distributional equation. The existence of the limit and its overall form were conjectured in the literature. However, our approach is different from those taken in earlier works on this subject and relies in great part on a comparison with the extremal landscape of the discrete Gaussian free field. Joint work with Aser Cortines and Oren Louidor.
Título: The infinitesimal generator of the stochastic Burgers equation
Palestrante: Massimiliano Gubinelli, University of Bonn
Data: 09/03/2021
Horário: 13:30h
Local: Transmissão online
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ID da reunião: 958 0581 3232
Resumo: In this talk I will discuss the martingale problem formulation of the Burgers equation (or KPZ equation) in one dimension or similar equations (multicomponent, fractional variants and other hydrodynamic models). The well posedness involves a detailed study of the space of test functions for the generator and of the solution of the associated Kolmogorov backward equation.
Joint work with Nicolas Perkowski and Mattia Turra. Based on the papers Massimiliano Gubinelli and Nicolas Perkowski, The Infinitesimal Generator of the Stochastic Burgers Equation’, Probability Theory and Related Fields 178, no. 3: 1067–1124, click HERE to read. M. Gubinelli and M. Turra, ‘Hyperviscous Stochastic Navier–Stokes Equations with White Noise Invariant Measure’, Stochastics and Dynamics 20, no. 06: 2040005, click HERE to read.
Título: Gaussian random permutations and the boson point process
Palestrante: Ines Armendariz (UBA)
Data: 01/03/2021
Horário: 15h - 16h (Rio de Janeiro local time)
Local: Transmissão online
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Resumo: We construct an infinite volume spatial random permutation associated to a Gaussian Hamiltonian, which is parametrized by the point density and the temperature. Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman (1953). Bose-Einstein condensation occurs for dimensions 3 or larger, above a critical density, and is manifest in this representation by the presence of cycles of macroscopic length. For subcritical densities we define the spatial random permutation as a Poisson process of finite unrooted loops of a random walk with Gaussian increments that we call Gaussian loop soup, analogous to the Brownian loop soup of Lawler and Werner (2004). We also construct Gaussian random interlacements, a Poisson process of doubly-infinite trajectories of random walks with Gaussian increments analogous to the Brownian random interlacements of Sznitman (2010). For dimensions greater than or equal to 3 and supercritical densities, we define the spatial permutation as the superposition of independent realizations of the Gaussian loop soup at critical density and Gaussian random interlacements at the remaining density. We show some properties of these spatial permutations, in particular that the point marginal is the boson point process, for any point density.
