Dear colleagues,
I would like to invite you to the following online seminar organized by the Probability group of the University of Pisa. The two talks will be accessible under the link
Best regards,
Giacomo
Tuesday, May 11, 16:00
Speaker: Michael Högele (Universidad de los Andes)
Title: Cutoff thermalization for Ornstein-Uhlenbeck systems with small Lévy noise in the Wasserstein distance
Abstract: This talk presents recent results on cutoff thermalization (also known as the cutoff phenomenon) for a general class of asymptotically exponentially stable Ornstein-Uhlenbeck systems under ε-small additive Lévy noise. The driving noise processes include Brownian motion, α-stable Lévy flights, finite intensity compound Poisson processes and red noises and may be highly degenerate. Window cutoff thermalization is shown under generic mild assumptions, that is, we see an asymptotically sharp ∞/0-collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure μ^ε along a time window centered in a precise ε-dependent time scale t_ε . In many interesting situations such as reversible (Lévy) diffusions it is possible to prove the existence of an explicit, universal, deterministic cutoff thermalization profile. The existence of this limit is characterized by the absence of non-normal growth patterns in terms of an orthogonality condition on a computable family of generalized eigenvectors of the matrix Q. With this piece of theory at hand this article provides a complete discussion of the cutoff phenomenon for the classical linear oscillator with friction subject to ε-small Brownian motion or α-stable Lévy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature.
Tuesday, May 11, 17:00
Speaker: Alessandra Caraceni (University of Oxford)
Title: Polynomial mixing time for edge flips on planar maps
Abstract: A long-standing problem proposed by David Aldous consists in giving a sharp upper bound for the mixing time of the so-called “triangulation walk”, a Markov chain defined on the set of all possible triangulations of the regular n-gon. A single step of the chain consists in performing a random edge flip, i.e. in choosing an (internal) edge of the triangulation uniformly at random and, with probability 1/2, replacing it with the other diagonal of the quadrilateral formed by the two triangles adjacent to the edge in question (with probability 1/2, the triangulation is left unchanged).
While it has been shown that the relaxation time for the triangulation walk is polynomial in n and bounded below by a multiple of n^{3/2}, the conjectured sharpness of the lower bound remains firmly out of reach in spite of the apparent simplicity of the chain. For edge flip chains on different models – such as planar maps, quadrangulations of the sphere, lattice triangulations and other geometric graphs – even less is known.
We shall discuss results concerning the mixing time of random edge flips on rooted quadrangulations of the sphere obtained in joint work with Alexandre Stauffer. A “growth scheme” for quadrangulations, which generates a uniform quadrangulation of the sphere by adding faces one at a time at appropriate random locations, can be combined with careful combinatorial constructions to build probabilistic canonical paths in a relatively novel way. This method has implications for a range of interesting edge-manipulating Markov chains on so-called Catalan structures, from “leaf translations” on plane trees to “edge rotations” on general planar maps.
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Giacomo Di Gesù
Dipartimento di Matematica
Università di Pisa
Largo Bruno Pontecorvo 5
56127 - Pisa, Italy
giacomo.digesu@unipi.itmailto:giacomo.digesu@unipi.it