Marches aléatoires sur des suites de graphes monotones
Date(s) : 20/10/2025 iCal
15h00 - 18h00
The jury consists of
- Perla Sousi — Professor, University of Cambridge — Reviewer
- Laurent Saloff-Coste — Professor, Cornell University — Reviewer
- Nina Gantert — Professor, Technical University of Munich — Chair of the Jury
- Laurent Miclo — Senior Researcher, Toulouse School of Economics and Institute of Mathematics — Examiner
- Fabienne Castell — Professor, Aix-Marseille University — Examiner
- Bruno Schapira — Professor, University of Lyon 1 — Examiner
- Ruojun Huang — Postdoctoral Researcher, Scuola Normale Superiore di Pisa — Examiner
- Pierre Mathieu — Professor, Aix-Marseille University — Thesis Supervisor
[
forall (x,y) in V, quad lim_{t to +infty} d_{TV}(mu_t^x, mu_t^y) = 0,.
]
This result is proved using an original tool: the accompanying sets process. This random object is derived from the evolving sets introduced by B. Morris and Y. Peres. It also allows us to provide a result within the framework of ratio theorems, but under the hypothesis that there exists a finite subset (A) of (V) such that the limit inferior of the sequence ((rho_t(A))_{t geq 0}) is strictly positive, instead of the usual tightness assumption. Additionally, in this same chapter, we study the convergence of the laws when the operators or invariant measures converge pointwise.
In the second chapter, we provide quantitative estimates on the merging of time-inhomogeneous Markov chains, assuming that the invariant measures can be ordered in an increasing manner. Markov chains arising from electrical networks provide a class of examples where the invariant measures are explicit. Our approach involves adapting classical functional inequalities: Poincaré, Nash, and hypercontractivity. In the proven bounds, alongside the classical terms, we find an additional term that takes the form of a function of the total masses. Let us illustrate our bounds using the spectral gap and the Poincaré inequality. Let ((K_t)_{t geq 1}) be a sequence of Markov transition operators on the same discrete set (V). Let ((pi_t)_{t geq 1}) be the sequence of their invariant measures, assumed to be finite. We denote by (Tilde{pi}_t) the probability measure derived from the finite measure (pi_t). We assume that for all (x in V), the sequence ((pi_t(x))_{t geq 1}) is non-decreasing. For each (t geq 1), let (gamma_t) be the Poincaré constant of the operator (K_t^*K_t), where (K_t^*) is the adjoint of (K_t) in the Hilbert space (ell^2(pi_t)). Then, we obtain:
[
forall (x,y) in V, quad d_{TV}(mu_t^x, mu_t^y) le frac{1}{2} sqrt{frac{pi_t(V)}{pi_1(V)}} left(frac{1}{sqrt{Tilde{pi}_1(x)}} + frac{1}{sqrt{Tilde{pi}_1(y)}}right) prod_{s=1}^t sqrt{1 – gamma_s}.
]
Emplacement
I2M Saint-Charles - Salle de séminaire
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