Relations entre isopérimétrie et trou spectral pour les chaînes de Markov finies

Abstract

Let G be a finite and connected graph, we will note by l(G) the maximum length of an injective path in G. We will show (by two dictinct proofs, one using sub-trees of G and the other based on multiflows of paths) that sup(P, μ) ∈ Script A sign (G) I (P, μ)/λ(P, μ) = l(G), where the supremum is taken over all Markovian kernels P reversible with respect to a probability μ and whose allowed transitions are given by the edges of G, and where I (P, μ) (respectively λ(P, μ)) is the isoperimetric constant (resp. the spectral gap) associated to the couple (P, μ). Then we will study more precisely the same supremum, but where the probability μ is also fixed. These results give optimal minorations of the spectral gap which are linear with respect to the isoperimetric constant (and not quadratic, as in the Cheeger inequality), and we will give several examples on infinite graphs.