Related papers: Regularized modified log-Sobolev inequalities, and…
For reversible Markov chains on finite state spaces, we show that the modified log-Sobolev inequality (MLSI) can be upgraded to a log-Sobolev inequality (LSI) at the surprisingly low cost of degrading the associated constant by $\log…
Consider the switch chain on the set of $d$-regular bipartite graphs on $n$ vertices with $3\leq d\leq n^{c}$, for a small universal constant $c>0$. We prove that the chain satisfies a Poincar\'e inequality with a constant of order $O(nd)$;…
We establish universal modified log-Sobolev inequalities for reversible Markov chains on the boolean lattice $\{0,1\}^n$, under the only assumption that the invariant law $\pi$ satisfies a form of negative dependence known as the stochastic…
We show that the modified log-Sobolev constant for a natural Markov chain which converges to an $r$-homogeneous strongly log-concave distribution is at least $1/r$. Applications include a sharp mixing time bound for the bases-exchange walk…
We consider a generic modified logarithmic Sobolev inequality (mLSI) of the form $\mathrm{Ent}_{\mu}(e^f) \le \tfrac{\rho}{2} \mathbb{E}_\mu e^f \Gamma(f)^2$ for some difference operator $\Gamma$, and show how it implies two-level…
We introduce a framework for obtaining tight mixing times for Markov chains based on what we call restricted modified log-Sobolev inequalities. Modified log-Sobolev inequalities (MLSI) quantify the rate of relative entropy contraction for…
The chain rule lies at the heart of the powerful Gamma calculus for Markov diffusions on manifolds, providing remarkable connections between several fundamental notions such as Bakry-\'Emery curvature, entropy decay, and hypercontractivity.…
We prove that in the context of general Markov semigroups Beckner inequalities with constants separated from zero as $p\to 1^+$ are equivalent to the modified log Sobolev inequality (previously only one implication was known to hold in this…
We consider continuous-time Markov chain on a finite state space X. We assume X can be clustered into several subsets such that the intra-transition rates within these subsets are of order $\mathcal{O}(\frac{1}{\epsilon})$ comparing to the…
We introduce a natural conjugate prior for the transition matrix of a reversible Markov chain. This allows estimation and testing. The prior arises from random walk with reinforcement in the same way the Dirichlet prior arises from…
Many finite-state reversible Markov chains can be naturally decomposed into "projection" and "restriction" chains. In this paper we provide bounds on the total variation mixing times of the original chain in terms of the mixing properties…
The problem of efficiently sampling from a set of (undirected, or directed) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the…
The continuum $\varphi^4_2$ and $\varphi^4_3$ measures are shown to satisfy a log-Sobolev inequality uniformly in the lattice regularisation under the optimal assumption that their susceptibility is bounded. In particular, this applies to…
This is an expository paper, focussing on the following scenario. We have two Markov chains, $\mathcal {M}$ and $\mathcal {M}'$. By some means, we have obtained a bound on the mixing time of $\mathcal {M}'$. We wish to compare $\mathcal…
Since 1997 a considerable effort has been spent to study the mixing time of switch Markov chains on the realizations of graphic degree sequences of simple graphs. Several results were proved on rapidly mixing Markov chains on unconstrained,…
We investigate the metastable behavior of reversible Markov chains on possibly countable infinite state spaces. Based on a new definition of metastable Markov processes, we compute precisely the mean transition time between metastable sets.…
The problem of efficiently sampling from a set of(undirected) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the sampling. The…
We consider a Markov chain on invertible $n\times n$ matrices with entries in $\mathbb{Z}_2$ which moves by picking an ordered pair of distinct rows and add the first one to the other, modulo $2$. We establish a logarithmic Sobolev…
We study Markov chains with non-negative sectional curvature on finite metric spaces. Neither reversibility, nor the restriction to a particular combinatorial distance are imposed. In this level of generality, we prove that a 1-step…
Measure-valued Markov chains have raised interest in Bayesian nonparametrics since the seminal paper by (Math. Proc. Cambridge Philos. Soc. 105 (1989) 579--585) where a Markov chain having the law of the Dirichlet process as unique…