• Corpus ID: 119332123

Modified log-Sobolev inequalities for strong-Rayleigh measures

@article{Hermon2019ModifiedLI,
  title={Modified log-Sobolev inequalities for strong-Rayleigh measures},
  author={Jonathan Hermon and Justin Salez},
  journal={arXiv: Probability},
  year={2019}
}
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 \emph{stochastic covering property}. This condition is strictly weaker than the strong Rayleigh property, and is satisfied in particular by all determinantal measures, as well as any product measure over the set of bases of a balanced matroid. In the special case where… 
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References

SHOWING 1-10 OF 32 REFERENCES
Concentration of Lipschitz Functionals of Determinantal and Other Strong Rayleigh Measures
  • R. Pemantle, Y. Peres
  • Mathematics, Computer Science
    Combinatorics, Probability and Computing
  • 2013
TLDR
A continuous version for concentration of Lipschitz functionals of a determinantal point process is proved and is shown to be a strong Rayleigh condition.
Modified Logarithmic Sobolev Inequalities in Discrete Settings
Motivated by the rate at which the entropy of an ergodic Markov chain relative to its stationary distribution decays to zero, we study modified versions of logarithmic Sobolev inequalities in the
Elementary bounds on Poincaré and log-Sobolev constants for decomposable Markov chains
We consider finite-state Markov chains that can be naturally decomposed into smaller ``projection'' and ``restriction'' chains. Possibly this decomposition will be inductive, in that the restriction
LOGARITHMIC SOBOLEV INEQUALITIES FOR FINITE MARKOV CHAINS
This is an expository paper on the use of logarithmic Sobolev inequalities for bounding rates of convergence of Markov chains on finite state spaces to their stationary distributions. Logarithmic
Spectral gap and log-Sobolev constant for balanced matroids
  • M. Jerrum, Jung-Bae Son
  • Mathematics, Computer Science
    The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
  • 2002
TLDR
Tight lower bounds are computed on the log-Sobolev constant of a class of inductively defined Markov chains, which contains the bases-exchange walks for balanced matroids studied by Feder and Mihail and improved upper bounds for the mixing time of a variety of Markov Chains are obtained.
Log-Concave Polynomials, Entropy, and a Deterministic Approximation Algorithm for Counting Bases of Matroids
TLDR
It is proved that the multivariate generating polynomial of the bases of any matroid is log-concave as a function over the positive orthant, and a general framework for approximate counting in discrete problems, based on convex optimization is developed.
Entropy dissipation estimates for inhomogeneous zero-range processes
Introduced by Lu and Yau (CMP, 1993), the martingale decomposition method is a powerful recursive strategy that has produced sharp log-Sobolev inequalities for homogeneous particle systems. However,
Modified log-Sobolev Inequalities for Strongly Log-Concave Distributions
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 an asymptotically
Rayleigh Matroids
TLDR
It is shown that a binary matroid is Rayleigh if and only if it does not contain $\mathcal{S}_{8}$ as a minor, which provides the first complete proof in print that $S8$ is the only minor-minimal binary non-balanced matroid.
Negative dependence and the geometry of polynomials
We introduce the class of strongly Rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers
...
1
2
3
4
...