Exponential decay of entropy in the random transposition and Bernoulli-Laplace models

Gao, Fuqing; Quastel, Jeremy

doi:10.1214/aoap/1069786512

Cited by 40 publications

(42 citation statements)

References 16 publications

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“…Proof of Proposition 1.9: It follows from Gao and Quastel (2003, Theorem 1) that we have for any f :…”

Section: Proofs and Auxiliary Resultsmentioning

confidence: 91%

Modified log-Sobolev inequalities and two-level concentration

Sambale¹,

Sinulis²

2021

ALEA

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We consider a generic modified logarithmic Sobolev inequality (mLSI) of the form Ent µ (e f ) ≤ ρ 2 E µ e f Γ(f ) 2 for some difference operator Γ, and show how it implies two-level concentration inequalities akin to the Hanson-Wright or Bernstein inequality. This can be applied to the continuous (e. g. the sphere or bounded perturbations of product measures) as well as discrete setting (the symmetric group, finite measures satisfying an approximate tensorization property, . . . ).Moreover, we use modified logarithmic Sobolev inequalities on the symmetric group S n and for slices of the hypercube to prove Talagrand's convex distance inequality, and provide concentration inequalities for locally Lipschitz functions on S n . Some examples of known statistics are worked out, for which we obtain the correct order of fluctuations, which is consistent with central limit theorems.

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“…Proof of Proposition 1.9: It follows from Gao and Quastel (2003, Theorem 1) that we have for any f :…”

Section: Proofs and Auxiliary Resultsmentioning

confidence: 91%

Modified log-Sobolev inequalities and two-level concentration

Sambale¹,

Sinulis²

2021

ALEA

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“…Remark 1.11. We prove Theorem 1.9 by a suitable version of the martingale method already employed in the estimation of the log-Sobolev and modified log-Sobolev constants for the interchange process on the complete graph [36,28,29]. It is remarkable that in those cases, which correspond to the = 2 case of the above theorem, the method does not allow one to compute exactly the constants but only to give an estimate that is tight up to a constant factor.…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

“…In the binary mean field case α A = 1 |A|=2 , this is known as the Bernoulli-Laplace model [24]. The log-Sobolev constant for this process was estimated in [36], while its modified log-Sobolev constant was estimated in [28,29,26]. As for the labeled case, these estimates are tight up to constant factors but the exact value of these constants remains unknown.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Entropy inequalities for random walks and permutations

Alexandre¹,

Caputo²

2021

Preprint

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We consider a new functional inequality controlling the rate of relative entropy decay for random walks, the interchange process and more general block-type dynamics for permutations. The inequality lies between the classical logarithmic Sobolev inequality and the modified logarithmic Sobolev inequality, roughly interpolating between the two as the size of the blocks grows. Our results suggest that the new inequality may have some advantages with respect to the latter well known inequalities when multi-particle processes are considered. We prove a strong form of tensorization for independent particles interacting through synchronous updates. Moreover, for block dynamics on permutations we compute the optimal constants in all mean field settings, namely whenever the rate of update of a block depends only on the size of the block. Along the way we establish the independence of the spectral gap on the number of particles for these mean field processes. As an application of our entropy inequalities we prove a new subadditivity estimate for permutations, which implies a sharp upper bound on the permanent of arbitrary matrices with nonnegative entries, thus resolving a well known conjecture.

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“…The smallest ρ 0 in (2.1) is called modified logarithmic Sobolev (or entropy) constant, see e.g. [BT06] and the definition of β in [GQ03]. It is known that the modified logarithmic Sobolev constant can be used to bound the mixing time for the total variation distance of (the distribution of) a Markov semigroup and its trend to equilibrium, see for example [BT06, Theorem 2.4].…”

Section: Mixing Timesmentioning

confidence: 99%

Logarithmic Sobolev inequalities for finite spin systems and applications

Sambale¹,

Sinulis²

2020

Bernoulli

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We derive sufficient conditions for a probability measure on a finite product space (a spin system) to satisfy a (modified) logarithmic Sobolev inequality. We establish these conditions for various examples, such as the (vertex-weighted) exponential random graph model, the random coloring and the hard-core model with fugacity.This leads to two separate branches of applications. The first branch is given by mixing time estimates of the Glauber dynamics. The proofs do not rely on coupling arguments, but instead use functional inequalities. As a byproduct, this also yields exponential decay of the relative entropy along the Glauber semigroup. Secondly, we investigate the concentration of measure phenomenon (particularly of higher order) for these spin systems. We show the effect of better concentration properties by centering not around the mean, but a stochastic term in the exponential random graph model. From there, one can deduce a central limit theorem for the number of triangles from the CLT of the edge count. In the Erdös-Rényi model the first order approximation leads to a quantification and a proof of a central limit theorem for subgraph counts.

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Exponential decay of entropy in the random transposition and Bernoulli-Laplace models

Cited by 40 publications

References 16 publications

Modified log-Sobolev inequalities and two-level concentration

Modified log-Sobolev inequalities and two-level concentration

Entropy inequalities for random walks and permutations

Logarithmic Sobolev inequalities for finite spin systems and applications

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