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On Poincaré and logarithmic Sobolev inequalities for a class of singular Gibbs measures. (English) Zbl 1459.60047

Klartag, Bo’az (ed.) et al., Geometric aspects of functional analysis. Israel seminar (GAFA) 2017–2019. Volume 1. Cham: Springer. Lect. Notes Math. 2256, 219-246 (2020).
The authors present several proofs of Poincaré and logarithmic Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interactions. These models in particular include the distributions of the eigenvalues of \(\beta\)-Hermite ensembles as the most prominent examples.
It is also shown that under mild additional conditions (which are satisfied for \(\beta\)-Hermite ensembles), the inequalities are sharp precisely precisely for affine-linear functions in the Poincaré case and for exponentials of affine linear functions in the logarithmic Sobolev case respectively. Besides these main results, further related results are discussed, e.g., on the multivariate Hermite polynomials of Lassalle which are related to the \(\beta\)-Hermite ensembles.
For the entire collection see [Zbl 1446.00029].

MSC:

60E15 Inequalities; stochastic orderings
60B20 Random matrices (probabilistic aspects)

Software:

MOPS
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Full Text: DOI arXiv

References:

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