A sharp inequality for the variance with respect to the Ewens Sampling Formula

Mixing time of Metropolis chain based on random transposition walk converging to multivariate Ewens distribution

We prove sharp rates of convergence to the Ewens equilibrium distribution for a family of Metropolis algorithms based on the random transposition shuffle on the symmetric group, with starting point at the identity. The proofs rely heavily on the theory of symmetric Jack polynomials, developed initially by Jack [Proc. Roy. Soc. Edinburgh Sect. A 69 (1970/1971) 1-18], Macdonald [Symmetric Functions and Hall Polynomials (1995) New York] and Stanley [Adv. Math. 77 (1989) 76-115]. This completes the analysis started by Diaconis and Hanlon in [Contemp. Math. 138 (1992) 99-117]. In the end we also ex…

整数格上的精确Gagliardo-Nirenberg不等式与对数Sobolev不等式

Sharp Gagliardo-Nirenberg inequality and logarithmic Sobolev inequality on integer lattices

本文在整数格上获得了精确的Gagliardo-Nirenberg不等式并刻画了其刚性，进而作为推论推导出整数格上精确的对数Sobolev不等式。

A sharp log-Sobolev inequality for the multislice

We determine the log-Sobolev constant of the multi-urn Bernoulli-Laplace diffusion model with arbitrary parameters, up to a small universal multiplicative constant. Our result extends a classical estimate of Lee and Yau (1998) and confirms a conjecture of Filmus, O'Donnell and Wu (2018). Among other applications, we completely quantify the "small-set expansion" phenomenon on the multislice, and obtain sharp mixing-time estimates for the colored exclusion process on various graphs.

不含Fano子式的二元拟阵的严格基质量不等式

A Sharp Basis-Mass Inequality for Binary Matroids without a Fano Minor

针对不含F₇子式的二元拟阵，证明了基生成多项式最大值的严格上界2^r/(r+1)，明确等号成立条件及该类拟阵的最大子式闭性质。

On configurations where the Loomis-Whitney inequality is nearly sharp and applications to the Furstenberg set problem

In this paper, we consider the so-called "Furstenberg set problem" in high dimensions. First, following Wolff's work on the two dimensional real case, we provide "reasonable" upper bounds for the problem for $\mathbb{R}$ or $\mathbb{F}_p$. Next we study the "critical" case and improve the "trivial" exponent by $Ω(\frac{1}{n^2})$ for $\mathbb{F}_p^n$. Our key tool to obtain this lower bound is a theorem about how things behave when the Loomis-Whitney inequality is nearly sharp, as it helps us to reduce the problem down to dimension two.