分离距离的极限轮廓

平坦 тор、球体和射影空间上布朗运动分离轮廓中的截断

Cutoff in separation profile for the flat torus, sphere, and projective spaces

本文研究了布朗运动在平坦 torus、球体及实、复、四元数射影空间上的分离轮廓截断问题，证明其遵循特定的 Gumbel 分布尾部分布。

Separation profiles, isoperimetry, growth and compression

We give lower and upper bounds for the separation profile (introduced by Benjamini, Schramm & Timár) for various graphs using the isoperimetric profile, growth and Hilbertian compression. For graphs which have polynomial isoperimetry and growth, we show that the separation profile $\mathrm{Sep}(n)$ is also bounded by powers of $n$. For many amenable groups, we show a lower bound in $n/ \log(n)^a$ and, for any group which has a non-trivial compression exponent in an $L^p$-space, an upper bound in $n/ \log(n)^b$. We show that solvable groups of exponential growth cannot have a separation profile…

The distance profile of rooted and unrooted simply generated trees

It is well-known that the height profile of a critical conditioned Galton-Watson tree with finite offspring variance converges, after a suitable normalization, to the local time of a standard Brownian excursion. In this work, we study the distance profile, defined as the profile of all distances between pairs of vertices. We show that after a proper rescaling the distance profile converges to a continuous random function that can be described as the density of distances between random points in the Brownian continuum random tree. We show that this limiting function a.s. is Hölder continuous of…

Distances between pairs of vertices and vertical profile in conditioned Galton--Watson trees

We consider a conditioned Galton-Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length. We give two proofs of this result, one probabilistic and the other using generating functions and singularity analysis. Moreover, the second proof yields a more general estimate for generating functions, which is used to prove a conjecture by Bousquet-Melou and Janson saying that the vertical profile of a randomly labelled conditioned Galton-Watson tree converges in distribution, after suitable normalization, to th…

有色top-$m$-to-random洗牌的显式截止轮廓

Explicit cutoff profiles for colored top-$m$-to-random shuffles

研究花环积$C_p\wr S_n$上的$p$色top-$m$-to-random洗牌，利用Nakano-Sadahiro-Sakurai基元素$B_m$得到精确的嵌套集占用混合，将似然比简化为单一统计量$L_p$，从而导出分离度、$L^\infty(U)$、全变差、$L^q(U)$、$\chi^2$和相对熵的精确公式，并给出截止轮廓。

Separation cutoffs for random walk on irreducible representations

Random walk on the irreducible representations of the symmetric and general linear groups is studied. A separation distance cutoff is proved and the exact separation distance asymptotics are determined. A key tool is a method for writing the multiplicities in the Kronecker tensor powers of a fixed representation as a sum of non-negative terms. Connections are made with the Lagrange-Sylvester interpolation approach to Markov chains.