The interchange process with reversals on the complete graph

带任意集合的符号交换过程的谱间隙

Spectral gap for the signed interchange process with arbitrary sets

本文研究了带任意集合的符号交换过程的谱间隙，扩展了允许的生成元并解决了相关猜想。

Cycle structure of the interchange process and representation theory

Consider the process of random transpositions on the complete graph. We use representation theory to give an exact, simple formula for the expected number of cycles of size k at time t, in terms of an incomplete Beta function. Using this we show that the expected number of cycles of size k jumps from 0 to its equilibrium value, 1/k, at the time where the giant component of the associated random graph first exceeds k. Consequently we deduce a new and simple proof of Schramm's theorem on random transpositions, that giant cycles emerge at the same time as the giant component in the random graph. …

Phase transition for the interchange and quantum Heisenberg models on the Hamming graph

We study a family of random permutation models on the Hamming graph $H(2,n)$ (i.e., the $2$-fold Cartesian product of complete graphs), containing the interchange process and the cycle-weighted interchange process with parameter $θ> 0$. This family contains the random walk representation of the quantum Heisenberg ferromagnet. We show that in these models the cycle structure of permutations undergoes a \textit{phase transition} -- when the number of transpositions defining the permutation is $\leq c n^2$, for small enough $c > 0$, all cycles are microscopic, while for more than $\geq C n^2$ tra…

由对称群上的连续时间随机游走控制的噪声敏感性

Noise Sensitivity Governed by Continuous-Time Random Walks on the Symmetric Group

研究对称群上布尔函数的噪声敏感性，其中噪声由马尔可夫链诱导，比较不同噪声源，并证明长循环指示函数在完全图、离散环面和星图上的交换过程中是噪声敏感的。

The Root solution to the multi-marginal embedding problem: an optimal stopping and time-reversal approach

We provide a complete characterisation of the Root solution to the Skorokhod embedding problem (SEP) by means of an optimal stopping formulation. Our methods are purely probabilistic and the analysis relies on a tailored time-reversal argument. This approach allows us to address the long-standing question of a multiple marginals extension of the Root solution of the SEP. Our main result establishes a complete solution to the n-marginal SEP using first hitting times of barrier sets by the time-space process. The barriers are characterised by means of a recursive sequence of optimal stopping pro…

Quantum spins and random loops on the complete graph

We present a systematic analysis of quantum Heisenberg-, XY- and interchange models on the complete graph. These models exhibit phase transitions accompanied by spontaneous symmetry breaking, which we study by calculating the generating function of expectations of powers of the averaged spin density. Various critical exponents are determined. Certain objects of the associated loop models are shown to have properties of Poisson--Dirichlet distributions.

Large cycles in random permutations related to the Heisenberg model

We study the weighted version of the interchange process where a permutation receives weight $θ^{\#\mathrm{cycles}}$. For $θ=2$ this is Tóth's representation of the quantum Heisenberg ferromagnet on the complete graph. We prove, for $θ>1$, that large cycles appear at `low temperature'.

Exact eigenspectrum of the symmetric simple exclusion process on the complete, complete bipartite, and related graphs

We show that the infinitesimal generator of the symmetric simple exclusion process, recast as a quantum spin-1/2 ferromagnetic Heisenberg model, can be solved by elementary techniques on the complete, complete bipartite, and related multipartite graphs. Some of the resulting infinitesimal generators are formally identical to homogeneous as well as mixed higher spins models. The degeneracies of the eigenspectra are described in detail, and the Clebsch-Gordan machinery needed to deal with arbitrary spin-s representations of the SU(2) is briefly developed. We mention in passing how our results fi…

无需学习时间反演的桥接评分匹配方法

Score matching for bridges without learning time-reversals

本文提出一种通过直接学习评分项来构建桥接扩散过程的新算法，无需先学习时间反演，实验表明其性能优于基于时间反演的现有方法。

On a class of orthogonal-invariant quantum spin systems on the complete graph

We study a two-parameter family of quantum spin systems on the complete graph, which is the most general model invariant under the complex orthogonal group. In spin $S=\frac{1}{2}$ it is equivalent to the XXZ model, and in spin $S=1$ to the bilinear-biquadratic Heisenberg model. The paper is motivated by the work of Björnberg, whose model is invariant under the (larger) complex general linear group. In spin $S=\frac{1}{2}$ and $S=1$ we give an explicit formula for the free energy for all values of the two parameters, and for spin $S>1$ for when one of the parameters is non-negative. This allow…