F. Alberto Grunbaum
The study of several naturally arising "nearest neighbours" random walks benefits from the study of the associated orthogonal polynomials and their orthogonality measure. I consider extensions of this approach to a larger class of random walks. This raises a number of open problems.
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Matrix-Valued Orthogonal Polynomials Related to a Class of Random Walk
本文将Karlin-McGregor框架推广到三对角矩阵多项式型转移，研究对应矩阵值正交多项式与正交测度，并用于分析含长程转移的广义Ehrenfest模型。
In memoriam Pal Revesz(1934-2022) The random walk from orthogonality to anisotropy
本文是对已故匈牙利数学家Pál Révész的生平与学术贡献的纪念。
We give bounds in total variation distance for random walks associated to pure central states on free orthogonal quantum groups. As a consequence, we prove that the analogue of the uniform plane Kac walk on this quantum group has a cut-off at $N\ln(N)/2(1-\cos(θ))$. This is the first result of this type for genuine compact quantum groups. We also obtain similar results for mixtures of rotations and quantum permutations.
We prove that a uniformized variant of both the Rosenthal walk \cite{Rosenthal} and the Kac random walk \cite{Kac} on SO(n) mixes in $\cO(n^3)$ steps in total variation distance. The proof also extends easily to Rosenthal walk with fixed angle $θ\neq π$. To the best of our knowledge, this is the first polynomial time bound for both walks. The techniques employed are mainly from representation theory of SO(n). But a crucial new ingredient is the interpretation of the Fourier coefficients of the character ratio as counting the number of particle cascade paths arising from the classical branching…
Recent work on recurrence in quantum walks has provided a representation of Schur functions in terms of unitary operators. We propose a generalization of Schur functions by extending this operator representation to arbitrary operators on Banach spaces. Such generalized Schur functions meet the formal structure of first return generating functions, thus we call them FR-functions. We derive general properties of FR-functions, among them a generalization of the renewal equation already known for random and quantum walks, as well as splitting properties which extend useful factorizations of Schur …
Examples of stable-like random walks on groups of polynomial growth
本文研究多项式体积增长群上的长跳随机游走，通过证明最优伪 Poincaré 不等式，证实其类稳定行为并得到详细估计。
The local time of random walks associated with Gegenbauer polynomials $P_n^{(α)}(x),\ x\in [-1,1]$ is studied in the recurrent case: $α\in\ [-\frac{1}{2},0]$. When $α$ is nonzero, the limit distribution is given in terms of a Mittag-Leffler distribution. The proof is based on a local limit theorem for the random walk associated with Gegenbauer polynomials. As a by-product, we derive the limit distribution of the local time of some particular birth and death Markov chains on $\bbN$.
We construct a coupling between the random walk composed of Lévy area increments from a $d$-dimensional Brownian motion and a random walk composed of quadratic polynomials of Gaussian random variables. This coupling construction is used to produce a new pathwise approximation scheme for stochastic differential equations in the preprint [Flint-Lyons-2015]. The coupling arguments of the present paper are based extensively on the recent coupling results of Davie concerning a multidimensional variant of the Komlós-Major-Tusnády theorem and Wasserstein estimates for polynomial perturbations of Gaus…
We study a family of bivariate orthogonal polynomials associated to the deltoid curve. These polynomials arise when classifying bivariate diffusion operators that have discrete spectral decomposition given by orthogonal polynomials with respect to some compactly-supported probability measure on the interior of the deltoid curve.
Amnesic Elephant Random Walk with Polynomially Decaying Step Sizes
本文提出具有多项式衰减步长的遗忘型大象随机游走，研究步长指数与记忆参数对长期行为的影响，确定了两个临界阈值，并揭示了亚扩散、经典扩散及局域化等不同相态。
In this paper we specify some facts about the sequence of polynomials associated to a \CSK family and we prove that quadratic variance function is characterized by the property of orthogonality of these polynomials.
Let $G$ be a finitely generated group of polynomial volume growth equipped with a word-length $|\cdot|$. The goal of this paper is to develop techniques to study the behavior of random walks driven by symmetric measures $μ$ such that, for any $ε>0$, $\sum|\cdot|^εμ=\infty$. In particular, we provide a sharp lower bound for the return probability in the case when $μ$ has a finite weak-logarithmic moment.
We describe some examples of classical and explicit h-transforms as particular cases of a general mechanism, which is related to the existence of symmetric diffusion operators having orthogonal polynomials as spectral decomposition.
Comparison of Dirichlet forms for stable-like random walks on groups of polynomial volume growth
本研究针对多项式体积增长群上由特定测度驱动的类稳定随机游动，建立了其关联狄利克雷形式等价性的几何与代数判定准则，用于比较两类随机游动的行为。
We consider the use of random walks as an approach to obtain connection coefficients for higher-order Bernoulli and Euler polynomials. In particular, we consider the cases of a $1$-dimensional linear reflected Brownian motion and of a $3$-dimensional Bessel process. Considering the successive hitting times of two, three, and four fixed levels by these random walks yields non-trivial identities that involve higher-order Bernoulli and Euler polynomials.
We prove a certain duality relation for orthogonal polynomials defined on a finite set. The result is used in a direct proof of the equivalence of two different ways of computing the correlation functions of a discrete orthogonal polynomial ensemble.
The recently found hypergeometric multiple orthogonal polynomials on the step-line by Lima and Loureiro are shown to be random walk polynomials. It is proven that the corresponding Jacobi matrix and its transpose, which are nonnegative matrices and describe higher recurrence relations, can be normalized to two stochastic matrices, dual to each other. Using the Christoffel-Darboux formula on the step-line and the Poincaré theory for non-homogeneous recurrence relations it is proven that both stochastic matrices are related by transposition in the large $n$ limit. These random walks are beyond b…
Given a non-negative Jacobi matrix describing higher order recurrence relations for multiple orthogonal polynomials of type~II and corresponding linear forms of type I, a general strategy for constructing a pair of stochastic matrices, dual to each other, is provided. The corresponding Markov chains (or 1D random walks) allow, in one transition, to reach for the N-th previous states, to remain in the state or reach for the immediately next state. The dual Markov chains allow, in one transition, to reach for the N-th next states, to remain in the state or reach for immediately previous state. T…
We study the higher-order Euler polynomials and give the corresponding monic orthogonal polynomials, which are Meixner-Pollaczek polynomials with certain arguments and constant factors. Moreover, through a general connection between moments of random variables and the generalized Motzkin numbers, we can obtain a new recurrence formula and a matrix representation for the higher-order Euler polynomials, interpreting them as weighted lattice paths.
The Fisher information of the classical orthogonal polynomials with respect to a parameter is introduced, its interest justified and its explicit expression for the Jacobi, Laguerre, Gegenbauer and Grosjean polynomials found.
We characterize, up to a conjecture, probability distributions of all order finite moments having ultraspherical type generating functions for orthogonal polynomials.
Planar Orthogonal Polynomials as Type I Multiple Orthogonal Polynomials
本文证明关于修正高斯测度的平面正交多项式在整数指数条件下也是围道上的第一类多重正交多项式，并由此导出若干等价的 Riemann-Hilbert 问题，同时用新技术建立了 Lee-Yang 基本恒等式。
Quantum walks, the discrete wave equation and Chebyshev polynomials
本文受Terence Tao启发，提出从图上离散波动方程推导量子游走，将其描述为对随机游走转移矩阵应用Chebyshev多项式，从而解耦量子起源并证明格上扩散行为的二次改进。
We prove a projection formula for the four-parameter family of orthogonal polynomials that are a reparameterization of the polynomials in the Askey-Wilson class. By carefully analyzing the recurrence relations we manage to avoid using the explicit expression for the orthogonality measure, which would be cumbersome due to the complexity of the reparameterization.
A model of random walk on knot diagrams is used to study the Alexander polynomial and the colored Jones polynomial of knots. In this context, the inverse of the Alexander polynomial of a knot plays the role of an Ihara-Selberg zeta function of a directed weighted graph, counting with weights cycles of random walk on a 1-string link whose closure is the knot in question. The colored Jones polynomial then counts with weights families of ``self-avoiding'' cycles of random walk on the cabling of the 1-string link. As a consequence of such interpretations of the Alexander and colored Jones polynomi…
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