We study a discrete random walk on a one-dimensional finite lattice, where each state has different probabilities to move one step forward, backward, staying for a moment or being absorbed. We obtain expected number of arrivals and expected time until absorption using a new concept: Fibonacci matrices.
We consider three matrix models of order 2 with one random entry $ε$ and the other three entries being deterministic. In the first model, we let $ε\sim\textrm{Bernoulli}\left(\frac{1}{2}\right)$. For this model we develop a new technique to obtain estimates for the top Lyapunov exponent in terms of a multi-level recursion involving Fibonacci-like sequences. This in turn gives a new characterization for the Lyapunov exponent in terms of these sequences. In the second model, we give similar estimates when $ε\sim\textrm{Bernoulli}\left(p\right)$ and $p\in [0,1]$ is a parameter. Both of these mode…
A class of even walks and divergence of high moments of large Wigner random matrices
我们研究截断Wigner n × n随机矩阵的高矩，通过使用其表示为加权偶数闭游动集合W上的和。我们构造W的子集W'，使得对于任意阶n^{1/6+ n}的截断，只要矩阵元素的概率分布使得它的第十二阶矩不存在，则相应的和在大n的极限和与n^{2/3}成比例的矩数的极限中发散，n>0。这使得我们可以提出一个假设，即第十二阶矩的有限性是大型Wigner随机矩阵高阶矩的普遍上界的必要条件。
The purpose of this note is twofold: firstly to improve the known results on variation of extreme eigenvalues of birth and death matrices and random walk matrices; and secondly to progress towards the solution of a thirty years old open problem concerning the variation of eigenvalues of these matrices. Keywords: Birth and death matrices, random walk matrices, eigenvalues, monotonicity
A pointed graph $(Γ,v_0)$ induces a family of transition matrices in Wildberger's construction of a hermitian hypergroup via a random walk on $Γ$ starting from $v_0$. We will give a necessary condition for producing a hermitian hypergroup as we assume a weaker condition than the distance-regularity for $(Γ,v_0)$. The condition obtained in this paper connects the transition matrices and the adjacency matrices associated with $Γ$.
It is well known that there are close connections between non-intersecting processes in one dimension and random matrices, based on the reflection principle. There is a generalisation of the reflection principle for more general (e.g. planar) processes, due to S. Fomin, in which the non-intersection condition is replaced by a condition involving loop-erased paths. In the context of independent Brownian motions in suitable planar domains, this also has close connections to random matrices. An example of this was first observed by Sato and Katori (Phys. Rev. E, 83, 2011). We present further exam…
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…
We consider random walks on the cone of $m \times m$ positive definite matrices, where the underlying random matrices have orthogonally invariant distributions on the cone and the Riemannian metric is the measure of distance on the cone. By applying results of Khare and Rajaratnam (Ann. Probab., 45 (2017), 4101--4111), we obtain inequalities of Hoffmann-Jørgensen type for such random walks on the cone. In the case of the Wishart distribution $W_m(a,I_m)$, with index parameter $a$ and matrix parameter $I_m$, the identity matrix, we derive explicit and computable bounds for each term appearing i…
Consider the random walk on the $n \times n$ upper triangular matrices with ones on the diagonal and elements over $\mathbb{F}_p$ where we pick a row at random and either add it or subtract it from the row directly above it. The main result of this paper is to prove that the dependency of the mixing time on $p$ is $p^2$. This is proven by combining super-character theory and comparison theory arguments.
We consider UL and LU stochastic factorizations of the transition probability matrix of a random walk on the integers, which is a doubly infinite tridiagonal stochastic Jacobi matrix. We give conditions on the free parameter of both factorizations in terms of certain continued fractions such that this stochastic factorization is always possible. By inverting the order of the factors (also known as a Darboux transformation) we get new families of random walks on the integers. We identify the spectral matrices associated with these Darboux transformations (in both cases) which are basically conj…
We consider discrete (time and space) random walks confined to the quarter plane, with jumps only in directions $(i,j)$ with $i+j \geq 0$ and small negative jumps, i.e., $i,j \geq -1$. These walks are called singular, and were recently intensively studied from a combinatorial point of view. In this paper, we show how the compensation approach introduced in the 90ies by Adan, Wessels and Zijm may be applied to compute positive harmonic functions with Dirichlet boundary conditions. In particular, in case the random walks have a drift with positive coordinates, we derive an explicit formula for t…
In this work we set up the generating function of the ultimate time survival probability $φ(u+1)$, where $$φ(u)=\mathbb{P}\left(\sup_{n\geqslant 1}\sum_{i=1}^{n}\left(X_i-κ\right)<u\right)$$ and $u\in\mathbb{N}_0,\,κ\in\mathbb{N}$, and the random walk $\left\{\sum_{i=1}^{n}X_i,\,n\in\mathbb{N}\right\}$ consists of independent and identically distributed random variables $X_i$, which are non-negative and integer valued. We also give expressions of $φ(u)$ via the roots of certain polynomials. Based on the proven theoretical statements, we give several examples on $φ(u)$ and its generating functi…
We study point process convergence for sequences of iid random walks. The objective is to derive asymptotic theory for the extremes of these random walks. We show convergence of the maximum random walk to the Gumbel distribution under the existence of a $(2+δ)$th moment. We make heavily use of precise large deviation results for sums of iid random variables. As a consequence, we derive the joint convergence of the off-diagonal entries in sample covariance and correlation matrices of a high-dimensional sample whose dimension increases with the sample size. This generalizes known results on the …
Let $A_kA_{k-1}\cdots A_1$ be product of some nonnegative 2-by-2 matrices. In general, its elements are hard to evaluate. Under some conditions, we show that $\forall i,j\in\{1,2\},$ $(A_kA_{k-1}\cdots A_1)_{i,j}\sim c\varrho(A_k)\varrho(A_{k-1})\cdots \varrho(A_1)$ as $k\rightarrow\infty,$ where $\varrho(A_n)$ is the spectral radius of the matrix $A_n$ and $c\in(0,\infty)$ is some constant, so that the elements of $A_kA_{k-1}\cdots A_1$ can be estimated. As applications, consider the maxima of certain excursions of (2,1) and (1,2) random walks with asymptotically zero drifts. We get some deli…
This article rigorously analyzes the meeting time between pursuers and evaders performing random walks on digraphs. There exist several bounds on the expected meeting time between random walkers on graphs in the literature, however, closed-form expressions are limited in scope. By utilizing the notion that multiple random walks on a common graph can be understood as a single random walk on the Kronecker product graph, we are able to provide the first analytic expression for the meeting time in terms of the transition matrices of the random walkers when modeled by either discrete-time Markov ch…
We consider a randomized urn model with objects of finitely many colors. The replacement matrices are random, and are conditionally independent of the color chosen given the past. Further, the conditional expectations of the replacement matrices are close to an almost surely irreducible matrix. We obtain almost sure and $L^1$ convergence of the configuration vector, the proportion vector and the count vector. We show that first moment is sufficient for i.i.d.\ replacement matrices independent of past color choices. This significantly improves the similar results for urn models obtained in Athr…
We classify extremal traces on the seven direct limit algebras of noncrossing partitions arising from the classification of free partition quantum groups of Banica-Speicher (arXiv:0808.2628) and Weber (arXiv:1201.4723). For the infinite-dimensional Temperley-Lieb-algebra (corresponding to the quantum group $O^+_N$) and the Motzkin algebra ($B^+_N$), the classification of extremal traces implies a classification result for well-known types of central random lattice paths. For the $2$-Fuss-Catalan algebra ($H_N^+$) we solve the classification problem by computing the \emph{minimal or exit bounda…
We define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph $G$, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW by using a determinant expression for the generalized weighted zeta function of $G$. As applications, we give the spectrum of the transition probability matrices for the CRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs. Furthermore, we consider another type of the CRW on a graph.
