Related papers: An Exceptional Convolutional Recurrence
Fractional revival occurs between two vertices in a graph if a continuous-time quantum walk unitarily maps the characteristic vector of one vertex to a superposition of the characteristic vectors of the two vertices. This phenomenon is…
A class of random discrete distributions $P$ is introduced by means of a recursive splitting of unity. Assuming supercritical branching, we show that for partitions induced by sampling from such $P$ a power growth of the number of blocks is…
We recast homogeneous linear recurrence sequences with fixed coefficients in terms of partial Bell polynomials, and use their properties to obtain various combinatorial identities and multifold convolution formulas. Our approach relies on a…
In this paper, we are concerned with mean hitting time $\langle\mathcal{H}\rangle$ for random walks on recursive growth tree networks that are built based on an arbitrary tree as the seed via implementing various primitive graphic…
Poly-trees are singly connected causal networks in which variables may arise from multiple causes. This paper develops a method of recovering ply-trees from empirically measured probability distributions of pairs of variables. The method…
We give four examples of families of orthogonal polynomials for which the coefficients in the recurrence relation satisfy a discrete Painlev\'e equation. The first example deals with Freud weights $|x|^\rho \exp(-|x|^m)$ on the real line,…
We introduce a new model of random tree that grows like a random recursive tree, except at some exceptional "doubling events" when the tree is replaced by two copies of itself attached to a new root. We prove asymptotic results for the size…
Every classical orthogonal polynomial system $p_n(x)$ satisfies a three-term recurrence relation of the type \[ p_{n+1}(x)=(A_nx+B_n)p_n(x)-C_np_{n-1}(x)~ (n=0,1,2,\ldots, p_{-1}\equiv 0), \] with $C_nA_nA_{n-1}>0$. Moreover, Favard's…
Distinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton-Watson trees. For example, let $\mathcal{T}_1$ be the…
We study the recurrence coefficients of the orthogonal polynomials with respect to a semi-classical extension of the Krawtchouk weight. We derive a coupled discrete system for these coefficients and show that they satisfy the fifth…
One class of random walks with infinite memory, so called elephant random walks, are simple models describing anomalous diffusion. We present a surprising connection between these models and bond percolation on random recursive trees. We…
Revivals of the coherent states of a deformed, adiabatically and cyclically varying oscillator Hamiltonian are examined. The revival time distribution is exactly that of Poincar\'{e} recurrences for a rotation map: only three distinct…
Begin continuous time random walks from every vertex of a graph and have particles coalesce when they collide. We use a duality relation with the voter model to prove the process is site recurrent on bounded degree graphs, and for…
Using combinatorial techniques, we derive a recurrence identity that expresses an exponential power sum with negative powers in terms of another exponential power sum with positive powers. Consequently, we derive a formula for the power sum…
Consider a class of null-recurrent randomly biased walks on a super-critical Gaton-Watson tree. We obtain the rates of convergence of the local times and the quenched local probability for the biased walk in the sub-diffusive case. These…
We investigate a quadratic dynamical system known as nonlinear recombinations. This system models the evolution of a probability measure over the Boolean cube, converging to the stationary state obtained as the product of the initial…
Random spanning trees are among the most prominent determinantal point processes. We give four examples of random spanning trees on ladder-like graphs whose rungs form stationary renewal processes or regenerative processes of order two,…
New differential-recurrence properties of dual Bernstein polynomials are given which follow from relations between dual Bernstein and orthogonal Hahn and Jacobi polynomials. Using these results, a fourth-order differential equation…
By means of the generating function method, a linear recurrence relation is explicitly resolved. The solution is expressed in terms of the Stirling numbers of both the first and the second kind. Two remarkable pairs of combinatorial…
We consider Reinforced Random Walks where transition probabilities are a function of the proportion of times the walk has traversed an edge. We give conditions for recurrence or transience. A phase transition is observed, similar to…