Related papers: On $\mathbf{2\times2}$ determinants originating fr…
In this paper we study two natural models of \textit{random temporal} graphs. In the first, the \textit{continuous} model, each edge $e$ is assigned $l_e$ labels, each drawn uniformly at random from $(0,1]$, where the numbers $l_e$ are…
A Pick function of $d$ variables is a holomorphic map from $\Pi^d$ to $\Pi$, where $\Pi$ is the upper halfplane. Some Pick functions of one variable have an asymptotic expansion at infinity, a power series $\sum_{n=1}^\infty \rho_n z^{-n}$…
In this note, we investigate the paratrophic determinants attached to the multiplicative semigroup $\mathbb{Z}/N\mathbb{Z}$. We show that, via discrete Fourier, cosine, and sine transforms, these determinants factor into products of group…
The six Painlev\'e transcendants which originally appeared in the studies of ordinary differential equations have been found numerous applications in physical problems. The well-known examples among which include symmetry reduction of the…
We develop a terminal-defect method for the double Dixie cup problem and use it to prove the finite-variance extremality conjecture of Doumas and Papanicolaou. For every \(m\ge1\) and \(N\ge2\), among all positive coupon probability vectors…
Let $X$ be a real valued random variable with an unbounded distribution $F$ and let $Y$ be a nonnegative valued random variable with a unbounded distribution $G$, which satisfy that \begin{eqnarray*} P(X>x|Y=y)\sim h(y)P(X>x)…
We prove the following Return Times Theorem along the sequence of prime times, the first extension of the Return Times Theorem to arithmetic sequences: For every probability space, $(\Omega,\nu)$, equipped with a measure-preserving…
The survival probability of immobile targets, annihilated by a population of random walkers on inhomogeneous discrete structures, such as disordered solids, glasses, fractals, polymer networks and gels, is analytically investigated. It is…
We calculate, for a branching random walk $X_n(l)$ to a leaf $l$ at depth $n$ on a binary tree, the positive integer moments of the random variable $\frac{1}{2^{n}}\sum_{l=1}^{2^n}e^{2\beta X_n(l)}$, for $\beta\in\mathbb{R}$. We obtain…
Let $(Z_n)$ be a supercritical branching process in an independent and identically distributed random environment $\xi$. We show the exact decay rate of the probability $\mathbb{P}(Z_n=j | Z_0 = k)$ as $n \to \infty$, for each $j \geq k,$…
We consider the extreme value statistics of correlated random variables that arise from a Langevin equation. Recently, it was shown that the extreme values of the Ornstein-Uhlenbeck process follow a different distribution than those…
This paper is concerned with an initial-boundary value problem of the two-dimensional inhomogeneous primitive equations with density-dependent viscosity. The global well-posedness of strong solutions is established, provided the initial…
For a large class of non-negative initial data, the solutions to the quasilinear viscous Hamilton-Jacobi equation $\partial\_t u-\Delta\_p u+|\nabla u|^q=0$ in $(0,\infty)\times\real^N$ are known to vanish identically after a finite time…
This paper studies the asymptotic properties of weighted sums of the form $Z_n=\sum_{i=1}^n a_i X_i$, in which $X_1, X_2, \ldots, X_n$ are i.i.d.~random variables and $a_1, a_2, \ldots, a_n$ correspond to either eigenvalues or singular…
Numerically we solve the microscopic deterministic equations of motion with random initial states for the two-dimensional $\phi^4$ theory. Scaling behavior of the persistence probability at criticality is systematically investigated and the…
Given an autoregressive process X of order p (i.e. X_n = a_1 X_{n-1} + ...+ a_p X_{n_p} + Y_n where the random variables Y_1, Y_2, ... are i.i.d.), we study the asymptotic behaviour of the probability that the process does not exceed a…
A recently developed model of random walks on a $D$-dimensional hyperspherical lattice, where $D$ is {\sl not} restricted to integer values, is extended to include the possibility of creating and annihilating random walkers. Steady-state…
In a companion paper \cite{jon-fei}, we established asymptotic formulae for the joint moments of derivatives of the characteristic polynomials of CUE random matrices. The leading order coefficients of these asymptotic formulae are expressed…
We consider the Hankel determinant generated by the moments of the even weight function ${\rm e}^{-x^2}(A+B\theta(x^2-a^2)), x\in(-\infty,+\infty), a>0, A\ge0, A+B\ge0$. It is intimately related to the gap probability of the Gaussian…
E. Heine in the 19th century studied a system of orthogonal polynomials associated with the weight $\left[x(x-\alpha)(x-\beta)\right]^{-\frac{1}{2}}$, $x\in[0,\alpha]$, $0<\alpha<\beta$. A related system was studied by C. J. Rees in 1945,…