We study a non-directed polymer model in random environments. The polymer is modeled by a simple symmetric random walk S on Z(d) with d >= 2, and the random environment is modeled by i.i.d. random variables whose tail probability decays polynomially. The interaction between the polymer and the random environment is captured by a Gibbs transform: at time N, the law of S is tilted by the factor exp(Sigma(xERN)(beta omega(x) - h)), where R-N is the range of S up to time N, beta >= 0 is the inverse temperature, and h is an element of R is an external field. By appropriately tunin beta = beta(N) and h = h(N), we establish the phase diagram, analyze the fluctuations of S under theg Gibbs transform, and derive the scaling limits of the (logarithmic) partition function. This paper is a follow-up work of Berger et al. (2022b). The main novelty and challenge arise from tuning the external field h, which brings in various range penalties, unlike in Berger et al. (2022b), where h is fixed and serves only as a centering term for the random environment.
In this paper, we study the random sum S-Zn of independent and identically distributed random variables {X-i}, where {Z(n)} is a supercritical branching process in an i.i.d. environment xi . We assume X-1 is in the domain of attraction of an alpha-stable law with 1 < alpha < 2 and EX1 = 0. We shall prove the large deviations of SZ(n), first in the case that Z(1) has linear fractional distribution, then in some general case.
We consider the biased random walk on a tree constructed from the set of finite self-avoiding walks on a lattice (self-avoiding tree), and use it to construct probability measures on infinite self-avoiding walks. The limit measure (if it exists) obtained when the bias converges to its critical value is conjectured to coincide with the weak limit of the uniform SAW. In this paper, we prove that the critical value of the biased random walk on the self-avoiding tree is equal to the inverse of the connective constant of the lattice. Along the way, we describe the relationship between the biased random walk on the self-avoiding tree and the percolation on the lattice.
The aim of this paper is to study intertwining relations for Laguerre processes with inverse temperature beta >= 1 and parameter alpha > -1. We introduce a Markov kernel that depends on both beta and alpha, and establish new intertwining relations for the beta-Laguerre processes using this kernel. A key observation is that Jack symmetric polynomials are eigenfunctions of our Markov kernel, which allows us to apply a method established by Ramanan and Shkolnikov. Additionally, as a by-product, we derive an integral formula for multivariate Laguerre polynomials and multivariate hypergeometric functions associated with Jack polynomials.
Consider the p-multiple range R(p) N which counts the number of points visited exactly p >= 1 times by a one-dimensional simple symmetric random walk starting at [alpha N], for alpha is an element of (0, 1), up to the time of exit from DN = {0, 1, ... , N}. We show that R(p) N / log(N) converges weakly to the law of an exponential random variable with mean 1/2. Moreover, we show, by the method of moments, that the collection of scaled multiple ranges {R(p) N / log(N) : p >= 1} in the limit is totally correlated.
