Random Polymer Models

Abstract

RANDOM POLYMER MODELS

The quenched critical point of a diluted disordered polymer model

“…The existence of this limit, both P(dω)-a.s. and in L 1 (P), and the fact that f (β, p) is nonrandom are proved in [8] via super-additivity arguments. Notice that trivially Z β, p N ,ω ≥ 1 and hence f (β, p) ≥ 0 for all β, p. Zero is in fact the contribution to the free energy of the paths that never touch the wall: indeed, by restricting to the set of random walk trajectories that stay strictly positive until time N , one has By a standard coupling on the environment, it is clear that the function p → f (β, p) is nondecreasing.…”

Concentration under scaling limits for weakly pinned Gaussian random walks

“…We sometimes denote the partition functions as Z When d = 1 and m = 0, the Markov chain (φ i ∈ R (or ∈ R + )) i∈D N may be interpreted as the heights of interfaces located in a plane measured from the position i on a reference line (x-axis), so that the system is called (1 + 1)-dimensional interface model with δ-pinning at 0, see [3,5,8,15]. See [14] for a relation to the polymer models.…”

“…The Markov chain, being transient at ε = 0, turns to be recurrent when the strength ε of the attractive force toward 0 increases and exceeds the critical value ε c or ε + c ; see [14] for random walks with discrete values. The asymptotic behavior of the free energies ξ ε and ξ ε,+ for ε close to their critical values is studied in Appendix A.…”

“…In particular, these will imply the statements in Theorem 1.1. A similar method is used in [3,14]. We will omit the subscript r of the partition functions, for example, Z 0,0,ε N ,r is simply denoted by Z 0,0,ε N in this section.…”

Long memory in a linear stochastic Volterra differential equation

“…The discrete analogue of Eq. (1.2) with positive summable kernel of infinite moment corresponds to the renewal sequence of a null-recurrent Markov chain [20], and under similar additional assumptions on the kernel, the hyperbolic decay of the sequence relies upon well-known results by Garsia and Lamperti [18] and Isaac [23]. Our second class of results in this paper employ the convergence rate of the resolvent r to investigate the long memory properties of the solution of the Itô-Volterra differential equation (1.1) and its discrete analogue.…”