Pinning by a sparse potential

随机钉扎模型与稀疏随机环境中随机游走的关联

A connection between the random pinning model and random walks in sparse random environments

本研究建立稀疏随机环境下一维随机游走与随机钉扎模型的关联，证明二者配分函数与返回原点平均次数的等价性，进而得到两类退火设定下返回次数的可积性结论。

The zeros of the partition function of the pinning model

We aim at understanding for which (complex) values of the potential the pinning partition function vanishes. The pinning model is a Gibbs measure based on discrete renewal processes with power law inter-arrival distributions. We obtain some results for rather general inter-arrival laws, but we achieve a substantially more complete understanding for a specific one parameter family of inter-arrivals. We show, for such a specific family, that the zeros asymptotically lie on (and densely fill) a closed curve that, unsurprisingly, touches the real axis only in one point (the critical point of the m…

Pinning of polymers and interfaces by random potentials

We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. Disorder is introduced by, for example, having the interaction vary from one monomer to another, as a constant $u$ plus i.i.d. mean-0 randomness. There is a critical value of $u$ above which the polymer is pinned, placing a positive fraction of its monomers at 0 with high probability. This critical point may differ for the quenched, annealed and deterministic cases. We show that self-averaging occurs, meani…

Random potentials for pinning models with Laplacian interactions

We consider a statistical mechanics model for biopolymers. Sophisticated polymer chains, such as DNA, have stiffness when they stretch chains. The Laplacian interaction is used to describe the stiffness. Also, the surface between two media has an attraction force, and the force will pull the chain back to the surface. In this paper, we deal with the random potentials when the monomers interact with the random media. Although these models are different from the pinning models studied before, the result about the gap between the annealed critical point and the quenched critical point stays the s…

Pinning of a renewal on a quenched renewal

We introduce the pinning model on a quenched renewal, which is an instance of a (strongly correlated) disordered pinning model. The potential takes value 1 at the renewal times of a quenched realization of a renewal process $σ$, and $0$ elsewhere, so nonzero potential values become sparse if the gaps in $σ$ have infinite mean. The "polymer" -- of length $σ_N$ -- is given by another renewal $τ$, whose law is modified by the Boltzmann weight $\exp(β\sum_{n=1}^N \mathbf{1}_{\{σ_n\inτ\}})$. Our assumption is that $τ$ and $σ$ have gap distributions with power-law-decay exponents $1+α$ and $1+\tilde…

通过稀疏定位实现熵独立

Entropic independence via sparse localization

本文提出稀疏定位框架，通过稀疏定位假设ℓ₂独立性，实现熵稳定性和熵独立性，并证明在有限度图中均匀分布的独立集熵近似守恒。