Surface transition in the collapsed phase of a self-interacting walk adsorbed along a hard wall

A sharp asymptotics of the partition function for the collapsed interacting partially directed self-avoiding walk

In the present paper, we investigate the collapsed phase of the interacting partially-directed self-avoiding walk (IPDSAW) that was introduced in Zwanzig and Lauritzen (1968). We provide sharp asymptotics of the partition function inside the collapsed phase, proving rigorously a conjecture formulated in Guttmann (2015) and Owczarek et al. (1993). As a by-product of our result, we obtain that, inside the collapsed phase, a typical IPDSAW trajectory is made of a unique macroscopic bead, consisting of a concatenation of long vertical stretches of alternating signs, outside which only finitely man…

交互式部分定向自回避行走。从相变到坍缩相的几何结构

Interacting partially directed self avoiding walk. From phase transition to the geometry of the collapsed phase

在本文中，我们研究了一个模型的$1+1$维自相互作用和部分定向自回避行走，通常简称IPDSAW。系统的相互作用强度和自由能分别用$β$和$f$表示。已知IPDSAW在$β_c$处经历塌陷转变。我们提供了精确的渐近自由能接近临界，也就是说，我们表明，$f(β_c-ε)\sim γε^{3/2}$，其中$γ$是明确计算和解释相关的连续模型。我们还建立了坍缩相$(β>β_c)$内随机游动的一些路径性质。我们证明了聚合物所采用的几何构象是由一系列长的垂直伸展，相互吸引，形成一个独特的宏观珠，我们确定了水平延伸的随机游动内的崩溃阶段，我们建立了一个确定性的武尔夫形状的宏观珠的重新缩放的信封收敛。

受垂直壁吸引的相互作用部分有向自回避行走的相图

Phase diagram of the interacting partially directed self-avoiding walk attracted by a vertical wall

本文研究受垂直壁吸引的相互作用部分有向自回避行走，证明塌缩相内存在表面转变，并给出配分函数的精确渐近，严格证实了物理学家的猜想。

交互部分定向自回避行走：尺度限制

Interacting partially directed self avoiding walk : scaling limits

本文致力于研究$1+1$维自相互作用和部分定向自回避行走，通常简称为IPDSAW，并在\cite{ZL68}中介绍了Zwanzig和Lauritzen，以研究浸在不良溶剂中的均聚物的塌陷转变。在\cite{POBG93}中，物理学家展示了关于路径的一些几何特征随着其长度$L$发散而典型增长率的数值结果。从这个角度来看，感兴趣的量是路径在水平轴上的投影（也称为水平扩展）和垂直轴上，这对于定义路径的下包络和上包络是有用的。借助一种新的随机游走表示，我们在\cite{CNGP13}中证明了路径在其塌陷状态下像$\sqrt{L}$一样水平增长，一旦由$\sqrt{L}$垂直和水平地重新缩放，其上包络和下包络收敛到某些确定的Wulff形状。在本文中，我们把路径的几何研究的几个步骤。在扩展的制度，我们证明了一个大数定律的聚合物的水平延伸重新缩放其总长度$L$，我们提供了一个精确的渐近的分拆函数，我们表明，它的下，上信封，一旦重新缩放的时间由$L$和空间由$\sqrt{L}$，收敛到相同的布朗运动。在临界状态下，我们确定的极限分布的水平扩展重新缩放的$L^{2/3}$，我们表明，过剩的分拆函数衰减为$L^{2/3}$与一个明确的前因子。在崩溃的制度，我们确定了联合限制分布的波动的上限和下限的信封周围的相关限制伍尔夫形状，重新缩放的时间由$\sqrt{L}$和空间由$L^{1/4}$。

Interacting partially directed self-avoiding walk: a probabilistic perspective

We review some recent results obtained in the framework of the 2-dimensional Interacting Self-Avoiding Walk (ISAW). After a brief presentation of the rigorous results that have been obtained so far for ISAW we focus on the Interacting Partially Directed Self-Avoiding Walk (IPDSAW), a model introduced in Zwanzig and Lauritzen (1968) to decrease the mathematical complexity of ISAW. In the first part of the paper, we discuss how a new probabilistic approach based on a random walk representation (see Nguyen and Pétrélis (2013)) allowed for a sharp determination of the asymptotics of the free energ…

Collapse transition of the interacting prudent walk

This article is dedicated to the study of the 2-dimensional interacting prudent self-avoiding walk (referred to by the acronym IPSAW) and in particular to its collapse transition. The interaction intensity is denoted by $β>0$ and the set of trajectories consists of those self-avoiding paths respecting the prudent condition, which means that they do not take a step towards a previously visited lattice site. The IPSAW interpolates between the interacting partially directed self-avoiding walk (IPDSAW) that was analyzed in details in, e.g., [Zwanzig and Lauritzen, 1968], [Brak et al., 1992], [Carm…