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
摘要：
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。
英文摘要：
In this paper, we investigate a model for a $1+1$ dimensional self-interacting and partially directed self-avoiding walk, usually referred to by the acronym IPDSAW. The interaction intensity and the free energy of the system are denoted by $β$ and $f$, respectively. The IPDSAW is known to undergo a collapse transition at $β_c$. We provide the precise asymptotic of the free energy close to criticality, that is we show that $f(β_c-ε)\sim γε^{3/2}$ where $γ$ is computed explicitly and interpreted in terms of an associated continuous model. We also establish some path properties of the random walk inside the collapsed phase $(β>β_c)$. We prove that the geometric conformation adopted by the polymer is made of a succession of long vertical stretches that attract each other to form a unique macroscopic bead, we identify the horizontal extension of the random walk inside the collapsed phase and we establish the convergence of the rescaled envelope of the macroscopic bead towards a deterministic Wulff shape.
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