[Paper Review] The intermediate disorder regime for a directed polymer model on a hierarchical lattice

Tom Alberts, Jeremy Clark+1 more|arXiv (Cornell University)|Aug 19, 2015

Stochastic processes and statistical mechanics25 references3 citations

TL;DR

This paper studies the intermediate disorder regime in a directed polymer model on a hierarchical diamond lattice with branching number $b$ and segment number $s$. For $b < s$, it establishes weak convergence of the normalized partition function to a non-degenerate limit under scaling $\beta_n = \hat{\beta}(b/s)^{n/2}$, while for $b = s$, it identifies a critical $\kappa_b$ such that the variance of the partition function vanishes for $\hat{\beta} \leq \kappa_b$ and diverges otherwise, proving a central limit theorem in the former case.

ABSTRACT

We study a directed polymer model defined on a hierarchical diamond lattice, where the lattice is constructed recursively through a recipe depending on a branching number $b\in \mathbb{N}$ and a segment number $s\in \mathbb{N}$. When $b\leq s$ previous work [27] has established that the model exhibits strong disorder for all positive values of the inverse temperature $β$, and thus weak disorder reigns only for $β=0$ (infinite temperature). Our focus is on the so-called intermediate disorder regime in which the inverse temperature $β\equiv β_{n}$ vanishes at an appropriate rate as the size $n$ of the system grows. Our analysis requires separate treatment for the cases $b0$, the normalized partition function of the system converges weakly as $n o \infty$ to a distribution $\mathbf{L}(\widehatβ)$ depending continuously on the parameter $\widehatβ$. In the case $b=s$ we find a critical point in the behavior of the model when the inverse temperature is scaled as $β_{n}=\widehatβ/n$; for an explicitly computable critical value $κ_{b} &gt; 0$ the variance of the normalized partition function converges to zero with large $n$ when $\widehatβ\leq κ_{b}$ and grows without bound when $\widehatβ&gt;κ_{b}$. Finally, we prove a central limit theorem for the normalized partition function when $\widehatβ\leq κ_{b}$.

Motivation & Objective

• To analyze the intermediate disorder regime in directed polymer models on hierarchical diamond lattices where the inverse temperature $\beta_n$ scales with system size $n$.
• To determine the behavior of the normalized partition function under different scalings of $\beta_n$ for $b < s$ and $b = s$.
• To identify a critical threshold $\kappa_b$ for $b = s$ that separates vanishing and diverging variance of the partition function.
• To establish a central limit theorem for the normalized partition function when $\hat{\beta} \leq \kappa_b$ in the $b = s$ case.

Proposed method

• Uses recursive construction of the hierarchical diamond lattice with branching number $b$ and segment number $s$ to model the polymer paths.
• Applies a Gibbsian reweighting via inverse temperature $\beta_n$ that decays with system size $n$ to access the intermediate disorder regime.
• Analyzes the normalized partition function $\widetilde{W}_n(\beta_n; D_n)$ and its variance through recursive maps $\widetilde{M}_n^m(0)$ and related functions $r_m^{(n)}$.
• Employs Taylor expansions and asymptotic analysis to approximate the behavior of $r_n^{(n)}$ and derive the leading-order term $1 - r_n \sim \frac{b+1}{3(b-1)}\frac{\log n}{n}$.
• Uses Riemann sum approximation and bounds on error terms to derive the asymptotic behavior of $\widetilde{M}_n^n(0)$, which governs the variance of the partition function.
• Applies a second-order Taylor expansion to the inverse tangent function to relate the recursive dynamics of $r_m$ to the scaling of $\beta_n$.

Experimental results

Research questions

• RQ1 How does the normalized partition function behave in the intermediate disorder regime when $b < s$?
• RQ2 What is the critical threshold $\kappa_b$ for $b = s$ that separates vanishing and diverging variance of the normalized partition function?
• RQ3 Does the normalized partition function satisfy a central limit theorem when $\hat{\beta} \leq \kappa_b$ for $b = s$?
• RQ4 What is the precise asymptotic scaling of the variance of the normalized partition function in the $b = s$ case?

Key findings

• For $b < s$, the normalized partition function converges weakly to a non-degenerate limit $\mathbf{L}(\hat{\beta})$ when $\beta_n = \hat{\beta}(b/s)^{n/2}$, with $\hat{\beta} > 0$.
• For $b = s$, the variance of the normalized partition function converges to zero as $n \to \infty$ when $\hat{\beta} \leq \kappa_b$, and diverges when $\hat{\beta} > \kappa_b$, where $\kappa_b > 0$ is explicitly computable.
• A central limit theorem holds for the normalized partition function when $\hat{\beta} \leq \kappa_b$ in the $b = s$ case.
• The asymptotic behavior of $\widetilde{M}_n^n(0)$ is $\widetilde{M}_n^n(0) = \frac{6}{b+1}\frac{n}{\log n} + O\left(\frac{n}{\log^2 n}\right)$, which governs the variance in the $b = s$ case.
• The critical threshold $\kappa_b$ is derived from the recursive dynamics of the system and is independent of the environment distribution under weak moment assumptions.
• The analysis reveals that $1 - r_n \sim \frac{b+1}{3(b-1)}\frac{\log n}{n}$, which underpins the variance scaling and the existence of the critical $\kappa_b$.