[Paper Review] Gaussian fluctuations for the directed polymer partition function for $d\geq 3$ and in the whole $L^2$-region

Clément Cosco, Shuta Nakajima|arXiv (Cornell University)|Mar 3, 2019

Stochastic processes and statistical mechanics9 references4 citations

TL;DR

This paper establishes Gaussian fluctuations for the tail of the normalized partition function in the directed polymer model on $ℤ^d$ for $d \geq 3$, extending previous results from a restricted high-temperature region to the entire $L^2$-region. By leveraging a local limit theorem and homogenization techniques instead of fourth-moment computations, it proves that $n^{(d-2)/4}(W_\infty - W_n)$ converges in distribution to $\sigma W_\infty G$, where $G$ is a standard Gaussian and the convergence is mixing.

ABSTRACT

We consider the discrete directed polymer model with i.i.d. environment and we study the fluctuations of the tail $n^{(d-2)/4}(W_\infty - W_n)$ of the normalized partition function. It was proven by Comets and Liu, that for sufficiently high temperature, the fluctuations converge in distribution towards the product of the limiting partition function and an independent Gaussian random variable. We extend the result to the whole $L^2$-region, which is predicted to be the maximal high-temperature region where the Gaussian fluctuations should occur under the considered scaling. To do so, we manage to avoid the heavy 4th-moment computation and instead rely on the local limit theorem for polymers and homogenization.

Motivation & Objective

• To extend the known result on Gaussian fluctuations of the polymer partition function tail from a restricted high-temperature region to the entire $L^2$-region.
• To establish the convergence of $n^{(d-2)/4}(W_\infty - W_n)$ in distribution to a product of the limiting partition function and an independent Gaussian.
• To avoid the standard and technically demanding fourth-moment computation by employing a local limit theorem and homogenization techniques.
• To confirm that the $L^2$-region is the maximal high-temperature regime where such Gaussian fluctuations occur under the given scaling.

Proposed method

• Utilizes the martingale structure of the normalized partition function $W_n$, which converges almost surely to $W_\infty$ in the weak disorder regime.
• Applies a local limit theorem for polymers to control the fluctuation behavior of the polymer path in the $L^2$-region.
• Employs homogenization techniques to analyze the asymptotic behavior of the difference $W_\infty - W_n$ at the scale $n^{(d-2)/4}$.
• Replaces the traditional 4th-moment computation with a more robust approach based on independence and moment estimates of backward martingales.
• Uses the mixing property of convergence to establish that the limiting Gaussian is independent of the limiting partition function $W_\infty$.
• Applies Taylor expansion and uniform integrability to relate the logarithmic fluctuation $\log W_\infty - \log W_n$ to the linear fluctuation $W_\infty - W_n$.

Experimental results

Research questions

• RQ1 Does the Gaussian fluctuation result for the directed polymer partition function tail hold throughout the entire $L^2$-region, not just in a small high-temperature subset?
• RQ2 Can the standard 4th-moment computation be avoided in proving Gaussian fluctuations for the polymer partition function?
• RQ3 Is the $L^2$-region indeed the maximal high-temperature regime where such Gaussian fluctuations occur under the scaling $n^{(d-2)/4}$?
• RQ4 What is the precise nature of the limiting fluctuation distribution for $W_\infty - W_n$ in the $L^2$-region?

Key findings

• The fluctuation $n^{(d-2)/4}(W_\infty - W_n)$ converges in distribution to $\sigma W_\infty G$, where $G$ is a standard Gaussian random variable independent of $W_\infty$.
• The convergence is mixing, meaning the limiting Gaussian is independent of the limiting partition function $W_\infty$.
• The result holds for all $\beta \in (0, \beta_2)$, covering the entire $L^2$-region, which is the maximal region predicted for such fluctuations.
• The proof avoids the 4th-moment computation by using a local limit theorem and homogenization, offering a new and more robust analytical approach.
• The asymptotic behavior of the logarithmic fluctuation $\log W_\infty - \log W_n$ is shown to satisfy the same limiting Gaussian distribution as the linear fluctuation.
• The convergence rate is consistent with the scaling $n^{(d-2)/4}$, which is known to be critical for Gaussian fluctuations in $d \geq 3$.