[Paper Review] Tightness conditions for polymer measures

Francesco Caravenna, Giambattista Giacomin+1 more|ArXiv.org|Feb 12, 2007

Point processes and geometric inequalities12 references3 citations

TL;DR

This paper establishes sufficient conditions for tightness in $C([0,1])$ of diffusively rescaled polymer measures with a decoupling between zero level sets and excursions. It proves that tightness holds if the rescaled bulk and final excursion laws are tight and a uniform integrability condition on the maximum squared height is satisfied, enabling scaling limit results for polymer and interface models.

ABSTRACT

We give sufficient conditions for tightness in the space C([0,1]) for sequences of probability measures which enjoy a suitable decoupling between zero level set and excursions. Applications of our results are given in the context of (homogeneous, periodic and disordered) random walk models for polymers and interfaces.

Motivation & Objective

• To establish sufficient conditions for tightness in $C([0,1])$ of sequences of polymer measures under diffusive rescaling.
• To analyze the scaling limit behavior of random walk models for polymers and interfaces with pinning, wetting, or copolymer features.
• To identify conditions under which the rescaled path measures converge weakly, focusing on the decoupling between zero set and excursion laws.
• To provide a general framework applicable to both homogeneous and disordered (e.g., copolymer) models.

Proposed method

• Define a class of probability measures $\boldsymbol{\mathrm{P}}_N$ on $\mathbb{R}^N$ where the zero level set is governed by $p_N$, and excursions between zeros are i.i.d. under $P_t$ or $P_t^f$.
• Apply diffusive rescaling via the map $X^N: \mathbb{R}^N \to C([0,1])$, defining $\boldsymbol{\mathrm{Q}}_N := \boldsymbol{\mathrm{P}}_N \circ (X^N)^{-1}$.
• Use the modified continuity modulus $\widetilde{\Gamma}(\delta)$ to reduce tightness to control over excursions within the same path component.
• Leverage conditional independence of excursions given the zero set to decouple the analysis across components.
• Establish tightness by verifying that $\sup_N \boldsymbol{\mathrm{Q}}_N(\widetilde{\Gamma}(\delta) > \gamma) \to 0$ as $\delta \to 0$, relying on uniform integrability of the maximum height.
• Prove the key condition (2.1) via a pathwise estimate using the reflection principle and local central limit theorem bounds on first-passage probabilities.

Experimental results

Research questions

• RQ1 Under what conditions is the sequence of rescaled polymer measures $\boldsymbol{\mathrm{Q}}_N$ tight in $C([0,1])$?
• RQ2 How can the decoupling between zero level sets and excursions be exploited to prove tightness without assumptions on the zero set law $p_N$?
• RQ3 What uniform integrability condition ensures that large excursions do not disrupt convergence in distribution?
• RQ4 Can the framework be applied to disordered and periodic copolymer models, and wetting/pinning models?
• RQ5 Does the maximum height of excursions, normalized by $\sqrt{N}$, satisfy a uniform integrability condition necessary for tightness?

Key findings

• Tightness of $\boldsymbol{\mathrm{Q}}_N$ in $C([0,1])$ holds if the rescaled bulk excursion laws $Q_N$ and final excursion laws $Q_N^f$ are tight.
• The key condition (2.1) — uniform integrability of $\max_{0\leq i\leq n} y_i^2 / n$ under the excursion law — is sufficient for tightness.
• For symmetric random walks with $\mathbb{P}(S_1 = +1) \in (0,1/2)$, the condition (2.1) holds due to exponential tail bounds on first-passage times.
• The bound $f_n(a) \leq C / (1 + a^2)$ on the conditional probability of large maximum height ensures uniform integrability.
• The result applies directly to disordered and periodic copolymer models, wetting models, and pinning models based on random walks.
• The proof relies on the reflection principle and local central limit theorem estimates to control the tail behavior of the maximum height under constrained excursion laws.