Decay of correlations for the massless hierarchical Liouville model in infinite volume

TL;DR

This paper investigates the decay of correlations in a massless hierarchical Liouville model by analyzing the asymptotics of negative exponential moments of multiplicative chaos on a tree, revealing weak correlation decay under certain tilts.

Contribution

It establishes the first order asymptotics of negative exponential moments of multiplicative chaos in hierarchical models and demonstrates weak correlation decay under specific tilts, extending understanding of such stochastic processes.

Findings

Asymptotic behavior of exponential moments of multiplicative chaos is characterized.

Weak decay of correlations is proven under the exponential tilt of the model.

Methods apply to related branching random walk models, revealing similar asymptotics.

Abstract

Let $(A_v)_{v\in \mathcal{T}}$ be the balanced Gaussian Branching Random Walk on a $d$-ary tree $\mathcal{T}$ and let $M^A$ be the multiplicative chaos with parameter $\gamma \in (0, \sqrt{2\log d})$ constructed from $A$. In this work we establish the precise first order asymptotics of negative exponential moment of $M^A$, i.e.\ we prove that for $t_k = \lambda p_\gamma^k$ with $\lambda>0$ and $p_\gamma$ an explicit constant depending only on $\gamma$, we have as $k \to \infty$, \begin{equation} -\frac{1}{d^k} \log \mathbb{E}[e^{-\lambda p_\gamma^k M^A } ] \to h(\lambda), \end{equation} where $h\colon (0,\infty)\to \mathbb{R}$ is a non-explicit positive continuous function. This result allows us to study the law of $A$ tilted by $e^{-t_k M^A}$ for particular values of $\lambda$, with $k\to \infty$. In this setting we prove that the normalized $L^1$ norm of $A$ in generation $k-a$ is bounded and converges to $0$ when first $k\to \infty$ and then $a\to 0$. As an application we prove that in this setting, under the tilt $e^{-t_k M^A}$ and with $k\to \infty$, the Branching Random Walk $A$ exhibits a weak decay of correlation, which is not present in the non-tilted model. Our methods also apply to the usual Branching Random Walk $(S_v)_{v\in \mathcal{T}}$ and with $M^A$ replaced by $\frac{1}{2}(M^+ + M^- )$, where $M^+$ and $M^-$ are the multiplicative chaoses with parameter $\gamma \in (0, \sqrt{2\log d})$ constructed from $S$ and $-S$. In that case we prove that, as $k\to \infty$, \begin{equation} -\frac{1}{d^k} \log \mathbb{E}[e^{- \frac{\lambda p_\gamma^k}{2}( M^+ + M^-) }] \to \tilde h(\lambda), \end{equation} where $\tilde h\colon (0,\infty)\to \mathbb{R}$ is again a non-explicit positive continuous function.