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

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.