Decay of Correlations for the Massless Hierarchical Liouville Model in Infinite Volume

Abstract

Abstract Let $$A=(A_v)_{v\in \mathcal {T}}$$ A = ( A v ) v ∈ T be the balanced Gaussian Branching Random Walk on a d -ary tree $$\mathcal {T}$$ T and let $$M^A$$ M A be the multiplicative chaos with parameter $$\gamma \in (0, \sqrt{2\log d})$$ γ ∈ ( 0 , 2 log d ) constructed from A . In this work we establish the precise first order asymptotics of negative exponential moment of $$M^A$$ M A , i.e. we prove that for $$t_k = \lambda {p}^k$$ t k = λ p k with $$\lambda >0$$ λ > 0 and $${p}$$ p an explicit constant depending only on $$\gamma $$ γ and d , we have as $$k \rightarrow \infty $$ k → ∞ , $$\begin{aligned} -\frac{1}{d^k} \log \mathbb {E}[e^{-\lambda {p}^k M^A} ] \rightarrow h(\lambda ), \end{aligned}$$ - 1 d k log E [ e - λ p k M A ] → h ( λ