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

Source 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 ) , where $$h:(0,\infty )\rightarrow \mathbb {R}$$ h : ( 0 , ∞ ) → R is a non-explicit positive continuous function. This result allows us to study the law of A tilted by $$e^{-t_k M^A}$$ e - t k M A for particular values of $$\lambda$$ λ , with $$k\rightarrow \infty$$ k → ∞ . In this setting we prove that the normalized $$L^1$$ L 1 norm of A in generation $$k-a$$ k - a is bounded and converges to 0 when first $$k\rightarrow \infty$$ k → ∞ and then $$a\rightarrow 0$$ a → 0 . As an application we prove that in this setting, under the tilt $$e^{-t_k M^A}$$ e - t k M A and with $$k\rightarrow \infty$$ k → ∞ , the Branching Random Walk A exhibits a weak decay of correlations, which is not present in the non-tilted model. Our methods also apply to the usual Branching Random Walk $$(S_v)_{v\in \mathcal {T}}$$ ( S v ) v ∈ T and with $$M^A$$ M A replaced by $$\frac{1}{2}(M^{+}+ M^{-})$$ 1 2 ( M + + M - ) , where $$M^{+}$$ M + and $$M^{-}$$ M - are the multiplicative chaoses with parameter $$\gamma \in (0, \sqrt{2\log d})$$ γ ∈ ( 0 , 2 log d ) constructed from S and $$-S$$ - S . In that case we prove that, as $$k\rightarrow \infty$$ k → ∞ , $$\begin{aligned} -\frac{1}{d^k} \log \mathbb {E}[e^{- \frac{\lambda {p}^k}{2}( M^{+}+ M^{-}) }] \rightarrow \tilde{h}(\lambda ), \end{aligned}$$ - 1 d k log E [ e - λ p k 2 ( M + + M - ) ] → h ~ ( λ ) , where $$\tilde{h}:(0,\infty )\rightarrow \mathbb {R}$$ h ~ : ( 0 , ∞ ) → R is again a non-explicit positive continuous function. Our models are motivated by Euclidean field theory and can be seen as hierarchical versions of the massless Liouville and the sinh-Gordon field theory in infinite volume. From this perspective our analysis sheds new light on the existence and the decay or correlations in these models, which are among the major open questions in this area.