Ofer Zeitouni

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.

We compute the second order asymptotics of the maximum of the absolute value of the log-characteristic polynomial of random Jacobi matrices whose coefficients satisfy some exponential integrability condition. In particular, by the triadiagonal representation of Dumitriu and Eldelman of Gaussian $β$ Ensembles, this result partially confirms the Fydorov-Simm conjecture.

We establish a strong coupling between the Liouville model and the Gaussian free field on the two dimensional torus in the $L^1$ phase $β\in (0, 8π)$, such that the difference of the two fields is a Hölder continuous function. The coupling originates from a Polchinski renormalisation group approach, which was previously used to prove analogous results for other Euclidean field theories in dimension two. Our main observations for the Liouville model are that the Polchinski flow has a definite sign and can be controlled well thanks to an FKG argument. The coupling allows to relate extreme values of the Liouville model and the Gaussian free field, and as an application we show that the global maximum of the Liouville field converges in distribution to a randomly shifted Gumbel distribtion.

We study the shape of the outer envelope of a branching Brownian motion (BBM) in $$\mathbb {R}^d$$ R d , $$d\ge 2$$ d ≥ 2 . We focus on the extremal particles: those whose norm is within O (1) of the maximal norm amongst the particles alive at time t . Our main result is a scaling limit, with exponent 3/2, for the outer-envelope of the BBM around each extremal particle (the front ); the scaling limit is a continuous random surface given explicitly in terms of a Bessel(3) process. Towards this end, we introduce a point process that captures the full landscape around each extremal particle and show convergence in distribution to an explicit point process. This complements the global description of the extremal process given in Berestycki et. al. (Ann. Probab. 52 (2024), no. 3, 955–982), where the local behavior at directions transversal to the radial component of the extremal particles is not addressed.

We consider the recursion $X_{n+1}=\sum_{i=0}^n ε_{n,i}X_{n-i}$, where $ε_{n,i}$ are i.i.d. (Bernoulli) random variables taking values in $\{-1,1\}$, and $X_0=1$, $X_{-j}=0$ for $j>0$. We prove that almost surely, $n^{-1}\log |X_n|\to \bar γ>0$, where $\bar γ$ is an appropriate Lyapunov exponent. This answers a question of Viswanath and Trefethen (\textit{SIAM J. Matrix Anal. Appl. 19:564--581, 1998}).

We study the maximum $φ_N^*$ of the partition function of the two dimensional (subcritical) Gaussian directed polymer over an $\sqrt N \times \sqrt N$ box. We show that $φ_N^*/\log N$ converges towards a constant $σ^*$, which we identify to be the same as for the maximum of a branching random walk with a slowly varying variance profile as studied in Fang-Zeitouni, J. Stat. Phys. 2012 and (in the context of the generalized random energy model) in Bovier-Kurkova, Ann. Inst. H. Poincare 2004.

We consider the extremes of the logarithm of the characteristic polynomial of matrices from the C $\beta $ E ensemble. We prove convergence in distribution of the centered maxima (of the real and imaginary parts) toward the sum of a Gumbel variable and another independent variable, which we characterize as the total mass of a ‘derivative martingale’. We also provide a description of the landscape near extrema points.