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Ofer Zeitouni

Publications and source records attributed to Ofer Zeitouni.

At least 19 recordsLinked to original sources
arXiv2026

The law of (1+1)D SOS with an area tilt in a wedge

Motivated by the study of the level lines of the $(2+1)$D Solid-On-Solid (SOS) model above a floor, near the corners of the box, we derive the limit law of an ensemble of $K$ curves from a $(1+1)$D SOS model, with an area tilt, and above a wedge-shaped floor in $\{-N,\ldots,N\}$. We show that there exist explicit critical points $\alpha_0=1>\alpha_1>\ldots>\alpha_K>0$ such that, for each $r \geq 1$, along the intervals $\pm(\alpha_{r} N,\alpha_{r-1} N)$, the bottom $K+1-r$ curves, rescaled by $(N^{2/3},N^{1/3})$, tend to the law of a Geometrically-Area-Tilted Ensemble of non-crossing Brownian paths (Brownian GATE), independently across those $2K$ intervals. All other curves, centered and rescaled by $(N,\sqrt{N})$, tend to a product of $K$ suitable Brownian bridges.

math.PR↗
arXiv2026

Large Deviations for Iterated Sums and Integrals

We describe large deviations for normalized multiple iterated sums and integrals of the form $\bbS_N^{(ν)}(t)=N^{-ν}\sum_{0\leq k_1<...<k_ν\leq Nt}ξ(k_1)\otimes\cdots\otimesξ(k_ν)$, $t\in[0,T]$ and $\bbS_N^{(ν)}(t)=N^{-ν}\int_{0\leq s_1\leq...\leq s_ν\leq Nt}ξ(s_1)\otimes\cdots\otimesξ(s_ν)ds_1\cdots ds_ν$, where $\{ξ(k)\}_{-\infty<k<\infty}$ and $\{ξ(s)\}_{-\infty<s<\infty}$ are centered bounded stationary vector processes whose sums or integrals satisfy a trajectorial large deviations principle.

math.PR↗
arXiv2025

Maximum of the characteristic polynomial of random Jacobi matrices

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.

math.PR↗
arXiv2025

The Liouville model in the $L^1$ phase: coupling and extreme values

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.

math.PR↗
arXiv2025

The shape of the front of multidimensional branching Brownian motion

We study the shape of the outer envelope of a branching Brownian motion (BBM) in $\mathbb{R}^d$, $d\geq 2$. 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.

math.PR↗
arXiv2025

Exponential growth of random infinite Fibonacci sequences

We consider the recursion $X_{n+1}=\sum_{i=0}^n \epsilon_{n,i}X_{n-i}$, where $\epsilon_{n,i}$ 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, $n^{-1}\log |X_n|\to \bar \gamma>0$, where $\bar \gamma$ is an appropriate Lyapunov exponent. This answers a question of Viswanath and Trefethen (\textit{SIAM J. Matrix Anal. Appl. 19:564--581, 1998}).

math.PR↗
arXiv2025

The maximum of the two dimensional Gaussian directed polymer in the subcritical regime

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.

math.PR↗
arXiv2024

The extremal landscape for the C$β$E ensemble

We consider the extremes of the logarithm of the characteristic polynomial of matrices from the C$β$E ensemble. We prove convergence in distribution of the centered maxima (of the real and imaginary parts) towards 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.

math.PR↗
arXiv2024

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

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 $t_k = \lambda p^k$ with $\lambda>0$ and $p$ 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^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 correlations, which is not present in the non-tilted model. Our methods also apply to the usual Branching Random Walk $(S_v)_{v\in \mathcal{T}}$ 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^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.

math.PR↗
arXiv2024

Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices

We determine to leading order the maximum of the characteristic polynomial for Wigner matrices and $β$-ensembles. In the special case of Gaussian-divisible Wigner matrices, our method provides universality of the maximum up to tightness. These are the first universal results on the Fyodorov--Hiary--Keating conjectures for these models, and in particular answer the question of optimal rigidity for the spectrum of Wigner matrices. Our proofs combine dynamical techniques for universality of eigenvalue statistics with ideas surrounding the maxima of log-correlated fields and Gaussian multiplicative chaos.

math.PR↗
arXiv2024

Tightness of the maximum of Ginzburg-Landau fields

We consider the discrete Ginzburg-Landau field with potential satisfying a uniform convexity condition, in the critical dimension $d=2$, and prove that its maximum over boxes of sidelength $N$, centered by an explicit $N$-dependent centering, is tight.

math.PR↗
arXiv2023

Voting models and tightness for a family of recursion equations

We consider recursion equations of the form $u_{n+1}(x)=Q[u_n](x),~n\ge 1,~x\in R$, with a non-local operator $Q[u](x)= g( u\ast q)$, where $g$ is a polynomial, satisfying $g(0)=0$, $g(1)=1$, $g((0,1)) \subseteq (0,1)$, and $q$ is a (compactly supported) probability density with $\ast$ denoting convolution. Motivated by a line of works for nonlinear PDEs initiated by Etheridge, Freeman and Penington (2017), we show that for general $g$, a probabilistic model based on branching random walk can be given to the solution of the recursion, while in case $g$ is also strictly monotone, a probabilistic threshold-based model can be given. In the latter case, we provide a conditional tightness result. We analyze in detail the bistable case and prove for it convergence of the solution shifted around a linear in $n$ centering.

math.PR↗
arXiv2023

The extremal point process of branching Brownian motion in $\mathbb{R}^d$

We consider a branching Brownian motion in $\mathbb{R}^d$ with $d \geq 1$ in which the position $X_t^{(u)}\in \mathbb{R}^d$ of a particle $u$ at time $t$ can be encoded by its direction $θ^{(u)}_t \in \mathbb{S}^{d-1}$ and its distance $R^{(u)}_t$ to 0. We prove that the {\it extremal point process} $\sum δ_{θ^{(u)}_t, R^{(u)}_t - m_t^{(d)}}$ (where the sum is over all particles alive at time $t$ and $m^{(d)}_t$ is an explicit centring term) converges in distribution to a randomly shifted decorated Poisson point process on $\mathbb{S}^{d-1} \times \mathbb{R}$. More precisely, the so-called {\it clan-leaders} form a Cox process with intensity proportional to $D_\infty(θ) e^{-\sqrt{2}r} ~\mathrm{d} r ~\mathrm{d} θ$, where $D_\infty(θ)$ is the limit of the derivative martingale in direction $θ$ and the decorations are i.i.d. copies of the decoration process of the standard one-dimensional branching Brownian motion. This proves a conjecture of Stasiński, Berestycki and Mallein (Ann. Inst. H. Poincaré 57:1786--1810, 2021), and builds on that paper and on Kim, Lubetzky and Zeitouni (arXiv:2104.07698).

math.PR↗
arXiv2023

Moments of partition functions of 2D Gaussian polymers in the weak disorder regime -- I

Let $W_N(β) = \mathrm{E}_0\left[e^{ \sum_{n=1}^N βω(n,S_n) - Nβ^2/2}\right]$ be the partition function of a two-dimensional directed polymer in a random environment, where $ω(i,x), i\in \mathbb{Z}_+, x\in \mathbb{Z}^2$ are i.i.d.\ standard normal and $\{S_n\}$ is the path of a random walk. With $β=β_N=\hatβ\sqrt{π/\log N}$ and $\hat β\in (0,1)$ (the subcritical window), $\log W_N(β_N)$ is known to converge in distribution to a Gaussian law of mean $-λ^2/2$ and variance $λ^2$, with $λ^2=\log \big(1/(1-\hatβ^2\big)$ (Caravenna, Sun, Zygouras, Ann. Appl. Probab. (2017)). We study in this paper the moments $\mathbb{E} [W_N( β_N)^q]$ in the subcritical window, for $q=O(\sqrt{\log N})$. The analysis is based on ruling out triple intersections

math.PR↗
arXiv2023

Moments of partition functions of 2D Gaussian polymers in the weak disorder regime -- II

Let $W_N(β) = \mathrm{E}_0\left[e^{ \sum_{n=1}^N βω(n,S_n) - Nβ^2/2}\right]$ be the partition function of a two-dimensional directed polymer in a random environment, where $ω(i,x), i\in \mathbb{N}, x\in \mathbb{Z}^2$ are i.i.d. standard normal and $\{S_n\}$ is the path of a random walk. With $β=β_N=\widehatβ \sqrt{π/\log N}$ and $\widehatβ\in (0,1)$ (the subcritical window), $\log W_N(β_N)$ is known to converge in distribution to a Gaussian law of mean $-λ^2/2$ and variance $λ^2$, with $λ^2=\log ((1-\widehatβ^2)^{-1})$ (Caravenna, Sun, Zygouras, Ann. Appl. Probab. (2017)). We study in this paper the moments $\mathbb E [W_N( β_N)^q]$ in the subcritical window, and prove a lower bound that matches for $q=O(\sqrt{\log N})$ the upper bound derived by us in Cosco, Zeitouni, arXiv:2112.03767 [math.PR]. The analysis is based on appropriate decouplings and a Poisson convergence that uses the method of ''two moments suffice''.

math.PR↗
arXiv2023

On-Site Potential Creates Complexity in Systems with Disordered Coupling

We calculate the average number of critical points $\overline{\mathcal{N}}$ of the energy landscape of a many-body system with disordered two-body interactions and a weak on-site potential. We find that introducing a weak nonlinear on-site potential dramatically increases $\overline{\mathcal{N}}$ to exponential in system size and give a complete picture of the organization of critical points. Our results extend solvable spin-glass models to physically more realistic models and are of relevance to glassy systems, nonlinear oscillator networks and many-body interacting systems.

cond-mat.dis-nn↗