arXiv subjects

Eyal Lubetzky

Publications and source records attributed to Eyal Lubetzky.

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 $α_0=1>α_1>\ldots>α_K>0$ such that, for each $r \geq 1$, along the intervals $\pm(α_{r} N,α_{r-1} N)$, 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

Tilted Solid-On-Solid is liquid: scaling limit of SOS with a potential on a slope

The $(2+1)$D Solid-On-Solid (SOS) model famously exhibits a roughening transition: on an $N\times N$ torus with the height at the origin rooted at $0$, the variance of $h(x)$, the height at $x$, is $O(1)$ at large inverse-temperature $β$, vs $\asymp \log |x|$ at small $β$ (as in the Gaussian free field (GFF)). The former--rigidity at large $β$--is known for a wide class of $|\nablaϕ|^p$ models ($p=1$ being SOS) yet is believed to fail once the surface is on a slope (tilted boundary conditions). It is conjectured that the slope would destabilize the rigidity and induce the GFF-type behavior of the surface at small $β$. The only rigorous result on this is by Sheffield '05: for these models of integer height functions, if the slope $θ$ is irrational, then Var$(h(x))\to\infty$ with $|x|$ (with no known quantitative bound). We study a family of SOS surfaces at a large enough fixed $β$, on an $N\times N$ torus with a nonzero boundary condition slope $θ$, perturbed by a potential $V$ on an $ε_β$-fraction of sites (arbitrarily small). Our main result is (a) the measure on the height gradients $\nabla h$ has a limit $μ_\infty$ as $N\to\infty$; and (b) the scaling limit of a sample from $μ_\infty$ converges to a full plane GFF. In particular, we recover the asymptotics Var$(h(x))\sim c\log|x|$. To our knowledge, this is the first example of a tilted $|\nablaϕ|^p$ model, or a perturbation thereof, where the limit is recovered at large $β$. The proof looks at random monotone surfaces that approximate the SOS surface, and shows that (i) these form a weakly interacting dimer model, and (ii) the renormalization framework of Giuliani, Mastropietro and Toninelli '17 leads to the GFF limit. New ingredients are needed in both parts, including a nontrivial extension of [GMT17] from finite interactions to ones with exponential decay in the radius.

math.PR↗

arXiv2026

The limiting law of the Discrete Gaussian level lines

Consider the $(2+1)$D Discrete Gaussian (ZGFF, integer-valued Gaussian free field) model in an $L\times L$ box above a hard floor. Bricmont, El-Mellouki and Fröhlich (1986) established that, at low enough temperature, this random surface exhibits entropic repulsion: the floor propels the average height to be poly-logarithmic in $L$. The second author, Martinelli and Sly (2016) showed that, for all but exceptional values of $L$, the surface has a plateau whose height concentrates on an explicit integer $H(L)$, and fills nearly the full square. It was conjectured there that the boundary of this plateau -- the top level-line of the surface -- should have random fluctuations of $L^{1/3+o(1)}$. We confirm this conjecture of [LMS16] and further recover the limiting law of the top level-line: there exists an explicit sequence $N=L^{1-o(1)}$ such that the distance of the top level-line from $I$, the interval of length $N^{2/3}$ centered along the side boundary, converges, after rescaling it by $N^{1/3}$ and the width of the interval by $N^{2/3}$, to a Ferrari--Spohn diffusion. In particular, the level-line fluctuations at, say, the center of $I$, have a limit law involving the Airy function rescaled by $N^{1/3}$. This gives the first example of one of the $(2+1)$D $|\nabla ϕ|^p$ models (approximating 3D Ising and crystal formation) where a Ferrari--Spohn limit law of its level-lines is confirmed (ZGFF is the case $p=2$). More generally, we find the joint limit law of any finite number of top level-lines: rescaling their distances from the side boundary, each by its $(N_n^{2/3},N_n^{1/3})$, yields a product of Ferrari--Spohn laws. These new results extend to the full universality class of $|\nablaϕ|^p$ models for any fixed $p>1$.

math.PR↗

arXiv2026

The limit shape and emergence of the Discrete Gaussian level lines

Consider the $(2+1)$D Discrete Gaussian model (ZGFF) on an $L\times L$ box with a hard floor at height zero and zero boundary conditions, at low temperature. The second author, Martinelli and Sly (2016) showed that the surface has a plateau, filling nearly the full square, at height either $H$ or $H+1$ for an explicit function $H(L)$. In a companion paper, we studied the local laws of the top level lines near the four sides of the box, and showed that after rescaling each by $(L^{2/3-o(1)},L^{1/3-o(1)})$, they converge to a product of Ferrari--Spohn diffusions. Two key features of the top level lines remained unaddressed: their global limit shape, and the critical window marking the transition from a top plateau at height $H$ to one at height $H+1$. These features are intrinsically linked: deriving the global limit of the top level line is needed for determining whether it is preferable to be at height $H$ or $H+1$ near criticality. This work completes this picture as follows. First, we obtain the global limit of the top level lines: for every fixed $n$, the $n$-th from-the-top level line converges in Hausdorff distance to a deterministic shape $\mathscr{L}_n$ that features the Wulff shape at scale $N_n=L^{1-o(1)}$ near the four corners of the box. Second, we identify, for every $h$, the point of emergence of a macroscopic $h$ level line: the probability of this event is monotone increasing in $L$ (up to a $o(1)$ error), and undergoes a sharp transition from near $0$ to near $1$ in a critical window of width $\leq L^{1/2+o(1)}$ around a side length $L=L_c^{(h)}$. This transition is discontinuous in that, once a macroscopic level $h$ emerges, it immediately occupies nearly all the box, and the above global and local scaling limits (Wulff, Ferrari--Spohn) hold for it. The new results extend to the $(2+1)$D $|\nablaϕ|^p$-models (ZGFF is the case $p=2$) for every fixed $p> 1$.

math.PR↗

arXiv2025

Logarithmic delocalization of low temperature 3D Ising and Potts interfaces above a hard floor

We study the entropic repulsion of the low temperature 3D Ising and Potts interface in an $n\times n \times n$ box with blue boundary conditions on its bottom face (the hard floor), and red boundary conditions on its other five faces. For Ising, Frohlich and Pfister proved in 1987 that the typical interface height above the origin diverges (non-quantitatively), via correlation inequalities special to the Ising model; no such result was known for Potts. We show for both the Ising and Potts models that the entropic repulsion fully overcomes the potentially attractive interaction with the floor, and obtain a logarithmically diverging lower bound on the typical interface height. This is complemented by a conjecturally sharp upper bound of $\lfloor ξ^{-1}\log n\rfloor$ where $ξ$ is the rate function for a point-to-plane non-red connection under the infinite volume red measure. The proof goes through a coupled random-cluster interface to overcome the potential attractive interaction with the boundary, and a coupled fuzzy Potts model to reduce the upper bound to a simpler setting where the repulsion is attained by conditioning a no-floor interface to lie in the upper half-space.

math.PR↗

arXiv2025

The noisy voter model with general initial conditions

We study the noisy voter model with $q\geq 2$ states and noise probability $θ$ on arbitrary bounded-degree $n$-vertex graphs $G$ with subexponential growth of balls (e.g., finite subsets of $\mathbb{Z}^d$). Cox, Peres and Steif (2016) showed for the binary case $q=2$ (and a wider class of chains) that, when starting from a worst-case initial state, this Markov chain has total variation cutoff at $t_n=\frac1{2θ}\log n$. The second author and Sly (2021) analyzed faster initial conditions for Glauber dynamics for the 1D Ising model, which is the noisy voter for $q=2$ and $G=\mathbb{Z}/n\mathbb{Z}$. They showed that the ``alternating'' initial state is the fastest one if $θ\geq \frac23$, and conjectured that this holds for all values of the noise $θ$. Here we show that for every graph $G$ as above and all $θ,q$ and initial states $x_0$, the noisy voter model exhibits cutoff at an explicit function of the autocorrelation of the model started at $x_0$. Consequently, for $G=\mathbb{Z}/n\mathbb{Z}$ and $q=2$ (Glauber dynamics for the 1D Ising model), we confirm the conjecture of [LS21] that the alternating initial condition is asymptotically fastest for all $θ$. Analogous results hold in $\mathbb{Z}_n^d$ for $q=2$ and all $d\geq 1$ (``checkerboard'' initial conditions are fastest) as well as for $d=1$ and all $q\geq 2$ (``rainbow'' initial conditions are fastest).

math.PR↗

arXiv2024

On level line fluctuations of SOS surfaces above a wall

We study the low temperature $(2+1)$D Solid-On-Solid model on $[[1, L ]]^2$ with zero boundary conditions and nonnegative heights (a floor at height $0$). Caputo et al. (2016) established that this random surface typically admits either $\mathfrak h $ or $\mathfrak h+1$ many nested macroscopic level line loops $\{\mathcal L_i\}_{i\geq 0}$ for an explicit $\mathfrak h\asymp \log L$, and its top loop $\mathcal L_0$ has cube-root fluctuations: e.g., if $ρ(x)$ is the vertical displacement of $\mathcal L_0$ from the bottom boundary point $(x,0)$, then $\max ρ(x) = L^{1/3+o(1)}$ over $x\in I_0:=L/2+[[-L^{2/3},L^{2/3}]]$. It is believed that rescaling $ρ$ by $L^{1/3}$ and $I_0$ by $L^{2/3}$ would yield a limit law of a diffusion on $[-1,1]$. However, no nontrivial lower bound was known on $ρ(x)$ for a fixed $x\in I_0$ (e.g., $x=\frac L2$), let alone on $\minρ(x)$ in $I_0$, to complement the bound on $\maxρ(x)$. Here we show a lower bound of the predicted order $L^{1/3}$: for every $ε>0$ there exists $δ>0$ such that $\min_{x\in I_0} ρ(x) \geq δL^{1/3}$ with probability at least $1-ε$. The proof relies on the Ornstein--Zernike machinery due to Campanino-Ioffe-Velenik, and a result of Ioffe, Shlosman and Toninelli (2015) that rules out pinning in Ising polymers with modified interactions along the boundary. En route, we refine the latter result into a Brownian excursion limit law, which may be of independent interest. We further show that in a $ K L^{2/3}\times K L^{2/3}$ box with boundary conditions $\mathfrak h-1,\mathfrak h,\mathfrak h,\mathfrak h$ (i.e., $\mathfrak h-1$ on the bottom side and $\mathfrak h$ elsewhere), the limit of $ρ(x)$ as $K,L\to\infty$ is a Ferrari--Spohn diffusion.

math.PR↗

arXiv2024

Metastability cascades and prewetting in the SOS model

We study Glauber dynamics for the low temperature $(2+1)$D Solid-On-Solid model on a box of side-length $n$ with a floor at height $0$ (inducing entropic repulsion) and a competing bulk external field $λ$ pointing down (the prewetting problem). In 1996, Cesi and Martinelli showed that if the inverse-temperature $β$ is large enough, then along a decreasing sequence of critical points $(λ_c^{(k)})_{k=0}^{K_β}$ the dynamics is torpid: its inverse spectral gap is $O(1)$ when $λ\in (λ_c^{(k+1)},λ_c^{(k)})$ whereas it is $\exp[Θ(n)]$ at each $λ_c^{(k)}$ for each $k\leq K_β$, due to a coexistence of rigid phases at heights $k+1$ and $k$. Our focus is understanding (a) the onset of metastability as $λ_n\uparrowλ_c^{(k)}$; and (b) the effect of an unbounded number of layers, as we remove the restriction $k\le K_β$, and even allow for $λ_n\to 0$ towards the $λ= 0$ case which has $O(\log n)$ layers and was studied by Caputo et al. (2014). We show that for any $k$, possibly growing with $n$, the inverse gap is $\exp[\tildeΘ(1/|λ_n-λ_c^{(k)}|)]$ as $λ\uparrow λ_c^{(k)}$ up to distance $n^{-1+o(1)}$ from this critical point, due to a metastable layer at height $k$ on the way to forming the desired layer at height $k+1$. By taking $λ_n = n^{-α}$ (corresponding to $k_n\asymp \log n$), this also interpolates down to the behavior of the dynamics when $λ=0$. We complement this by extending the fast mixing to all $λ$ uniformly bounded away from $(λ_c^{(k)})_{k=0}^\infty$. Together, these results provide a sharp understanding of the predicted infinite sequence of dynamical phase transitions governed by the layering phenomenon.

math.PR↗

arXiv2024

Critical wetting in the (2+1)D Solid-On-Solid model

In this note, we study the low temperature $(2+1)$D SOS interface above a hard floor with critical pinning potential $λ_w= \log (\frac{1}{1-e^{-4β}})$. At $λ<λ_w$ entropic repulsion causes the surface to delocalize and be rigid at height $\frac1{4β}\log n+O(1)$; at $λ>λ_w$ it is localized at some $O(1)$ height. We show that at $λ=λ_w$, there is delocalization, with rigidity now at height $\lfloor \frac1{6β}\log n+\frac13\rfloor$, confirming a conjecture of Lacoin.

math.PR↗

arXiv2024

Entropic repulsion of 3D Ising interfaces conditioned to stay above a floor

We study the interface of the Ising model in a box of side-length $n$ in $\mathbb Z^3$ at low temperature $1/β$ under Dobrushin's boundary conditions, conditioned to stay in a half-space above height $h$ (a hard floor). Without this conditioning, Dobrushin showed in 1972 that typically most of the interface is flat at height $0$. With the floor, for small $h$, the model is expected to exhibit {\it entropic repulsion}, where the typical height of the interface lifts off of $0$. Detailed understanding of the SOS model -- a more tractable height function approximation of 3D Ising -- due to Caputo et al., suggests that there is a single integer value $-h_n^* \sim -c\log n$ of the floor height, delineating the transition between rigidity at height $0$ and entropic repulsion. We identify an explicit $h_n^*=( c_\star+o(1))\log n$ such that, for the typical Ising interface above a hard floor at $h$, all but an $ε(β)$-fraction of the sites are propelled to be above height $0$ if $h < h_n^*-1$, whereas all but an $ε(β)$-fraction of the sites remain at height $0$ if $h\geq h_n^*$. Further, $c_\star$ is such that the typical height of the unconditional maximum is $(2c_\star + o(1))\log n$; this confirms scaling predictions from the SOS approximation.

math.PR↗

arXiv2024

Extrema of 3D Potts interfaces

The interface between the plus and minus phases in the low temperature 3D Ising model has been intensely studied since Dobrushin's pioneering works in the early 1970's established its rigidity. Advances in the last decade yielded the tightness of the maximum of the interface of this Ising model on the cylinder of side length $n$, around a mean that is asymptotically $c\log n$ for an explicit $c$ (temperature dependent). In this work, we establish analogous results for the 3D Potts and random cluster (FK) models. Compared to 3D Ising, the Potts model and its lack of monotonicity form obstacles for existing methods, calling for new proof ideas, while its interfaces (and associated extrema) exhibit richer behavior. We show that the maxima and minima of the interface bounding the blue component in the 3D Potts interface, and those of the interface bounding the bottom component in the 3D FK model, are governed by 4 different large deviation rates, whence the corresponding global extrema feature 4 distinct constants $c$ as above. Due to the above obstacles, our methods are initially only applicable to 1 of these 4 interface extrema, and additional ideas are needed to recover the other 3 rates given the behavior of the first one.

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↗

arXiv2022

The maximum of branching Brownian motion in $\mathbb{R}^d$

We show that in branching Brownian motion (BBM) in $\mathbb{R}^d$, $d\geq 2$, the law of $R_t^*$, the maximum distance of a particle from the origin at time $t$, converges as $t\to\infty$ to the law of a randomly shifted Gumbel random variable.

math.PR↗

arXiv2022

On the limiting law of line ensembles of Brownian polymers with geometric area tilts

We study the line ensembles of non-crossing Brownian bridges above a hard wall, each tilted by the area of the region below it with geometrically growing pre-factors. This model, which mimics the level lines of the $(2+1)$D SOS model above a hard wall, was studied in two works from 2019 by Caputo, Ioffe and Wachtel. In those works, the tightness of the law of the top $k$ paths, for any fixed $k$, was established under either zero or free boundary conditions, which in the former setting implied the existence of a limit via a monotonicity argument. Here we address the open problem of a limit under free boundary conditions: we prove that as the interval length, followed by the number of paths, go to $\infty$, the top $k$ paths converge to the same limit as in the free boundary case, as conjectured by Caputo, Ioffe and Wachtel.

math.PR↗

arXiv2022

The threshold for stacked triangulations

A \emph{stacked triangulation} of a $d$-simplex $\mathbf{o}=\{1,\ldots,d+1\}$ ($d\geq 2$) is a triangulation obtained by repeatedly subdividing a $d$-simplex into $d+1$ new ones via a new vertex (the case $d=2$ is known as an Appolonian network). We study the occurrence of such a triangulation in the Linial--Meshulam model, i.e., for which $p$ does the random simplicial complex $Y\sim \mathcal{Y}_d(n,p)$ contain the faces of a stacked triangulation of the $d$-simplex $\mathbf{o}$, with its internal vertices labeled in $[n]$. In the language of bootstrap percolation in hypergraphs, it pertains to the threshold for $K_{d+2}^{d+1}$, the $(d+1)$-uniform clique on $d+2$ vertices. Our main result identifies this threshold for every $d\geq 2$, showing it is asymptotically $(α_d n)^{-1/d}$, where $α_d$ is the growth rate of the Fuss--Catalan numbers of order $d$. The proof hinges on a second moment argument in the supercritical regime, and on Kalai's algebraic shifting in the subcritical regime.

math.CO↗

arXiv2021

Approximate domain Markov property for rigid Ising interfaces

Consider the Ising model on a centered box of side length $n$ in $\mathbb Z^d$ with $\mp$-boundary conditions that are minus in the upper half-space and plus in the lower half-space. Dobrushin famously showed that in dimensions $d\ge 3$, at low-temperatures the Ising interface (dual-surface separating the plus/minus phases) is rigid, i.e., it has $O(1)$ height fluctuations. Recently, the authors decomposed these oscillations into pillars and identified their typical shape, leading to a law of large numbers and tightness of their maximum. Suppose we condition on a height-$h$ level curve of the interface, bounding a set $S \subset \mathbb Z^{d-1}$, along with the entire interface outside the cylinder $S\times \mathbb Z$: what does the interface in $S\times \mathbb Z$ look like? Many models of random surfaces (e.g., SOS and DGFF) fundamentally satisfy the domain Markov property, whereby their heights on $S$ only depend on the heights on $S^c$ through the heights on $\partial S$. The Ising interface importantly does not satisfy this property; the law of the interface depends on the full spin configuration outside $S\times \mathbb Z$. Here we establish an approximate domain Markov property inside the level curves of the Ising interface. We first extend Dobrushin's result to this setting, showing the interface in $S\times \mathbb Z$ is rigid about height $h$, with exponential tails on its height oscillations. Then we show that the typical tall pillars in $S\times \mathbb Z$ are uniformly absolutely continuous with respect to tall pillars of the unconditional Ising interface. Using this we identify the law of large numbers, tightness, and Gumbel tail bounds on the maximum oscillations in $S\times \mathbb Z$ about height $h$, showing that these only depend on the conditioning through the cardinality of $S$.

math.PR↗

arXiv2020

Noise sensitivity of critical random graphs

We study noise sensitivity of properties of the largest components $({\cal C}_j)_{j\geq 1}$ of the random graph ${\cal G}(n,p)$ in its critical window $p=(1+λn^{-1/3})/n$. For instance, is the property "$|{\cal C}_1|$ exceeds its median size" noise sensitive? Roberts and Şengül (2018) proved that the answer to this is yes if the noise $ε$ is such that $ε\gg n^{-1/6}$, and conjectured the correct threshold is $ε\gg n^{-1/3}$. That is, the threshold for sensitivity should coincide with the critical window---as shown for the existence of long cycles by the first author and Steif (2015). We prove that for $ε\gg n^{-1/3}$ the pair of vectors $ n^{-2/3}(|{\cal C}_j|)_{j\geq 1}$ before and after the noise converges in distribution to a pair of i.i.d. random variables, whereas for $ε\ll n^{-1/3}$ the $\ell^2$-distance between the two goes to 0 in probability. This confirms the above conjecture: any Boolean function of the vector of rescaled component sizes is sensitive in the former case and stable in the latter. We also look at the effect of the noise on the metric space $n^{-1/3}({\cal C}_j)_{j\geq 1}$. E.g., for $ε\geq n^{-1/3+o(1)}$, we show that the joint law of the spaces before and after the noise converges to a product measure, implying noise sensitivity of any property seen in the limit, e.g., "the diameter of ${\cal C}_1$ exceeds its median."

math.PR↗

arXiv2020

Cycle lengths in sparse random graphs

We study the set ${\cal L}(G)$ of lengths of all cycles that appear in a random $d$-regular $G$ on $n$ vertices for a fixed $d\geq 3$, as well as in Erdős--Rényi random graphs on $n$ vertices with a fixed average degree $c>1$. Fundamental results on the distribution of cycle counts in these models were established in the 1980's and early 1990's, with a focus on the extreme lengths: cycles of fixed length, and cycles of length linear in $n$. Here we derive, for a random $d$-regular graph, the limiting probability that ${\cal L}(G)$ simultaneously contains the entire range $\{\ell,\ldots,n\}$ for $\ell\geq 3$, as an explicit expression $θ_\ell=θ_\ell(d)\in(0,1)$ which goes to $1$ as $\ell\to\infty$. For the random graph ${\cal G}(n,p)$ with $p=c/n$, where $c\geq C_0$ for some absolute constant $C_0$, we show the analogous result for the range $\{\ell,\ldots,(1-o(1))L_{\max}(G)\}$, where $L_{\max}$ is the length of a longest cycle in $G$. The limiting probability for ${\cal G}(n,p)$ coincides with $θ_\ell$ from the $d$-regular case when $c$ is the integer $d-1$. In addition, for the directed random graph ${\cal D}(n,p)$ we show results analogous to those on ${\cal G}(n,p)$, and for both models we find an interval of $c ε^2 n$ consecutive cycle lengths in the slightly supercritical regime $p=\frac{1+ε}n$.

math.CO↗