arXiv subjects

Benoît Laslier

Publications and source records attributed to Benoît Laslier.

7 recordsLinked to original sources

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

Harmonic functions on Tutte embeddings and linearized Monge-Ampère equation

We prove convergence of solutions of Dirichlet problems and Green's functions on Tutte harmonic embeddings to those of the linearized Monge--Ampère equation $\mathcal{L}_φh=0$. More precisely, we assume that piecewise linear Maxwell--Cremona potentials associated with the embeddings converge to a continuous potential $φ$ and the only assumption that we use is the uniform convexity of $φ$ or, equivalently, the uniform ellipticity of the operator $\mathcal{L}_φ$. Even if $φ$ is quadratic, this setup significantly generalizes known results for discrete harmonic functions on orthodiagonal tilings. Motivated by potential applications to the analysis of 2d lattice models on irregular graphs, we also study the situation in which the limits are harmonic in a different complex structure.

math-ph↗

arXiv2025

Convergence of dynamical stationary fluctuations

We present a general black box theorem that ensures convergence of a sequence of stationary Markov processes, provided a few assumptions are satisfied. This theorem relies on a control of the resolvents of the sequence of Markov processes, and on a suitable characterization of the resolvents of the limit. One major advantage of this approach is that it circumvents the use of the Boltzmann-Gibbs principle: for instance, we deduce in a rather simple way that the stationary fluctuations of the one-dimensional zero-range process converge to the stochastic heat equation. More importantly, it allows to establish results that were probably out of reach of existing methods: using the black box result, we are able to prove that the stationary fluctuations of a discrete model of ordered interfaces, that was considered previously in the statistical physics literature, converge to a system of reflected stochastic PDEs.

math.PR↗

arXiv2022

Dimer model and holomorphic functions on t-embeddings of planar graphs

We introduce the framework of discrete holomorphic functions on t-embeddings of weighted bipartite planar graphs; t-embeddings also appeared under the name Coulomb gauges in a recent paper arXiv:1810.05616. We argue that this framework is particularly relevant for the analysis of scaling limits of the height fluctuations in the corresponding dimer models. In particular, it unifies both Kenyon's interpretation of dimer observables as derivatives of harmonic functions on T-graphs and the notion of s-holomorphic functions originated in Smirnov's work on the critical Ising model. We develop an a priori regularity theory for such functions and provide a meta-theorem on convergence of the height fluctuations to the Gaussian Free Field. We also discuss how several more standard discretizations of complex analysis fit this general framework.

math.PR↗

arXiv2021

Bipartite dimer model: perfect t-embeddings and Lorentz-minimal surfaces

This is the second paper in the series devoted to the study of the dimer model on t-embeddings of planar bipartite graphs. We introduce the notion of perfect t-embeddings and assume that the graphs of the associated origami maps converge to a Lorentz-minimal surface $\mathrm{S}_ξ$ as $δ\to 0$. In this setup we prove (under very mild technical assumptions) that the gradients of the height correlation functions converge to those of the Gaussian Free Field defined in the intrinsic metric of the surface $\mathrm{S}_ξ$. We also formulate several open questions motivated by our work.

math.PR↗

arXiv2016

Critical exponents on Fortuin--Kasteleyn weighted planar maps

In this paper we consider random planar maps weighted by the self-dual Fortuin--Kasteleyn model with parameter $q \in (0,4)$. Using a bijection due to Sheffield and a connection to planar Brownian motion in a cone we obtain rigorously the value of the critical exponent associated with the length of cluster interfaces, which is shown to be $$ \frac{4}π \arccos \left( \frac{\sqrt{2 - \sqrt{q}}}{2} \right)=\frac{κ'}{8}. $$ where $κ' $ is the SLE parameter associated with this model. We also derive the exponent corresponding to the area enclosed by a loop which is shown to be 1 for all values of $q \in (0,4)$. Applying the KPZ formula we find that this value is consistent with the dimension of SLE curves and SLE duality.

math.PR↗

arXiv2012

Nanometer Scale Spectral Imaging of Quantum Emitters in Nanowires and Its Correlation to Their Atomically Resolved Structure

We report the spectral imaging in the UV to visible range with nanometer scale resolution of closely packed GaN/AlN quantum disks in individual nanowires using an improved custom-made cathodoluminescence system. We demonstrate the possibility to measure full spectral features of individual quantum emitters as small as 1 nm and separated from each other by only a few nanometers and the ability to correlate their optical properties to their size, measured with atomic resolution. The direct correlation between the quantum disk size and emission wavelength provides evidence of the quantum confined Stark effect leading to an emission below the bulk GaN band gap for disks thicker than 2.6 nm. With the help of simulations, we show that the internal electric field in the studied quantum disks is smaller than what is expected in the quantum well case. We show evidence of a clear dispersion of the emission wavelengths of different quantum disks of identical size but different positions along the wire. This dispersion is systematically correlated to a change of the diameter of the AlN shell coating the wire and is thus attributed to the related strain variations along the wire. The present work opens the way both to fundamental studies of quantum confinement in closely packed quantum emitters and to characterizations of optoelectronic devices presenting carrier localization on the nanometer scale.

cond-mat.mtrl-sci↗