arXiv subjects

Hubert Lacoin

Publications and source records attributed to Hubert Lacoin.

At least 37 records · Page 2Linked to original sources

arXiv2020

Solid-On-Solid interfaces with disordered pinning

We investigate the localization transition for a simple model of interface which interacts with an inhomonegeous defect plane. The interface is modeled by the graph of a function $ϕ: \mathbb Z^2 \to \mathbb Z$,and the disorder is given by a fixed realization of a field of IID centered random variables$(ω_x)_{x\in \mathbb Z^2}$. The Hamiltonian of the system depends on three parameters $α,β>0$ and $h\in \mathbb R$ which determine respectively the intensity of nearest neighbor interaction the amplitude of disorder and the mean value of the interaction with the substrate, and is given by the expression $$\mathcal H(ϕ):= β\sum_{x\sim y} |ϕ(x)-ϕ(y)|- \sum_{x} (αω_x+h){\bf 1}_{\{ϕ(x)=0\}}.$$ We focus on the large-$β$/rigid phase phase of the Solid-On-Solid (SOS) model. In that regime, we provide a sharp description of the phase transition in $h$ from a localized phase to a delocalized one corresponding respectivelly to a positive and vanishing fraction of points with $ϕ(x)=0$. We prove that the critical value for $h$ corresponds to that of the annealed model and is given by $h_c(α)= -\log \mathbb E[e^{αω}]$, and that near the critical point, the free energy displays the following critical behavior $$F_β(α,h_c+u )\stackrel{u\to 0+}{\sim} \max_{n\ge 1} \left\{θ_1 e^{-4βn} u- \frac{1}{2}θ^2_1 e^{-8βn} \frac{\mathrm{Var}\left[e^{αω}\right]}{\mathbb E \left[ e^{αω} \right]^2}\right\}.$$ The positive constant $θ_1(β)>0$ is defined by the asymptotic probability of spikes for the infinite volume SOS with $0$ boundary condition $θ_1(β):=\lim_{n\to \infty} e^{4βn}\mathbf P_β (ϕ({\bf 0})=n)$ ...

math.PR↗

arXiv2020

Mixing time of the adjacent walk on the simplex

By viewing the $N$-simplex as the set of positions of $N-1$ ordered particles on the unit interval, the adjacent walk is the continuous time Markov chain obtained by updating independently at rate 1 the position of each particle with a sample from the uniform distribution over the interval given by the two particles adjacent to it. We determine its spectral gap and prove that both the total variation distance and the separation distance to the uniform distribution exhibit a cutoff phenomenon, with mixing times that differ by a factor $2$. The results are extended to the family of log-concave distributions obtained by replacing the uniform sampling by a symmetric log-concave Beta distribution.

math.PR↗

arXiv2020

A probabilistic approach of ultraviolet renormalisation in the boundary Sine-Gordon model

The Sine-Gordon model is obtained by tilting the law of a log-correlated Gaussian field $X$ defined on a subset of $\mathbb{R}^d$ by the exponential of its cosine, namely $\exp(α\smallint \cos (βX))$. It is an important model in quantum field theory or in statistic physics like in the study of log-gases. In spite of its relatively simple definition, the model has a very rich phenomenology. While the integral $\smallint \cos (βX)$ can properly be defined when $β^2<d$ using the standard Wick normalisation of $\cos (βX)$, a more involved renormalization procedure is needed when $β^2\in [d,2d)$. In particular it exhibits a countable sequence of phase transition accumulating to the left of $β=\sqrt{2d}$, each transitions corresponding to the addition of an extra term in the renormalization scheme. The final threshold $β=\sqrt{2}$ corresponds to the Kosterlitz-Thouless (KT) phase transition of the $\log$-gas. In this paper, we present a novel probabilistic approach to renormalization of the two-dimensional boundary (or 1-dimensional) Sine-Gordon model up to the KT threshold $β=\sqrt{2}$. The purpose of this approach is to propose a simple and flexible method to treat this problem which, unlike the existing renormalization group techniques, does not rely on translation invariance for the covariance kernel of $X$ or the reference measure along which $\cos (βX)$ is integrated. To this purpose we establish by induction a general formula for the cumulants of a random variable. We apply this formula to study the cumulants of (approximations of) $\smallint \cos (βX)$. To control all terms produced by the induction proceedure, we prove a refinement of classical electrostatic inequalities, which allows to bound the energy of configurations in terms of the Wasserstein distance between $+$ and $-$ charges.

math.PR↗

arXiv2020

Cutoff at the entropic time for random walks on covered expander graphs

It is a fact simple to establish that the mixing time of the simple random walk on a d-regular graph $G_n$ with n vertices is asymptotically bounded from below by $d/ ((d-2)\log (d-1))\log n$. Such a bound is obtained by comparing the walk on $G_n$ to the walk on the infinite $d$-regular tree. If one can map another infinite transitive graph onto $G_n$, then we can improve the strategy by using a comparison with the random walk on this transitive graph (instead of that of the regular tree), and we obtain a lower bound of the form $1/h \log n$, where $h$ is the entropy rate associated with the walk on the transitive graph. We call this the entropic lower bound. It was recently proved that in the case of the tree, this entropic lower bound is sharp when graphs have minimal spectral radius and thus that in that case the random walk exhibit cutoff at the entropic time. In this paper, we provide a generalization of the result by providing a sufficient condition on the spectra the random walks on $G_n$ under which the random walk exhibit cutoff at the entropic time. It applies notably to anisotropic random walks on random $d$-regular graphs and to random walks on random $n$-lifts of a base graph (including non-reversible walks).

math.PR↗

arXiv2020

Spectral gap and cutoff phenomenon for the Gibbs sampler of $\nablaφ$ interfaces with convex potential

We consider the Gibbs sampler, or heat bath dynamics associated to log-concave measures on $\mathbb{R}^N$ describing $\nablaφ$ interfaces with convex potentials. Under minimal assumptions on the potential, we find that the spectral gap of the process is always given by $\mathrm{gap}_N=1-\cos(π/N)$, and that for all $ε\in(0,1)$, its $ε$-mixing time satisfies $T_N(ε)\sim \frac{\log N}{2\mathrm{gap}_N}$ as $N\to\infty$, thus establishing the cutoff phenomenon. The results reveal a universal behavior in that they do not depend on the choice of the potential.

math.PR↗

arXiv2020

Metastability for expanding bubbles on a sticky substrate

We study the dynamical behavior of a one dimensional interface interacting with a sticky unpenetrable substrate or wall. The interface is subject to two effects going in opposite directions. Contact between the interface and the substrate are given an energetic bonus while an external force with constant intensity pulls the interface away from the wall. Our interface is modeled by the graph of a one-dimensional nearest-neighbor path on $\mathbb{Z}_+$, starting at $0$ and ending at $0$ after $2N$ steps, the wall corresponding to level-zero the horizontal axis. At equilibrium each path $ξ=(ξ_x)_{x=0}^{2N}$, is given a probability proportional to $λ^{H(ξ)} \exp(\fracσ{N}A(ξ))$, where $H(ξ):=\#\{x \ : ξ_x=0\}$ and $A(ξ)$ is the area enclosed between the path $ξ$ and the $x$-axis. We then consider the classical heat-bath dynamics which equilibrates the value of each $ξ_x$ at a constant rate via corner-flip. Investigating the statics of the model, we derive the full phase diagram in $λ$ and $σ$ of this model, and identify the critical line which separates a localized phase where the pinning force sticks the interface to the wall and a delocalized one, for which the external force stabilizes $ξ$ around a deterministic shape at a macroscopic distance of the wall. On the dynamical side, we identify a second critical line, which separates a rapidly mixing phase (for which the system mixes in polynomial time) to a slow phase where the mixing time grows exponentially. In this slowly mixing regime we obtain a sharp estimate of the mixing time on the $\log$ scale, and provide evidences of a metastable behavior.

math.PR↗

arXiv2019

The disordered lattice free field pinning model approaching criticality

We continue the study, initiated in [Giacomin and Lacoin, JEMS 2018], of the localization transition of a lattice free field $ϕ=(ϕ(x))_{x \in Z^d}$, $d\ge 3$, in presence of a quenched disordered substrate. The presence of the substrate affects the interface at the spatial sites in which the interface height is close to zero. This corresponds to the Hamiltonian $$ \sum_{x\in Z^d }(βω_x+h)δ_x,$$ where $δ_x=1_{[-1,1]}(ϕ(x))$, and $(ω_x)_{x\in Z^d}$ is an IID centered field. A transition takes place when the average pinning potential $h$ goes past a threshold $h_c(β)$: from a delocalized phase $h h_c(β)$ where the field sticks to the substrate. In [Giacomin and Lacoin, JEMS 2018] the critical value of $h$ is identified and it coincides, up to the sign, with the $\log$-Laplace transform of $ω=ω_x$, that is $-h_c(β)=λ(β):=\log E[e^{βω}]$. Here we obtain the sharp critical behavior of the free energy approaching criticality: $$\lim_{u\searrow 0} \frac{ F(β,h_c(β)+u)}{u^2}= \frac{1}{2\, \textrm{Var}\left(e^{βω-λ(β)}\right)}.$$ Moreover, we give a precise description of the trajectories of the field in the same regime: the absolute value of the field is $\sqrt{2σ_d^2\vert\log(h-h_c(β))\vert}$ to leading order when $h\searrow h_c(β)$ except on a vanishing fraction of sites ($σ_d^2$ is the single site variance of the free field).

math-ph↗

arXiv2019

Cutoff phenomenon for the asymmetric simple exclusion process and the biased card shuffling

We consider the biased card shuffling and the Asymmetric Simple Exclusion Process (ASEP) on the segment. We obtain the asymptotic of their mixing times: our result show that these two continuous-time Markov chains display cutoff. Our analysis combines several ingredients including: a study of the hydrodynamic profile for ASEP, the use of monotonic eigenfunctions, stochastic comparisons and concentration inequalities.

math.PR↗

arXiv2019

The semiclassical limit of Liouville conformal field theory

A rigorous probabilistic construction of Liouville conformal field theory (LCFT) on the Riemann sphere was recently given by David-Kupiainen and the last two authors. In this paper, we focus on the connection between LCFT and the classical Liouville field theory via the semiclassical approach. LCFT depends on a parameter $γ\in (0,2)$ and the limit $γ\to 0$ corresponds to the semiclassical limit of the theory. Within this asymptotic and under a negative curvature condition (on the limiting metric of the theory), we determine the limit of the correlation functions and of the associated Liouville field. We also establish a large deviation result for the Liouville field: as expected, the large deviation functional is the classical Liouville action. As a corollary, we give a new (probabilistic) proof of the Takhtajan-Zograf theorem which relates the classical Liouville action (taken at its minimum) to Poincaré's accessory parameters. Finally, we gather conjectures in the positive curvature case (including the study of the so-called quantum spheres introduced by Duplantier-Miller-Sheffield).

math.PR↗

arXiv2017

Wetting and layering for Solid-on-Solid II: Layering transitions, Gibbs states, and regularity of the free energy

We consider the Solid-On-Solid model interacting with a wall, which is the statistical mechanics model associated with the integer-valued field $(ϕ(x))_{x\in \mathbb Z^2}$, and the energy functional $$V(ϕ)=β\sum_{x\sim y}|ϕ(x)-ϕ(y)|-\sum_{x}\left( h{\bf 1}_{\{ϕ(x)=0\}}-\infty{\bf 1}_{\{ϕ(x)<0\}} \right).$$ We prove that for $β$ sufficiently large, there exists a decreasing sequence $(h^*_n(β))_{n\ge 0}$, satisfying $\lim_{n\to\infty}h^*_n(β)=h_w(β),$ and such that: $(A)$ The free energy associated with the system is infinitely differentiable on $\mathbb R \setminus \left(\{h^*_n\}_{n\ge 1}\cup h_w(β)\right)$, and not differentiable on $\{h^*_n\}_{n\ge 1}$. $(B)$ For each $n\ge 0$ within the interval $(h^*_{n+1},h^*_n)$ (with the convention $h^*_0=\infty$), there exists a unique translation invariant Gibbs state which is localized around height $n$, while at a point of non-differentiability, at least two ergodic Gibbs state coexist. The respective typical heights of these two Gibbs states are $n-1$ and $n$. The value $h^*_n$ corresponds thus to a first order layering transition from level $n$ to level $n-1$. These results combined with those obtained in [23] provide a complete description of the wetting and layering transition for SOS.

math-ph↗

arXiv2017

Disorder and critical phenomena: the $α=0$ copolymer model

The generalized copolymer model is a disordered system built on a discrete renewal process with inter-arrival distribution that decays in a regularly varying fashion with exponent $1+ α\geq 1$. It exhibits a localization transition which can be characterized in terms of the free energy of the model: the free energy is zero in the delocalized phase and it is positive in the localized phase. This transition, which is observed when tuning the mean $h$ of the disorder variable, has been tackled in the physics literature notably via a renormalization group procedure that goes under the name of \emph{strong disorder renormalization}. We focus on the case $α=0$ -- the critical value $h_c(β)$ of the parameter $h$ is exactly known (for every strength $β$ of the disorder) in this case -- and we provide precise estimates on the critical behavior. Our results confirm the strong disorder renormalization group prediction that the transition is of infinite order, namely that when $h\searrow h_c(β)$ the free energy vanishes faster than any power of $h-h_c(β)$. But we show that the free energy vanishes much faster than the physicists' prediction.

math.PR↗

arXiv2017

Semiclassical limit of Liouville Field Theory

Liouville Field Theory (LFT for short) is a two dimensional model of random surfaces, which is for instance involved in $2d$ string theory or in the description of the fluctuations of metrics in $2d$ Liouville quantum gravity. This is a probabilistic model that consists in weighting the classical Free Field action with an interaction term given by the exponential of a Gaussian multiplicative chaos. The main input of our work is the study of the semiclassical limit of the theory, which is a prescribed asymptotic regime of LFT of interest in physics literature (see \cite{witten} and references therein). We derive exact formulas for the Laplace transform of the Liouville field in the case of flat metric on the unit disk with Dirichlet boundary conditions. As a consequence, we prove that the Liouville field concentrates on the solution of the classical Liouville equation with explicit negative scalar curvature. We also characterize the leading fluctuations, which are Gaussian and massive, and establish a large deviation principle. Though considered as an ansatz in the whole physics literature, it seems that it is the first rigorous probabilistic derivation of the semiclassical limit of LFT. On the other hand, we carry out the same analysis when we further weight the Liouville action with heavy matter operators. This procedure appears when computing the $n$-points correlation functions of LFT.

math.PR↗

arXiv2017

Wetting and layering for Solid-on-Solid I: Identification of the wetting point and critical behavior

We provide a complete description of the low temperature wetting transition for the two dimensional Solid-On-Solid model. More precisely we study the integer-valued field $(ϕ(x))_{x\in \mathbb Z^2}$, associated associated to the energy functional $$V(ϕ)=β\sum_{x\sim y}|ϕ(x)-ϕ(y)|-\sum_{x}\left(h{\bf 1}_{\{ϕ(x)=0\}}-\infty{\bf 1}_{\{ϕ(x)<0\}} \right).$$ It is known since the pioneering work of Chalker (J. Phys. A {\bf 15} (1982) 481-485) that for every $β$, there exists $h_{w}(β)>0$ delimiting a transition between a delocalized phase ($h h_{w}(β)$) where this proportion is positive. We prove in the present paper that for $β$ sufficiently large we have $$h_w(β)= \log \left(\frac{e^{4β}}{e^{4β}-1}\right).$$ Furthermore we provide a sharp asymptotic for the free energy at the vicinity of the critical point: We show that close to $h_w(β)$, the free energy is approximately piecewise affine and that the points of discontinuity for the derivative of the affine approximation forms a geometric sequence accumulating on the right of $h_w(β)$. This asymptotic behavior provides a strong evidence for the conjectured existence of countably many "layering transitions" at the vicinity of the critical point, corresponding to jumps for the typical height of the field.

math-ph↗

arXiv2016

Marginal relevance for the $γ$-stable pinning model

We investigate disorder relevance for the pinning of a renewal when the law of the random environment is in the domain of attraction of a stable law with parameter $γ\in (1,2)$. Assuming that the renewal jumps have power-law decay, we determine under which condition the critical point of the system modified by the introduction of a small quantity of disorder. In an earlier study of the problem, we have shown that the answer depends on the value of the tail exponent $α$ associated to the distribution of renewal jumps: when $α>1-γ^{-1}$ a small amount of disorder shifts the critical point whereas it does not when $α<1-γ^{-1}$. The present paper is focused on the boundary case $α=1-γ^{-1}$. We show that a critical point shifts occurs in this case, and obtain an estimate for its intensity.

math.PR↗

arXiv2016

Disorder relevance without Harris Criterion: the case of pinning model with $γ$-stable environment

We investigate disorder relevance for the pinning of a renewal whose inter-arrival law has tail exponent $α>0$ when the law of the random environment is in the domain of attraction of a stable law with parameter $γ\in (1,2)$. We prove that in this case, the effect of disorder is not decided by the sign of the specific heat exponent as predicted by Harris criterion but that a new criterion emerges to decide disorder relevance. More precisely we show that when $α>1-γ^{-1}$ there is a shift of the critical point at every temperature whereas when $α< 1-γ^{-1}$, at high temperature the quenched and annealed critical point coincide, and the critical exponents are identical.

math.PR↗

arXiv2016

The cutoff profile for the simple exclusion process on the circle

In this paper, we give a very accurate description of the way the simple exclusion process relaxes to equilibrium. Let $P_t$ denote the semi-group associated the exclusion on the circle with $2N$ sites and $N$ particles. For any initial condition $χ$, and for any $t\ge\frac{4N^2}{9π^2}\log N$, we show that the probability density $P_t(χ,\cdot)$ is given by an exponential tilt of the equilibrium measure by the main eigenfunction of the particle system. As $\frac{4N^2}{9π^2}\log N$ is smaller than the mixing time which is $\frac{N^2}{2π^2}\log N$, this allows to give a sharp description of the cutoff profile: if $d_N(t)$ denote the total-variation distance starting from the worse initial condition we have \[\lim_{N\to\infty}d_N\biggl(\frac{N^2}{2π^2}\log N+\frac{N^2}{π^2}s\biggr)=\operatorname {erf}\biggl(\frac{\sqrt{2}}πe^{-s}\biggr),\] where $\operatorname {erf}$ is the Gauss error function.

math.PR↗

arXiv2016

Disorder and wetting transition: the pinned harmonic crystal in dimension three or larger

We consider the Lattice Gaussian free field in $d+1$ dimensions, $d=3$ or larger, on a large box (linear size $N$) with boundary conditions zero. On this field two potentials are acting: one, that models the presence of a wall, penalizes the field when it enters the lower half space and one, the «pinning potential» , that rewards visits to the proximity of the wall. The wall can be soft, i.e. the field has a finite penalty to enter the lower half plane, or hard when the penalty is infinite. In general the pinning potential is disordered and it gives on average a reward h in $\mathbb{R}$ (a negative reward is a penalty): the energetic contribution when the field at site x visits the pinning region is $βω_x+h$, $\{ω_x\}_{x \in \mathbb{Z}^d}$ are IID centered and exponentially integrable random variables of unit variance and $β\ge 0$. In [E. Bolthausen, J.-D. Deuschel and O. Zeitouni, J. Math. Phys. 41 (2000), 1211-1223] it is shown that, when $β=0$ (that is, in the non disordered model), a delocalization-localization transition happens at $h=0$, in particular the free energy of the system is zero for $h \le 0$ and positive for $h>0$. We show that, for $β\neq 0$, the transition happens at $h=h_c(β):=- \log \mathbb{E} \exp(βω_x)$ and we find the precise asymptotic behavior of the logarithm of the free energy density of the system when $h \searrow h_c(β)$. In particular, we show that the transition is of infinite order in the sense that the free energy is smaller than any power of $h-h_c(β)$ in the neighborhood of the critical point and that disorder does not modify at all the nature of the transition. We also provide results on the behavior of the paths of the random field in the limit $N \to \infty$.

math-ph↗

arXiv2016

Total Variation and Separation Cutoffs are not equivalent and neither one implies the other

The cutoff phenomenon describes the case when an abrupt transition occurs in the convergence of a Markov chain to its equilibrium measure. There are various metrics which can be used to measure the distance to equilibrium, each of which corresponding to a different notion of cutoff. The most commonly used are the total-variation and the separation distances. In this note we prove that the cutoff for these two distances are not equivalent by constructing several counterexamples which display cutoff in total-variation but not in separation and with the opposite behavior, including lazy simple random walk on a sequence of uniformly bounded degree expander graphs. These examples give a negative answer to a question of Ding, Lubetzky and Peres.

math.PR↗