Le Doussal Research Articles

Landscape complexity beyond invariance and the elastic manifold

AbstractThis paper characterizes the annealed, topological complexity (both of total critical points and of local minima) of the elastic manifold. This classical model of a disordered elastic system captures point configurations with self‐interactions in a random medium. We establish the simple versus glassy phase diagram in the model parameters, with these phases separated by a physical boundary known as the Larkin mass, confirming formulas of Fyodorov and Le Doussal. One essential, dynamical, step of the proof also applies to a general signal‐to‐noise model of soft spins in an anisotropic well, for which we prove a negative‐second‐moment threshold distinguishing positive from zero complexity. A universal near‐critical behavior appears within this phase portrait, namely quadratic near‐critical vanishing of the complexity of total critical points, and cubic near‐critical vanishing of the complexity of local minima. These two models serve as a paradigm of complexity calculations for Gaussian landscapes exhibiting few distributional symmetries, that is, beyond the invariant setting. The two main inputs for the proof are determinant asymptotics for non‐invariant random matrices from our companion paper (Ben Arous, Bourgade, McKenna 2022), and the atypical convexity and integrability of the limiting variational problems.

Integrability in the weak noise theory

We consider the variational problem associated with the Freidlin–Wentzell Large Deviation Principle (LDP) for the Stochastic Heat Equation (SHE). For a general class of initial-terminal conditions, we show that a minimizer of this variational problem exists, and any minimizer solves a system of imaginary-time Nonlinear Schrödinger equations. This system is integrable. Utilizing the integrability, we prove that the formulas from the physics work (see Alexandre Krajenbrink and Pierre Le Doussal [Phys. Rev. Lett. 127 (2021), p. 8]) hold for every minimizer of the variational problem. As an application, we consider the Freidlin–Wentzell LDP for the SHE with the delta initial condition. Under a technical assumption on the poles of the reflection coefficients, we prove the explicit expression for the one-point rate function that was predicted in the physics works (see Pierre Le Doussal, Satya N. Majumdar, Alberto Rosso, and Grégory Schehr [Phys. Rev. Lett. 117 (2016), p. 070403]; Alexandre Krajenbrink and Pierre Le Doussal [Phys. Rev. Lett. 127 (2021), p. 8]). Under the same assumption, we give detailed pointwise estimates of the most probable shape in the upper-tail limit.

KPZ equation with a small noise, deep upper tail and limit shape

In this paper, we consider the KPZ equation under the weak noise scaling. That is, we introduce a small parameter $$\sqrt{\varepsilon }$$ in front of the noise and let $$\varepsilon \rightarrow 0$$ . We prove that the one-point large deviation rate function has a $$\frac{3}{2}$$ power law in the deep upper tail. Furthermore, by forcing the value of the KPZ equation at a point to be very large, we prove a limit shape of the solution of the KPZ equation as $$\varepsilon \rightarrow 0$$ . This confirms the physics prediction in Hartmann et al. (Phys Rev Res 1(3):032043, 2019), Kolokolov and Korshunov (Phys Rev B 75(14):140201, 2007, Phys Rev E 80(3):031107, 2009), Kamenev et al. (Phys Rev E 94(3):032108, 2016), Le Doussal et al. (Phys Rev Lett 117(7):070403, 2016) and Meerson et al. (Phys Rev Lett 116(7):070601, 2016).

Exact lower-tail large deviations of the KPZ equation

Consider the Hopf–Cole solution h(t,x) of the Kandar–Parisi–Zhang (KPZ) equation with the narrow wedge initial condition. Regarding t→∞ as a scaling parameter, we provide the first rigorous proof of the large deviation principle (LDP) for the lower tail of h(2t,0)+t 12, with speed t2 and an explicit rate function Φ −(z). This result confirms existing physics predictions made by Corwin (2011); Sasorov, Meerson, and Prolhac (2017); and Krajenbrink, Le Doussal, and Prolhac (2018). Our analysis utilizes a formula from Borodin and Gorin (2016) to convert the LDP for the KPZ equation to calculating an exponential moment of the Airy point process (PP). To estimate this exponential moment, we invoke the stochastic Airy operator (SAO) and use the Riccati transform, comparison techniques, and certain variational characterizations of the relevant functional.

Markov limits of steady states of the KPZ equation on an interval

This paper builds upon the research of Corwin and Knizel who proved the existence of stationary measures for the KPZ equation on an interval and characterized them through a Laplace transform formula. Bryc, Kuznetsov, Wang and Wesolowski found a probabilistic description of the stationary measures in terms of a Doob transform of some Markov kernels; essentially at the same time, another description connecting the stationary measures to the exponential functionals of the Brownian motion appeared in work of Barraquand and Le Doussal. Our first main result clarifies and proves the equivalence of the two probabilistic description of these stationary measures. We then use the Markovian description to give rigorous proofs of some of the results claimed in Barraquand and Le Doussal. We analyze how the stationary measures of the KPZ equation on finite interval behave at large scale. We investigate which of the limits of the steady states of the KPZ equation obtained recently by G. Barraquand and P. Le Doussal can be represented by Markov processes in spatial variable under an additional restriction on the range of parameters.

Short Time Large Deviations of the KPZ Equation

We establish the Freidlin–Wentzell Large Deviation Principle (LDP) for the Stochastic Heat Equation with multiplicative noise in one spatial dimension. That is, we introduce a small parameter $$ \sqrt{\varepsilon } $$ to the noise, and establish an LDP for the trajectory of the solution. Such a Freidlin–Wentzell LDP gives the short-time, one-point LDP for the KPZ equation in terms of a variational problem. Analyzing this variational problem under the narrow wedge initial data, we prove a quadratic law for the near-center tail and a $$ \frac{5}{2} $$ law for the deep lower tail. These power laws confirm existing physics predictions (Kolokolov and Korshunov in Phys Rev B 75(14):140201, 2007, Phys Rev E 80(3):031107, 2009; Meerson et al. in Phys Rev Lett 116(7):070601, 2016; Le Doussal et al. in Phys Rev Lett 117(7):070403, 2016; Kamenev et al. in Phys Rev E 94(3):032108, 2016).

Overlaps of a spherical spin glass model with microscopic external field

We examine the behavior of the 2-spin spherical Sherrington-Kirkpatrick model with an external field by analyzing the overlap of a spin with the external field. Previous research has noted that, at low temperature, this overlap exhibits dramatically different behavior in the presence of an external field as compared to the model with no external field. The transition between those two settings was examined in a recent physics paper by Baik, Collins-Woodfin, Le Doussal, and Wu as well as a recent math paper by Landon and Sosoe. Both papers focus on the setting in which the external field strength, $h$, approaches zero as the dimension, $N$, approaches infinity. In particular, the paper of Baik et al studies the overlap with a microscopic external field ($h\sim N^{-1/2}$) but without a rigorous proof. This paper aims to give a proof of that result. The proof involves representing the generating function of the overlap as a ratio of contour integrals and then analyzing the asymptotics of those contour integrals using results from random matrix theory.

Time and place of the maximum for one-dimensional diffusion bridges and meanders

For three constrained Brownian motions, the excursion, the meander, and the reflected bridge, the densities of the maximum and of the time to reach it were expressed as double series by Majumdar, Randon-Furling, Kearney, and Yor (2008). Some of these series were regularized by Abel summation. Similar results for Bessel processes were obtained by Schehr and Le Doussal (2010) using the real space renormalization group method. Here this work is reviewed, and extended from the point of view of one-dimensional diffusion theory to some other diffusion processes including skew Brownian bridges and generalized Bessel meanders. We discuss the limits of the application of this method for other diffusion processes.

Fredholm Determinant Solutions of the Painlevé II Hierarchy and Gap Probabilities of Determinantal Point Processes

AbstractWe study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We show that the logarithmic derivatives of the Fredholm determinants are directly related to solutions of the Painlevé II hierarchy. This confirms and generalizes a recent conjecture by Le Doussal, Majumdar, and Schehr [20]. In addition, we obtain asymptotics at $\pm \infty $ for the Painlevé transcendents and large gap asymptotics for the corresponding point processes.

The Periodic Schur Process and Free Fermions at Finite Temperature

We revisit the periodic Schur process introduced by Borodin in 2007. Our contribution is threefold. First, we provide a new simpler derivation of its correlation functions via the free fermion formalism. In particular, we shall see that the process becomes determinantal by passing to the grand canonical ensemble, which gives a physical explanation to Borodin's "shift-mixing" trick. Second, we consider the edge scaling limit in the simplest nontrivial case, corresponding to a deformation of the poissonized Plancherel measure on partitions. We show that the edge behavior is described, in a certain crossover regime different from that for the bulk, by the universal finite-temperature Airy kernel, which was previously encountered by Johansson and Le Doussal et al. in other models, and whose extreme value statistics interpolates between the Tracy-Widom GUE and the Gumbel distributions. We also define and prove convergence for a stationary extension of our model. Finally, we compute the correlation functions for a variant of the periodic Schur process involving strict partitions, Schur's P and Q functions, and neutral fermions.

Two-time height distribution for 1D KPZ growth: the recent exact result and its tail via replica

We consider the fluctuations in the stochastic growth of a one-dimensional interface of height described by the Kardar–Parisi–Zhang (KPZ) universality class. We study the joint probability distribution function (JPDF) of the interface heights at two-times t1 and , with droplet initial conditions at t = 0. In the large-time limit, this JPDF is expected to become a universal function of the time ratio , and of the (properly scaled) heights h(x,t1) and h(x,t2). Using the replica Bethe ansatz method for the KPZ equation, in De Nardis and Le Doussal (2017 J. Stat. Mech. 053212) we obtained a formula for the JPDF in the (partial) tail regime where h(x,t1) is large and positive, subsequently found to be in excellent agreement with the experimental and numerical data (De Nardis et al 2017 Phys. Rev. Lett. 118 125701). Here we show that our results are in perfect agreement with Johansson’s recent rigorous expression of the full JPDF (Johansson 2018 in preparation (arXiv:1802.00729)), thereby confirming the validity of our methods.

Maximum of an Airy process plus Brownian motion and memory in Kardar-Parisi-Zhang growth.

We obtain several exact results for universal distributions involving the maximum of the Airy_{2} process minus a parabola and plus a Brownian motion, with applications to the one-dimensional Kardar-Parisi-Zhang (KPZ) stochastic growth universality class. This allows one to obtain (i) the universal limit, for large time separation, of the two-time height correlation for droplet initial conditions, e.g., C_{∞}=lim_{t_{2}/t_{1}→+∞}h(t_{1})h(t_{2})[over ¯]^{c}/h(t_{1})^{2}[over ¯]^{c}, with C_{∞}≈0.623, as well as conditional moments, which quantify ergodicity breaking in the time evolution; (ii) in the same limit, the distribution of the midpoint position x(t_{1}) of a directed polymer of length t_{2}; and (iii) the height distribution in stationary KPZ with a step. These results are derived from the replica Bethe ansatz for the KPZ continuum equation, with a "decoupling assumption" in the large time limit. They agree and confirm, whenever they can be compared, with (i) our recent tail results for two-time KPZ with the work by de Nardis and Le Doussal [J. Stat. Mech. (2017) 0532121742-546810.1088/1742-5468/aa6bce], checked in experiments with the work by Takeuchi and co-workers [De Nardis et al., Phys. Rev. Lett. 118, 125701 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.125701] and (ii) a recent result of Maes and Thiery [J. Stat. Phys. 168, 937 (2017)JSTPBS0022-471510.1007/s10955-017-1839-2] on midpoint position.

Anomalous elasticity, fluctuations and disorder in elastic membranes

Motivated by freely suspended graphene and polymerized membranes in soft and biological matter we present a detailed study of a tensionless elastic sheet in the presence of thermal fluctuations and quenched disorder. The manuscript is based on an extensive draft dating back to 1993, that was circulated privately. It presents the general theoretical framework and calculational details of numerous results, partial forms of which have been published in brief Letters (Le Doussal and Radzihovsky, 1992; 1993). The experimental realization atom-thin graphene sheets (Novoselov et al., 2004) have driven a resurgence in this fascinating subject, making our dated predictions and their detailed derivations timely. To this end we analyze the statistical mechanics of a generalized D-dimensional elastic “membrane” embedded in d dimensions using a self-consistent screening approximation (SCSA), that has proved to be unprecedentedly accurate in this system, exact in three complementary limits: (i) d→∞, (ii) D→4, and (iii) D=d. Focusing on the critical “flat” phase, for a homogeneous two-dimensional (D=2) membrane embedded in three dimensions (d=3), we predict its universal roughness exponent ζ=0.590, length-scale dependent elastic moduli exponents η=0.821 and ηu=0.358, and an anomalous Poisson ratio, σ=−1∕3. In the presence of random uncorrelated heterogeneity the membrane exhibits a glassy wrinkled ground state, characterized by ζ′=0.775,η′=0.449,ηu′=1.101 and a Poisson ratio σ′=−1∕3. Motivated by a number of physical realizations (charged impurities, disclinations and dislocations) we also study power-law correlated quenched disorder that leads to a variety of distinct glassy wrinkled phases. Finally, neglecting self-avoiding interaction we demonstrate that at high temperature a “phantom” sheet undergoes a continuous crumpling transition, characterized by a radius of gyration exponent, ν=0.732 and η=0.535. Many of these universal predictions have received considerable support from simulations. We hope that this detailed presentation of the SCSA theory will be useful to further theoretical developments and corresponding experimental investigations on freely suspended graphene.

Log-gamma directed polymer with one free end via coordinate Bethe Ansatz

The discrete polymer model with random Boltzmann weights with homogeneous inverse gamma distribution, introduced by Seppäläinen, is studied in the case of a polymer with one fixed and one free end. The model with two fixed ends has been integrated by Thiery and Le Doussal, using coordinate Bethe Ansatz techniques and an analytic-continuation prescription. The probability distribution of the free energy has been obtained through the replica method, even though the moments of the partition sum do not exist at all orders due to the fat tail in the distribution of Boltzmann weights. To extend this approach to the polymer with one free end, we argue that the contribution to the partition sums in the thermodynamic limit is localised on parity-invariant string states. This situation is analogous to the case of the continuum polymer with one free end, related to the Kardar–Parisi–Zhang equation with flat boundary conditions and solved by Le Doussal and Calabrese. The expansion of the generating function of the partition sum in terms of numbers of strings can also be transposed to the log-gamma polymer model, with the induced Fredholm determinant structure. We derive the large-time limit of the rescaled cumulative distribution function, and relate it to the GOE Tracy–Widom distribution. The derivation is conjectural in the sense that it assumes completeness of a family of string states, and expressions of their norms, already useful in the fixed-end problem, and extends heuristically the order of moments of the partition sum to the complex plane.

Directed percolation with a conserved field and the depinning transition.

Conserved directed percolation (C-DP) and the depinning transition of a disordered elastic interface belong to the same universality class, as has been proven very recently by Le Doussal and Wiese [Phys. Rev. Lett. 114, 110601 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.110601] through a mapping of the field theory for C-DP onto that of the quenched Edwards-Wilkinson model. Here, we present an alternative derivation of the C-DP field theoretic functional, starting with the coherent-state path integral formulation of the C-DP and then applying the Grassberger transformation, which avoids the disadvantages of the so-called Doi shift. We revisit the aforementioned mapping with focus on a specific term in the field theoretic functional that has been problematic in the past when it came to assessing its relevance. We show that this term is redundant in the sense of the renormalization group.

Persistence of some additive functionals of Sinai’s walk

Nous nous intéressons à la marche de Sinai $(S_{n})_{n\in\mathbb{N}}$. Nous prouvons que la probabilité annealed que $\sum_{k=0}^{n}f(S_{k})$ soit strictement positive pour tout $n\in[1,N]$ est égale à $1/(\log N)^{(3-\sqrt{5})/2+o(1)}$, pour une large classe de fonctions $f$, et en particulier pour $f(x)=x$. L’exposant de persistance $\frac{3-\sqrt{5}}{2}$ est d’abord apparu dans un article non rigoureux de Le Doussal, Monthus et Fischer, avec des motivations venant de la physique. La preuve est basée sur des techniques de localisation pour la marche de Sinai et utilise des résultats de Cheliotis sur les changements de signe des fonds de vallées d’un mouvement Brownien indexé par $\mathbb{R}$.

Exact formulas for random growth with half-flat initial data

We obtain exact formulas for moments and generating functions of the height function of the asymmetric simple exclusion process at one spatial point, starting from special initial data in which every positive even site is initially occupied. These complement earlier formulas of E. Lee [J. Stat. Phys. 140 (2010) 635–647] but, unlike those formulas, ours are suitable in principle for asymptotics. We also explain how our formulas are related to divergent series formulas for half-flat KPZ of Le Doussal and Calabrese [J. Stat. Mech. 2012 (2012) P06001], which we also recover using the methods of this paper. These generating functions are given as a series without any apparent Fredholm determinant or Pfaffian structure. In the long time limit, formal asymptotics show that the fluctuations are given by the Airy$_{2\to1}$ marginals.

Matrix Optimization Under Random External Fields

We consider the quadratic optimization problem $$F_n^{W,h}:= \sup_{x \in S^{n-1}} ( x^T W x/2 + h^T x )\,, $$ with $W$ a (random) matrix and $h$ a random external field. We study the probabilities of large deviation of $F_n^{W,h}$ for $h$ a centered Gaussian vector with i.i.d. entries, both conditioned on $W$ (a general Wigner matrix), and unconditioned when $W$ is a GOE matrix. Our results validate (in a certain region) and correct (in another region), the prediction obtained by the mathematically non-rigorous replica method in Y. V. Fyodorov, P. Le Doussal, J. Stat. phys. 154 (2014).

High values of disorder-generated multifractals and logarithmically correlated processes

In the introductory section of the article we give a brief account of recent insights into statistics of high and extreme values of disorder-generated multifractals following a recent work by the first author with P. Le Doussal and A. Rosso (FLR) employing a close relation between multifractality and logarithmically correlated random fields. We then substantiate some aspects of the FLR approach analytically for multifractal eigenvectors in the Ruijsenaars–Schneider ensemble (RSE) of random matrices introduced by E. Bogomolny and the second author by providing an ab initio calculation that reveals hidden logarithmic correlations at the background of the disorder-generated multifractality. In the rest we investigate numerically a few representative models of that class, including the study of the highest component of multifractal eigenvectors in the Ruijsenaars–Schneider ensemble.

Poisson–Dirichlet statistics for the extremes of a log-correlated Gaussian field

We study the statistics of the extremes of a discrete Gaussian field with logarithmic correlations at the level of the Gibbs measure. The model is defined on the periodic interval $[0,1]$, and its correlation structure is nonhierarchical. It is based on a model introduced by Bacry and Muzy [Comm. Math. Phys. 236 (2003) 449-475] (see also Barral and Mandelbrot [Probab. Theory Related Fields 124 (2002) 409-430]), and is similar to the logarithmic Random Energy Model studied by Carpentier and Le Doussal [Phys. Rev. E (3) 63 (2001) 026110] and more recently by Fyodorov and Bouchaud [J. Phys. A 41 (2008) 372001]. At low temperature, it is shown that the normalized covariance of two points sampled from the Gibbs measure is either $0$ or $1$. This is used to prove that the joint distribution of the Gibbs weights converges in a suitable sense to that of a Poisson-Dirichlet variable. In particular, this proves a conjecture of Carpentier and Le Doussal that the statistics of the extremes of the log-correlated field behave as those of i.i.d. Gaussian variables and of branching Brownian motion at the level of the Gibbs measure. The method of proof is robust and is adaptable to other log-correlated Gaussian fields.