arXiv subjects

Pierre Le Doussal

Publications and source records attributed to Pierre Le Doussal.

At least 55 records · Page 3Linked to original sources

arXiv2023

Replica-symmetry breaking transitions in the large deviations of the ground-state of a spherical spin-glass

We derive, within the replica formalism, a generalisation of the Crisanti-Sommers formula to describe the large deviation function (LDF) ${\cal L}(e)$ for the speed-$N$ atypical fluctuations of the intensive ground-state energy $e$ of a generic spherical spin-glass in the presence of a random external magnetic field of variance $Γ$. We then analyse our exact formula for the LDF in much detail for the Replica symmetric, single step Replica Symmetry Breaking (1-RSB) and Full Replica Symmetry Breaking (FRSB) situations. Our main qualitative conclusion is that the level of RSB governing the LDF may be different from that for the typical ground state. We find that while the deepest ground-states are always controlled by a LDF of replica symmetric form, beyond a finite threshold $e\geq e_{t}$ a replica-symmetry breaking starts to be operative. These findings resolve the puzzling discrepancy between our earlier replica calculations for the $p=2$ spherical spin-glass and the rigorous results by Dembo and Zeitouni which we are able to reproduce invoking an 1-RSB pattern. Finally at an even larger critical energy $e_{c}\geq e_{t}$, acting as a "wall", the LDF diverges logarithmically, which we interpret as a change in the large deviation speed from $N$ to a faster growth. In addition, we show that in the limit $Γ\to 0$ the LDF takes non-trivial scaling forms (i) ${\cal L}(e) \sim G((e-e_c)/Γ)$ in the vicinity of the wall (ii) ${\cal L}(e) \sim Γ^{ην} F((e-e_{\rm typ})/Γ^ν)$ in the vicinity of the typical energy, characterised by two new exponents $η\geq 1$ and $ν$ characterising universality classes. Via matching the latter allows us to formulate several conjectures concerning the regime of {\it typical fluctuations}, identified as $e-e_{\rm typ} \sim N^{-1/η}$ and $Γ\sim N^{-1/(ην)}$.

cond-mat.stat-mech↗

arXiv2023

The weak noise theory of the O'Connell-Yor polymer as an integrable discretisation of the nonlinear Schrodinger equation

We investigate and solve the weak noise theory for the semi-discrete O'Connell-Yor directed polymer. In the large deviation regime, the most probable evolution of the partition function obeys a classical non-linear system which is a non-standard discretisation of the nonlinear Schrodinger equation with mixed initial-final conditions. We show that this system is integrable and find its general solution through an inverse scattering method and a novel Fredholm determinant framework that we develop. This allows to obtain the large deviation rate function of the free energy of the polymer model from its conserved quantities and to study its convergence to the large deviations of the Kardar-Parisi-Zhang equation. Our model also degenerates to the classical Toda chain, which further substantiates the applicability of our Fredholm framework.

cond-mat.stat-mech↗

arXiv2023

The KPZ equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall

We present an exact solution for the height distribution of the KPZ equation at any time $t$ in a half space with flat initial condition. This is equivalent to obtaining the free energy distribution of a polymer of length $t$ pinned at a wall at a single point. In the large $t$ limit a binding transition takes place upon increasing the attractiveness of the wall. Around the critical point we find the same statistics as in the Baik-Ben--Arous-Péché transition for outlier eigenvalues in random matrix theory. In the bound phase, we obtain the exact measure for the endpoint and the midpoint of the polymer at large time. We also unveil curious identities in distribution between partition functions in half-space and certain partition functions in full space for Brownian type initial condition.

cond-mat.stat-mech↗

arXiv2023

Clusters in the critical branching Brownian motion

Brownian particles that are replicated and annihilated at equal rate have strongly correlated positions, forming a few compact clusters separated by large gaps. We characterize the distribution of the particles at a given time, using a definition of clusters in terms a coarse-graining length recently introduced by some of us. We show that, in a non-extinct realization, the average number of clusters grows as $\sim t^{D_{\mathrm{f}}/2}$ where $D_{\mathrm{f}} \approx 0.22$ is the Haussdoff dimension of the boundary of the super-Brownian motion, found by Mueller, Mytnik, and Perkins. We also compute the distribution of gaps between consecutive particles. We find two regimes separated by the characteristic length scale $\ell = \sqrt{D/β}$ where $D$ is the diffusion constant and $β$ the branching rate. The average number of gaps greater than $g$ decays as $\sim g^{D_{\mathrm{f}}-2}$ for $g\ll \ell$ and $\sim g^{-D_{\mathrm{f}}}$ for $g \gg \ell$. Finally, conditioned on the number of particles $n$, the above distributions are valid for $g \ll \sqrt{n}$; the average number of gaps greater than $g \gg \sqrt{n}$ is much less than one, and decays as $\simeq 4 (g/\sqrt{n})^{-2}$, in agreement with the universal gap distribution predicted by Ramola, Majumdar, and Schehr. Our results interpolate between a dense super-Brownian motion regime and a large-gap regime, unifying two previously independent approaches.

cond-mat.stat-mech↗

arXiv2023

A stationary model of non-intersecting directed polymers

We consider the partition function $Z_{\ell}(\vec x,0\vert \vec y,t)$ of $\ell$ non-intersecting continuous directed polymers of length $t$ in dimension $1+1$, in a white noise environment, starting from positions $\vec x$ and terminating at positions $\vec y$. When $\ell=1$, it is well known that for fixed $x$, the field $\log Z_1(x,0\vert y,t)$ solves the Kardar-Parisi-Zhang equation and admits the Brownian motion as a stationary measure. In particular, as $t$ goes to infinity, $Z_1(x,0\vert y,t)/Z_1(x,0\vert 0,t) $ converges to the exponential of a Brownian motion $B(y)$. In this article, we show an analogue of this result for any $\ell$. We show that $Z_{\ell}(\vec x,0\vert \vec y,t)/Z_{\ell}(\vec x,0\vert \vec 0,t) $ converges as $t$ goes to infinity to an explicit functional $Z_{\ell}^{\rm stat}(\vec y)$ of $\ell$ independent Brownian motions. This functional $Z_{\ell}^{\rm stat}(\vec y)$ admits a simple description as the partition sum for $\ell$ non-intersecting semi-discrete polymers on $\ell$ lines. We discuss applications to the endpoints and midpoints distribution for long non-crossing polymers and derive explicit formulas in the case of two polymers. To obtain these results, we show that the stationary measure of the O'Connell-Warren multilayer stochastic heat equation is given by a collection of independent Brownian motions. This in turn is shown via analogous results in a discrete setup for the so-called log-gamma polymer and exploit the connection between non-intersecting log-gamma polymers and the geometric RSK correspondence found in arXiv:1110.3489. .

cond-mat.stat-mech↗

arXiv2023

Out of equilibrium dynamics of repulsive ranked diffusions: the expanding crystal

We study the non-equilibrium Langevin dynamics of $N$ particles in one dimension with Coulomb repulsive linear interactions. This is a dynamical version of the so-called jellium model (without confinement) also known as ranked diffusion. Using a mapping to the Lieb-Liniger model of quantum bosons, we obtain an exact formula for the joint distribution of the positions of the $N$ particles at time $t$, all starting from the origin. A saddle point analysis shows that the system converges at large time to a linearly expanding crystal. Properly rescaled, this dynamical state resembles the equilibrium crystal in a time dependent effective quadratic potential. This analogy allows to study the fluctuations around the perfect crystal, which, to leading order, are Gaussian. There are however deviations from this Gaussian behavior, which embody long-range correlations of purely dynamical origin, characterized by the higher order cumulants of, e.g., the gaps between the particles, that we calculate exactly. We complement these results using a recent approach by one of us in terms of a noisy Burgers equation. In the large $N$ limit, the mean density of the gas can be obtained at any time from the solution of a deterministic viscous Burgers equation. This approach provides a quantitative description of the dense regime at shorter times. Our predictions are in good agreement with numerical simulations for finite and large $N$.

cond-mat.stat-mech↗

arXiv2022

Stationary time correlations for fermions after a quench in the presence of an impurity

We consider the quench dynamics of non-interacting fermions in one dimension in the presence of a finite-size impurity at the origin. This impurity is characterized by general momentum-dependent reflection and transmission coefficients which are changed from ${\sf r}_0(k), {\sf t}_0(k)$ to ${\sf r}(k), {\sf t}(k)$ at time $t=0$. The initial state is at equilibrium with ${\sf t}_0(k)=0$ such that the system is cut in two independent halves with ${\sf r}_0^R(k)$, ${\sf r}_0^L(k)$ respectively to the right and to the left of the impurity. We obtain the exact large time limit of the multi-time correlations. These correlations become time translationally invariant, and are non-zero in two different regimes: (i) for $x=O(1)$ where the system reaches a non-equilibrium steady state (NESS) (ii) for $x \sim t$, i.e., the ray-regime. For a repulsive impurity these correlations are independent of ${\sf r}_0^R(k)$, ${\sf r}_0^L(k)$, while in the presence of bound states they oscillate and memory effects persist. We show that these nontrivial relaxational properties can be retrieved in a simple manner from the large time behaviour of the single particle wave functions.

cond-mat.stat-mech↗

arXiv2022

Stationary measures of the KPZ equation on an interval from Enaud-Derrida's matrix product ansatz representation

The stationary measures of the Kardar-Parisi-Zhang equation on an interval have been computed recently. We present a rather direct derivation of this result by taking the weak asymmetry limit of the matrix product ansatz for the asymmetric simple exclusion process. We rely on the matrix product ansatz representation of Enaud and Derrida, which allows to express the steady-state in terms of re-weighted simple random walks. In the continuum limit, its measure becomes a path integral (or re-weighted Brownian motion) of the form encountered in Liouville quantum mechanics, recovering the recent formula.

math-ph↗

arXiv2022

Exact position distribution of a harmonically-confined run-and-tumble particle in two dimensions

We consider an overdamped run-and-tumble particle in two dimensions, with self propulsion in an orientation that stochastically rotates by 90 degrees at a constant rate, clockwise or counter-clockwise with equal probabilities. In addition, the particle is confined by an external harmonic potential of stiffness $μ$, and possibly diffuses. We find the exact time-dependent distribution $P\left(x,y,t\right)$ of the particle's position, and in particular, the steady-state distribution $P_{\text{st}}\left(x,y\right)$ that is reached in the long-time limit. We also find $P\left(x,y,t\right)$ for a "free" particle, $μ=0$. We achieve this by showing that, under a proper change of coordinates, the problem decomposes into two statistically-independent one-dimensional problems, whose exact solution has recently been obtained. We then extend these results in several directions, to two such run-and-tumble particles with a harmonic interaction, to analogous systems of dimension three or higher, and by allowing stochastic resetting.

cond-mat.stat-mech↗

arXiv2022

The inverse scattering of the Zakharov-Shabat system solves the weak noise theory of the Kardar-Parisi-Zhang equation

We solve the large deviations of the Kardar-Parisi-Zhang (KPZ) equation in one dimension at short time by introducing an approach which combines field theoretical, probabilistic and integrable techniques. We expand the program of the weak noise theory, which maps the large deviations onto a non-linear hydrodynamic problem, and unveil its complete solvability through a connection to the integrability of the Zakharov-Shabat system. Exact solutions, depending on the initial condition of the KPZ equation, are obtained using the inverse scattering method and a Fredholm determinant framework recently developed. These results, explicit in the case of the droplet geometry, open the path to obtain the complete large deviations for general initial conditions.

cond-mat.stat-mech↗

arXiv2022

Clusters in an epidemic model with long-range dispersal

In presence of long range dispersal, epidemics spread in spatially disconnected regions known as clusters. Here, we characterize exactly their statistical properties in a solvable model, in both the supercritical (outbreak) and critical regimes. We identify two diverging length scales, corresponding to the bulk and the outskirt of the epidemic. We reveal a nontrivial critical exponent that governs the cluster number, the distribution of their sizes and of the distances between them. We also discuss applications to depinning avalanches with long range elasticity.

cond-mat.stat-mech↗

arXiv2022

Density profile of noninteracting fermions in a rotating $2d$ trap at finite temperature

We study the average density of $N$ spinless noninteracting fermions in a $2d$ harmonic trap rotating with a constant frequency $Ω$ and in the presence of an additional repulsive central potential $γ/r^2$. The average density at zero temperature was recently studied in Phys. Rev. A $\textbf{103}$, 033321 (2021) and an interesting multi-layered "wedding cake" structure with a "hole" at the center was found for the density in the large $N$ limit. In this paper, we study the average density at finite temperature. We demonstrate how this "wedding-cake" structure is modified at finite temperature. These large $N$ results warrant going much beyond the standard Local Density Approximation. We also generalize our results to a wide variety of trapping potentials and demonstrate the universality of the associated scaling functions both in the bulk and at the edges of the "wedding-cake".

cond-mat.stat-mech↗

arXiv2022

Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case

We study the Kardar-Parisi-Zhang equation on the half-line $x \geqslant 0$ with Neumann type boundary condition. Stationary measures of the KPZ dynamics were characterized in recent work: they depend on two parameters, the boundary parameter $u$ of the dynamics, and the drift $-v$ of the initial condition at infinity. We consider the fluctuations of the height field when the initial condition is given by one of these stationary processes. At large time $t$, it is natural to rescale parameters as $(u,v)=t^{-1/3}(a,b)$ to study the critical region. In the special case $a+b=0$, treated in previous works, the stationary process is simply Brownian. However, these Brownian stationary measures are particularly relevant in the bound phase ($a<0$) but not in the unbound phase. For instance, starting from the flat or droplet initial data, the height field near the boundary converges to the stationary process with $a>0$ and $b=0$, which is not Brownian. For $a+b\geqslant 0$, we determine exactly the large time distribution $F_{a,b}^{\rm stat}$ of the height function $h(0,t)$. As an application, we obtain the exact covariance of the height field in a half-line at two times $1\ll t_1\ll t_2$ starting from stationary initial data, as well as estimates, when starting from droplet initial data, in the limit $t_1/t_2\to 1$.

cond-mat.stat-mech↗

arXiv2022

Counting statistics for non-interacting fermions in a rotating trap

We study the ground state of $N \gg 1$ noninteracting fermions in a two-dimensional harmonic trap rotating at angular frequency $Ω>0$. The support of the density of the Fermi gas is a disk of radius $R_e$. We calculate the variance of the number of fermions ${\cal N}_R$ inside a disk of radius $R$ centered at the origin for $R$ in the bulk of the Fermi gas. We find rich and interesting behaviours in two different scaling regimes: (i) $Ω/ ω<1 $ and (ii) $1 - Ω/ ω= O(1/N)$, where $ω$ is the angular frequency of the oscillator. In the first regime (i) we find that ${\rm Var}\,{\cal N}_{R}\simeq\left(A\log N+B\right)\sqrt{N}$ and we calculate $A$ and $B$ as functions of $R/R_e$, $Ω$ and $ω$. We also predict the higher cumulants of ${\cal N}_{R}$ and the bipartite entanglement entropy of the disk with the rest of the system. In the second regime (ii), the mean fermion density exhibits a staircase form, with discrete plateaus corresponding to filling $k$ successive Landau levels, as found in previous studies. Here, we show that ${\rm Var}\,{\cal N}_{R}$ is a discontinuous piecewise linear function of $\sim (R/R_e) \sqrt{N}$ within each plateau, with coefficients that we calculate exactly, and with steps whose precise shape we obtain for any $k$. We argue that a similar piecewise linear behavior extends to all the cumulants of ${\cal N}_{R}$ and to the entanglement entropy. We show that these results match smoothly at large $k$ with the above results for $Ω/ω=O(1)$. These findings are nicely confirmed by numerical simulations. Finally, we uncover a universal behavior of ${\rm Var}\,{\cal N}_{R}$ near the fermionic edge. We extend our results to a three-dimensional geometry, where an additional confining potential is applied in the $z$ direction.

cond-mat.stat-mech↗

arXiv2022

The crossover from the Macroscopic Fluctuation Theory to the Kardar-Parisi-Zhang equation controls the large deviations beyond Einstein's diffusion

We study the crossover from the macroscopic fluctuation theory (MFT) which describes 1D stochastic diffusive systems at late times, to the weak noise theory (WNT) which describes the Kardar-Parisi-Zhang (KPZ) equation at early times. We focus on the example of the diffusion in a time-dependent random field, observed in an atypical direction which induces an asymmetry. The crossover is described by a non-linear system which interpolates between the derivative and the standard non-linear Schrodinger equations in imaginary time. We solve this system using the inverse scattering method for mixed-time boundary conditions introduced by us to solve the WNT. We obtain the rate function which describes the large deviations of the sample-to-sample fluctuations of the cumulative distribution of the tracer position. It exhibits a crossover as the asymmetry is varied, recovering both MFT and KPZ limits. We sketch how it is consistent with extracting the asymptotics of a Fredholm determinant formula, recently derived for sticky Brownian motions. The crossover mechanism studied here should generalize to a larger class of models described by the MFT. Our results apply to study extremal diffusion beyond Einstein's theory.

cond-mat.stat-mech↗

arXiv2022

More on the Brownian force model: avalanche shapes, tip driven, higher $d$

The Brownian force model (BFM) is the mean-field model for the avalanches of an elastic interface slowly driven in a random medium. It describes the spatio-temporal statistics of the velocity field, and, to some extent is analytically tractable. We extend our previous studies to obtain several observables for the BFM with short range elasticity, related to the local jump sizes $S(x)$ and to the avalanche spatial extension in $d=1$, or the avalanche span in $d>1$. In $d=1$ we consider both driving (i) by an imposed force (ii) by an imposed displacement "at the tip" and obtain in each case the mean spatial shape $\langle S(x) \rangle$ at fixed extension, or at fixed seed to edge distance. We find that near an edge $x_e$, $S(x) \simeq σ|x-x_e|^3$ where $σ$ has a universal distribution that we obtain. We also obtain the spatiotemporal shape near the edge. In $d>1$ we obtain (i) the mean shape $\langle S(x_1,x_\perp) \rangle$ for a fixed span, which exhibits a non-trivial dependence in the transverse distance to the seed $x_\perp$ (ii) the mean shape around a point which has not moved, $\langle S(x) \rangle_{S(0)=0}$, which vanishes at the center as $|x|^{b_d}$ with non trivial exponents, $b_1=4$, $b_2=2 \sqrt{2}$ and $b_3=\frac{1}{2} (\sqrt{17}-1)$. We obtain the probability distributions in any $d$ for the maximal radius of an avalanche and the minimal distance of approach to a given point, as well as the probability of not hitting a cone in $d=2$. These results equivalently apply to the continuum limit of some branching diffusions, as detailed in a companion paper.

cond-mat.stat-mech↗

arXiv2022

Equivalence of mean-field avalanches and branching diffusions: From the Brownian force model to the super-Brownian motion

We point out that the mean-field theory of avalanches in the dynamics of elastic interfaces, the so-called Brownian force model (BFM) developed recently in non-equilibrium statistical physics, is equivalent to the so-called super-Brownian motion (SBM) developed in probability theory, a continuum limit of branching processes related to {\it space-embedded} Galton-Watson trees. In particular the exact solvability property recently (re-)discovered from the field theory in mean-field avalanches (the "instanton equation") maps onto the so-called Dawson-Watanabe 1968 duality property. In the light of this correspondence we compare the results obtained independently in the two fields, and transport some of them from one field to the other. In particular, we discuss a scaling limit of the branching Brownian motion which maps onto the continuum field theory of mean-field avalanches

cond-mat.stat-mech↗

arXiv2022

Quench dynamics of noninteracting fermions with a delta impurity

We study the out-of-equilibrium dynamics of noninteracting fermions in one dimension and in continuum space, in the presence of a delta impurity potential at the origin whose strength $g$ is varied at time $t=0$. The system is prepared in its ground state with $g=g_0=+\infty$, with two different densities and Fermi wave-vectors $k_L$ and $k_R$ on the two half-spaces $x>0$ and $x<0$ respectively. It then evolves for $t>0$ as an isolated system, with a finite impurity strength $g$. We compute exactly the time dependent density and current. For a fixed position $x$ and in the large time limit $t \to \infty$, the system reaches a non-equilibrium stationary state (NESS). We obtain analytically the correlation kernel, density, particle current, and energy current in the NESS, and characterize their relaxation, which is algebraic in time. In particular, in the NESS, we show that, away from the impurity, the particle density displays oscillations which are the non-equilibrium analog of the Friedel oscillations. In the regime of "rays", $x/t=ξ$ fixed with $x, t \to \infty$, we compute the same quantities and observe the emergence of two light cones, associated to the Fermi velocities $k_L$ and $k_R$ in the initial state. Interestingly, we find non trivial quantum correlations between two opposite rays with velocities $ξ$ and $-ξ$ which we compute explicitly. We extend to a continuum setting and to a correlated initial state the analytical methods developed in a recent work of Ljubotina, Sotiriadis and Prosen, in the context of a discrete fermionic chain with an impurity. We also generalize our results to an initial state at finite temperature, recovering, via explicit calculations, some predictions of conformal field theory in the low energy limit.

cond-mat.stat-mech↗