Ground state energy fluctuations of pinned elastic manifolds

TL;DR

This work analyzes ground-state energy fluctuations of pinned elastic manifolds in a random environment using a replica framework to derive a Parisi-type variational formula for the cumulant generating function. It shows that typical fluctuations obey a central limit theorem with an explicit rescaled variance and characterizes the full large deviation function across replica-symmetric and replica-symmetry-breaking phases, including a massless limit where fluctuations exhibit superconcentration. Short-range disorder is treated in detail with RS, 1RSB, and FRSB branches and transitions, complemented by explicit results in several dimensions for exponential covariances; long-range disorder is discussed via the energy-difference variable. Collectively, the results deepen understanding of energy landscapes in disordered elastic manifolds and connect to broader spin-glass physics through Parisi-type optimization and phase-transition structure.

Abstract

We describe the atypical fluctuations of the ground state energy of the random elastic manifold, a disordered model defined on a lattice of linear size $L$ with internal dimension $0\leq d<4$ embedded in a medium of dimension $N\gg 1$. The ground-state energy results from a competition between confinement, elasticity and disorder. We obtain an exact description of the large deviation rate function with speed $NL^d$ and its different phases, corresponding to different patterns of replica symmetry breaking (RSB). Our results show that the ground-state energy satisfies a central limit theorem and we obtain an explicit expression for the rescaled variance. In the (massless) limit of zero confinement, this variance vanishes for short-range disorder and the ground-state energy displays super-concentration. From our results on the large deviation function, we characterise explicitly the left tail of the distribution of the typical fluctuations of the ground state energy. It displays an exponential tail for a one step RSB pattern while for a full RSB pattern it decays super-exponentially with a non trivial exponent $ξ$ that we compute explicitly.

Figures (3)

• Figure 1: Plot of the rescaled variance ${\cal V}_{\infty}(\mu)$ defined in Eq. \ref{['var_res']} as a function of the mass $\mu$ for the elastic manifold with internal dimension $d=3$, in the continuous limit $L\to\infty$ where $\nu({\bf k})=\mu+{\bf k}^2$ and for a short-range covariance of the disorder of the form $f_\gamma(q)=(1+q/(\gamma-1))^{1-\gamma}$ with $\gamma>1$ for $\gamma=2,4,\infty$. Below the RSB transition, i.e. for $\mu<\mu_c$ (with $\mu_c$ indicated by a dot), the variance is given by the FRSB expression (orange dotted, green dashed and red dashed dotted curves for $\gamma=2,4,\infty$ respectively) while it is given by the RS expression (solid blue curve) above the second order transition $\mu>\mu_c$.
• Figure 2: Phase diagram of the large deviation function ${\cal L}(\epsilon,\mu)=\lim_{L\to \infty}\Lambda_L(e_{\rm typ}+\epsilon,\mu)$ for the centred ground state energy $\epsilon_{\min}=e_{\min}-e_{\rm typ}$ of the elastic manifold $d=1$ with exponential correlation function $f_{\infty}(q)=\exp(-q)$ (see section \ref{['exp-1d']} for details). For any fixed value of $\mu$, the RS phase (white) and 1RSB phase (blue) are separated by a transition line. The transition occurs at $\epsilon=\epsilon_{\rm dis}$ (dashed black curve) for $\mu\leq \mu_{\rm dis}$ (indicated by the horizontal dotted blue line) and is of second order while for $\mu\geq \mu_{\rm dis}$ it is of third order and occurs at $\epsilon_{\rm AT}$ (solid black curve). The typical value of the centred energy $e_{\rm typ}$, where the LDF achieves its unique zero, corresponds to the vertical line $\epsilon=0$. For $\mu\geq \mu_{c}$ (indicated by the horizontal dotted red line), it is given by its RS expression while for $\mu<\mu_{c}$ it is given by its 1RSB expression. For $\mu=0$, one expects that the large deviation function of speed $N$ is only given by its RS expression and describes any $\epsilon\leq 0$. On the other hand, the fluctuations for $\epsilon\geq 0$ (indicated by the green line) are described by a large deviation regime with higher speed.
• Figure 3: Phase diagram of the large deviation function ${\cal L}(\epsilon,\mu)=\lim_{L\to \infty}\Lambda_L(e_{\rm typ}+\epsilon,\mu)$ for the centred ground state energy $\epsilon_{\min}=e_{\min}-e_{\rm typ}$ of the elastic manifold of internal dimension $d=3$ with elasticity term $\nu({\bf k})=\mu+{\bf k}^2$ and exponential correlation function $f_{\infty}(q)=\exp(-q)$ (see section \ref{['exp-3d']} for details). For any fixed value of $\mu$, the RS phase (white) and FRSB phase (orange) are separated by a transition line. The transition occurs at $\epsilon=\epsilon_{\rm AT}$ (solid black curve) and is of third order. The typical value of the centred energy $\epsilon_{\rm typ}$, where the LDF achieves its unique zero, is represented via the solid red curve. For $\mu\geq \mu_{c}$ (indicated by the horizontal dotted red line), it is given by its RS expression while for $\mu<\mu_{c}$ it is given by its FRSB expression. For $\mu=0$, in addition to the RS phase (for energies below the black dot) and the FRSB phase (for energies between the blakc and green dot), an additional phase appears (represented by the solid green line) where the large deviation of speed $NL^d$ is infinite and a large deviation regime with higher speed describes the fluctuations.