Manifolds in a high-dimensional random landscape: Complexity of stationary points and depinning

Abstract

We obtain explicit expressions for the annealed complexities associated respectively with the total number of (i) stationary points and (ii) local minima of the energy landscape for an elastic manifold with internal dimension d < 4 embedded in a random medium of dimension N ≫ 1 and confined by a parabolic potential with the curvature parameter µ. These complexities are found to both vanish at the critical value µc identified as the Larkin mass. For µ < µc the system is in complex phase corresponding to the replica symmetry breaking in its T = 0 thermodynamics. The complexities vanish respectively quadratically (stationary points) and cubically (minima) at µ − c . For d ≥ 1 they admit a finite "massless" limit µ = 0 which is used to provide an upper bound for the depinning threshold under an applied force. PACS numbers:Numerous physical systems can be modeled by a collection of points or particles coupled by an elastic energy, usually called an elastic manifold, submitted to a random potential (see [1] for a review). They are often called "disordered elastic systems" and generically exhibit pinning in their statics and depinning transitions and avalanches in their driven dynamics [2][3][4][5][6].The manifold can be parameterized by a N -componentis the sum of an elastic energy, given by the (discrete) Laplacian matrix −t 0 ∆ xy , t 0 > 0, a quadratic confining energy controlled by the curvature parameter µ 0 > 0 (or, alternatively, the "mass" m = √ µ 0 ) and a centered Gaussian random potential with covarianceparametrized by a function B(z). This random potential is thus uncorrelated in the internal space and statistically translational invariant in the embedding space. We use periodic boundary conditions, i.e. the Laplacian eigenmodes are plane waves ∼ e ikx with eigenvalues ∆(k).Examples are ∆(k) = 2(cos k − 1) with k = 2πn/L, n = 0, ..L − 1 in d = 1, and for the continuum model ∆(k) = −k 2 in any dimension with k ∈ R d .The energy landscape provided by the functional (1) is complex and necessarily high-dimensional, i.e. involves many interacting and competing degrees of freedom, leading to glassy behavior. This necessitates to use methods and ideas of statistical mechanics of disordered systems such that Replica Symmetry Breaking (RSB), Functional Renormalization Group, etc. for understanding their properties [8][9][10][11]. Among other approaches, the problem of characterizing complexity of random high-dimensional landscapes by understanding statistical structure of their stationary points (minima, maxima and saddles), defined in our model by the condition δH δu(x) = 0 for all x, has attracted considerable intrinsic interest in recent years in pure and applied mathematics, see e.g. [12][13][14][15][16], as well as in theoretical physics, see [17][18][19][20][21][22][23][24] and references to earlier works in [25].For the energy functional (1)-(2) in the simplest 'toy model' limit d = 0 with no elastic interactions (when x is essentially a single point, L d = 1), the mean number of stationary points, N tot , an...

Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates

“…Further assuming the self-averaging property of this large determinant (4) we have obtained exact formulae for both these annealed complexities extending the known rigorous results [43,44] for gradient random field to fields with both gradient and solenoidal components. We have shown that the total complexity is independent of the fraction τ of gradient components while the complexity of stable equilibria undergoes an additional transition: it is negative for τ ≤ τ 0 (c) and positive conversely.…”

“…These properties can be exploited for the essential simplification in Kac-Rice formulae (16) reducing them after standard calculations, see e.g. [29,43], to the following form…”

“…Recently, the framework of the "landscape topology trivialization transition" phenomena has been essentially extended to deal with a physical problem of longstanding interest -the depinning transition of elastic manifolds in a random potential [42,43]. Developing that line of research further the recent paper [44] revealed its intimate connection to the equilibria counting problem for gradient flows Eq.…”

Distribution of rare saddles in the p -spin energy landscape

“…High-dimensional systems are typically associated to complex, highly non-convex energy landscapes, in which the number of stationary points (local minima, maxima or saddles) increases steeply with the dimensionality. Classifying these points in terms of their energy, of their stability and of their location in the underlying configuration space is a topic that is of interest in a large variety of fields, including disordered systems [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], ecology and biology [16][17][18][19], neural networks [20,21], inference [22][23][24][25][26], game theory [27], string theory and cosmology [28,29]. In many of these contexts, a crucial motivation for determining the distribution of stationary points is to understand how the energy functional is explored dynamically, through algorithms that proceed via local moves in configuration space, biased towards lower-energy configurations.…”

Landscape complexity beyond invariance and the elastic manifold

“…We remark that Fyodorov and Le Doussal also exhibited a quadratic/cubic near‐critical behavior for this model but in a different scaling, varying μ 0 for fixed $$B^{\prime \prime }(0)$$ and t 0 [31].…”