Critical Jammed Phase of the Linear Perceptron

Abstract

Criticality in statistical physics naturally emerges at isolated points in the phase diagram. Jamming of spheres is not an exception: varying density, it is the critical point that separates the unjammed phase where spheres do not overlap and the jammed phase where they cannot be arranged without overlaps. The same remains true in more general constraint satisfaction problems with continuous variables (CCSP) where jamming coincides with the (protocol dependent) satisfiability transition point. In this work we show that by carefully choosing the cost function to be minimized, the region of criticality extends to occupy a whole region of the jammed phase. As a working example, we consider the spherical perceptron with a linear cost function in the unsatisfiable (UNSAT) jammed phase and we perform numerical simulations which show critical power laws emerging in the configurations obtained minimizing the linear cost function. We develop a scaling theory to compute the emerging critical exponents.

Critical energy landscape of linear soft spheres

“…Assuming that both processes occur with finite probability, we have θ + = θ − and γ + = γ − . Following [9,12], one arrives at the scaling relation γ + = 1/(2 + θ + ) controlling the critical exponents, which is verified by both our numerics and the mean field theory of [22].…”

“…Isostaticity and critical behavior in the force and gap distributions have been shown to appear in the unsatisfiable phase of the spherical perceptron optimization problem with linear cost function, which is a mean field model for linear spheres [22]. The main result of the present work is that these properties appear to survive in a robust way when we go to finite dimension.…”

“…The exact solution of the perceptron optimization problem with a linear cost function [22] provides a comprehensive mean-field framework that predicts the main features discussed in this paper, namely isostaticity, identity of exponents θ + = θ − , γ + = γ − , their numerical values and so on. This theory can be adapted straightforwardly to soft linear spheres in infinite dimension [27].…”

“…In this section we construct the mean field theory for such behavior. We consider the spherical perceptron optimization problem with linear cost function studied recently in [22], see also [41][42][43]. This model is in the same universality class of linear soft spheres.…”

Jamming with Tunable Roughness

“…Experiments [4] and numerical simulations [10,11] show that the contact number at φ J indeed satisfies z J ¼ 2d. Remarkably, recent numerical and theoretical progress unveiled that isostatic systems, which encompass some classes of neural networks [8,9,12,13] in addition to frictionless spherical particles, belong to the same universality class [14][15][16][17][18].…”