Surfing on minima of isostatic landscapes: avalanches and unjamming transition

TL;DR

This paper explores the high-dimensional landscape of isostatic configurations in optimization problems, revealing avalanche dynamics, critical exponents, and logarithmic scaling near the unjamming transition using an innovative adiabatic algorithm.

Contribution

It introduces an adiabatic algorithm to navigate isostatic landscapes and characterizes avalanche statistics and critical properties at the unjamming transition.

Findings

Avalanches follow a power-law size distribution with a specific critical exponent.

Energy and pressure exhibit logarithmic scaling near the unjamming point.

Number of violated soft constraints scales as a non-trivial power law.

Abstract

Recently, we showed that optimization problems, both in infinite as well as in finite dimensions, for continuous variables and soft excluded volume constraints, can display entire isostatic phases where local minima of the cost function are marginally stable configurations endowed with non-linear excitations [1,2]. In this work we describe an athermal adiabatic algorithm to explore with continuity the corresponding rough high-dimensional landscape. We concentrate on a prototype problem of this kind, the spherical perceptron optimization problem with linear cost function (hinge loss). This algorithm allows to "surf" between isostatic marginally stable configurations and to investigate some properties of such landscape. In particular we focus on the statistics of avalanches occurring when local minima are destabilized. We show that when perturbing such minima, the system undergoes plastic rearrangements whose size is power law distributed and we characterize the corresponding critical exponent. Finally we investigate the critical properties of the unjamming transition, showing that the linear interaction potential gives rise to logarithmic behavior in the scaling of energy and pressure as a function of the distance from the unjamming point. For some quantities, the logarithmic corrections can be gauged out. This is the case of the number of soft constraints that are violated as a function of the distance from jamming which follows a non-trivial power law behavior.