# Critical jammed phase of the linear perceptron

## TL;DR

This paper demonstrates that by selecting an appropriate cost function, the critical jammed phase in the spherical perceptron extends beyond a point to occupy an entire region, revealing new critical behavior in constraint satisfaction problems.

## Contribution

It introduces a method to extend the critical jammed phase in the spherical perceptron using a linear cost function, supported by numerical simulations and a scaling theory.

## Key findings

- Critical power laws emerge in configurations minimizing the linear cost function.
- The critical region extends throughout the entire jammed phase with the right cost function.
- A scaling theory successfully predicts the critical exponents.

## 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.