Critical Jammed Phase of the Linear Perceptron.

Area of Science:

• Statistical Physics
• Complex Systems
• Constraint Satisfaction Problems

Background:

• Criticality in statistical physics usually occurs at isolated points in phase diagrams.
• Jamming transitions in systems like spheres define boundaries between unjammed and jammed phases.
• Constraint satisfaction problems with continuous variables often exhibit jamming at satisfiability transition points.

Purpose of the Study:

• To demonstrate that criticality can be extended into a region of the jammed phase.
• To investigate the impact of carefully chosen cost functions on criticality.
• To analyze critical phenomena in the spherical perceptron model within the jammed phase.

Main Methods:

• Minimization of a linear cost function in the spherical perceptron model.
• Numerical simulations to observe emergent power laws.
• Development of a scaling theory to calculate critical exponents.

Main Results:

• Criticality was successfully extended to occupy a region within the jammed phase.
• Numerical simulations revealed critical power laws in configurations from cost function minimization.
• Emergent critical exponents were identified and theoretically computed.

Conclusions:

• The choice of cost function is crucial for extending criticality into jammed phases.
• Jamming and criticality can coexist and exhibit power-law behaviors.
• The spherical perceptron with a linear cost function serves as a valid model for studying these extended critical phenomena.