Phase Transitions – like death and taxes? Why we should care; what to do about it. Scott Kirkpatrick, Hebrew University, Jerusalem With thanks to: Uri Gordon, Erik Aurell, Johannes Schneider,…

Phase transitions are inevitable in Avogadro- scale engineering. System properties go from “good” to “not good enough” at the extremes of their parameter space  What happens in between these phases?  We have limited tools for understanding such transitions.  Physics of disordered materials: Sharp vs. smeared – Harris criterion Glass transitions  Combinatorics on large scales Sharp property crossovers – Friedgut’s theorem We should care because computing is HARD at phase boundaries.

SAT and 3-SAT as classic test cases Parameters: N variables, M constraining clauses, M/N constant For 3-SAT, phase diagram is known: M/N< 3.9“easy,” satisfiable (probably P) 3.9 < M/N < 4.27“hard,” but still satisfiable 4.27… < M/Nunsatisfiable, exponential cost Recent advances, applying message-passing, pushed boundary of solubility in the “hard-SAT” region from N = 300 to N = 10^7. General technique – soften the variables into beliefs or surveys

Surveys and Beliefs for the SAT problem Beliefs – probabilities that the spin is up or down  Avg. over satisfying configurations, as estimated by the local tree Surveys – probabilities that the spin is up or down  In all sat configurations.  This leaves a third possibility – spins that do both at different times. Equations for both are nearly identical. We can define hybrid methods, and they prove useful. Use beliefs or surveys to guide decimation – this solves problem.

Propagating surveys or beliefs in a “cavity”

Depth of decimation characterizes BP, SP and mixed-P

Study how SP, BP, m-P evolve by analyzing their hydrodynamics Use movies of N = 100,000 Look at three cases, SP, BP, and the hybrid-P:     (caution, large files)