2 research outputs found
What makes a phase transition? Analysis of the random satisfiability problem
In the last 30 years it was found that many combinatorial systems undergo
phase transitions. One of the most important examples of these can be found
among the random k-satisfiability problems (often referred to as k-SAT), asking
whether there exists an assignment of Boolean values satisfying a Boolean
formula composed of clauses with k random variables each. The random 3-SAT
problem is reported to show various phase transitions at different critical
values of the ratio of the number of clauses to the number of variables. The
most famous of these occurs when the probability of finding a satisfiable
instance suddenly drops from 1 to 0. This transition is associated with a rise
in the hardness of the problem, but until now the correlation between any of
the proposed phase transitions and the hardness is not totally clear. In this
paper we will first show numerically that the number of solutions universally
follows a lognormal distribution, thereby explaining the puzzling question of
why the number of solutions is still exponential at the critical point.
Moreover we provide evidence that the hardness of the closely related problem
of counting the total number of solutions does not show any phase
transition-like behavior. This raises the question of whether the probability
of finding a satisfiable instance is really an order parameter of a phase
transition or whether it is more likely to just show a simple sharp threshold
phenomenon. More generally, this paper aims at starting a discussion where a
simple sharp threshold phenomenon turns into a genuine phase transition
The spherical 2+p spin glass model: an analytically solvable model with a glass-to-glass transition
We present the detailed analysis of the spherical s+p spin glass model with
two competing interactions: among p spins and among s spins. The most
interesting case is the 2+p model with p > 3 for which a very rich phase
diagram occurs, including, next to the paramagnetic and the glassy phase
represented by the one step replica symmetry breaking ansatz typical of the
spherical p-spin model, other two amorphous phases. Transitions between two
contiguous phases can also be of different kind. The model can thus serve as
mean-field representation of amorphous-amorphous transitions (or transitions
between undercooled liquids of different structure). The model is analytically
solvable everywhere in the phase space, even in the limit where the infinite
replica symmetry breaking ansatz is required to yield a thermodynamically
stable phase.Comment: 21 pages, 18 figure