10 research outputs found
Intrinsic limitations of inverse inference in the pairwise Ising spin glass
We analyze the limits inherent to the inverse reconstruction of a pairwise
Ising spin glass based on susceptibility propagation. We establish the
conditions under which the susceptibility propagation algorithm is able to
reconstruct the characteristics of the network given first- and second-order
local observables, evaluate eventual errors due to various types of noise in
the originally observed data, and discuss the scaling of the problem with the
number of degrees of freedom
Random subcubes as a toy model for constraint satisfaction problems
We present an exactly solvable random-subcube model inspired by the structure
of hard constraint satisfaction and optimization problems. Our model reproduces
the structure of the solution space of the random k-satisfiability and
k-coloring problems, and undergoes the same phase transitions as these
problems. The comparison becomes quantitative in the large-k limit. Distance
properties, as well the x-satisfiability threshold, are studied. The model is
also generalized to define a continuous energy landscape useful for studying
several aspects of glassy dynamics.Comment: 21 pages, 4 figure
Geometrical organization of solutions to random linear Boolean equations
The random XORSAT problem deals with large random linear systems of Boolean
variables. The difficulty of such problems is controlled by the ratio of number
of equations to number of variables. It is known that in some range of values
of this parameter, the space of solutions breaks into many disconnected
clusters. Here we study precisely the corresponding geometrical organization.
In particular, the distribution of distances between these clusters is computed
by the cavity method. This allows to study the `x-satisfiability' threshold,
the critical density of equations where there exist two solutions at a given
distance.Comment: 20 page
Clustering of solutions in the random satisfiability problem
Using elementary rigorous methods we prove the existence of a clustered phase
in the random -SAT problem, for . In this phase the solutions are
grouped into clusters which are far away from each other. The results are in
agreement with previous predictions of the cavity method and give a rigorous
confirmation to one of its main building blocks. It can be generalized to other
systems of both physical and computational interest.Comment: 4 pages, 1 figur
Statistical mechanics of error exponents for error-correcting codes
Error exponents characterize the exponential decay, when increasing message
length, of the probability of error of many error-correcting codes. To tackle
the long standing problem of computing them exactly, we introduce a general,
thermodynamic, formalism that we illustrate with maximum-likelihood decoding of
low-density parity-check (LDPC) codes on the binary erasure channel (BEC) and
the binary symmetric channel (BSC). In this formalism, we apply the cavity
method for large deviations to derive expressions for both the average and
typical error exponents, which differ by the procedure used to select the codes
from specified ensembles. When decreasing the noise intensity, we find that two
phase transitions take place, at two different levels: a glass to ferromagnetic
transition in the space of codewords, and a paramagnetic to glass transition in
the space of codes.Comment: 32 pages, 13 figure
Clusters of solutions and replica symmetry breaking in random k-satisfiability
We study the set of solutions of random k-satisfiability formulae through the
cavity method. It is known that, for an interval of the clause-to-variables
ratio, this decomposes into an exponential number of pure states (clusters). We
refine substantially this picture by: (i) determining the precise location of
the clustering transition; (ii) uncovering a second `condensation' phase
transition in the structure of the solution set for k larger or equal than 4.
These results both follow from computing the large deviation rate of the
internal entropy of pure states. From a technical point of view our main
contributions are a simplified version of the cavity formalism for special
values of the Parisi replica symmetry breaking parameter m (in particular for
m=1 via a correspondence with the tree reconstruction problem) and new large-k
expansions.Comment: 30 pages, 14 figures, typos corrected, discussion of appendix C
expanded with a new figur
Are biological systems poised at criticality?
Many of life's most fascinating phenomena emerge from interactions among many
elements--many amino acids determine the structure of a single protein, many
genes determine the fate of a cell, many neurons are involved in shaping our
thoughts and memories. Physicists have long hoped that these collective
behaviors could be described using the ideas and methods of statistical
mechanics. In the past few years, new, larger scale experiments have made it
possible to construct statistical mechanics models of biological systems
directly from real data. We review the surprising successes of this "inverse"
approach, using examples form families of proteins, networks of neurons, and
flocks of birds. Remarkably, in all these cases the models that emerge from the
data are poised at a very special point in their parameter space--a critical
point. This suggests there may be some deeper theoretical principle behind the
behavior of these diverse systems.Comment: 21 page