255 research outputs found
Exactly solvable models of adaptive networks
A satisfiability (SAT-UNSAT) transition takes place for many optimization
problems when the number of constraints, graphically represented by links
between variables nodes, is brought above some threshold. If the network of
constraints is allowed to adapt by redistributing its links, the SAT-UNSAT
transition may be delayed and preceded by an intermediate phase where the
structure self-organizes to satisfy the constraints. We present an analytic
approach, based on the recently introduced cavity method for large deviations,
which exactly describes the two phase transitions delimiting this adaptive
intermediate phase. We give explicit results for random bond models subject to
the connectivity or rigidity percolation transitions, and compare them with
numerical simulations.Comment: 4 pages, 4 figure
Landscape of solutions in constraint satisfaction problems
We present a theoretical framework for characterizing the geometrical
properties of the space of solutions in constraint satisfaction problems,
together with practical algorithms for studying this structure on particular
instances. We apply our method to the coloring problem, for which we obtain the
total number of solutions and analyze in detail the distribution of distances
between solutions.Comment: 4 pages, 4 figures. Replaced with published versio
The cavity method for large deviations
A method is introduced for studying large deviations in the context of
statistical physics of disordered systems. The approach, based on an extension
of the cavity method to atypical realizations of the quenched disorder, allows
us to compute exponentially small probabilities (rate functions) over different
classes of random graphs. It is illustrated with two combinatorial optimization
problems, the vertex-cover and coloring problems, for which the presence of
replica symmetry breaking phases is taken into account. Applications include
the analysis of models on adaptive graph structures.Comment: 18 pages, 7 figure
Cusps and shocks in the renormalized potential of glassy random manifolds: How Functional Renormalization Group and Replica Symmetry Breaking fit together
We compute the Functional Renormalization Group (FRG) disorder- correlator
function R(v) for d-dimensional elastic manifolds pinned by a random potential
in the limit of infinite embedding space dimension N. It measures the
equilibrium response of the manifold in a quadratic potential well as the
center of the well is varied from 0 to v. We find two distinct scaling regimes:
(i) a "single shock" regime, v^2 ~ 1/L^d where L^d is the system volume and
(ii) a "thermodynamic" regime, v^2 ~ N. In regime (i) all the equivalent
replica symmetry breaking (RSB) saddle points within the Gaussian variational
approximation contribute, while in regime (ii) the effect of RSB enters only
through a single anomaly. When the RSB is continuous (e.g., for short-range
disorder, in dimension 2 <= d <= 4), we prove that regime (ii) yields the
large-N FRG function obtained previously. In that case, the disorder correlator
exhibits a cusp in both regimes, though with different amplitudes and of
different physical origin. When the RSB solution is 1-step and non- marginal
(e.g., d < 2 for SR disorder), the correlator R(v) in regime (ii) is
considerably reduced, and exhibits no cusp. Solutions of the FRG flow
corresponding to non-equilibrium states are discussed as well. In all cases the
regime (i) exhibits a cusp non-analyticity at T=0, whose form and thermal
rounding at finite T is obtained exactly and interpreted in terms of shocks.
The results are compared with previous work, and consequences for manifolds at
finite N, as well as extensions to spin glasses and related models are
discussed.Comment: v2: Note added in proo
Message passing for vertex covers
Constructing a minimal vertex cover of a graph can be seen as a prototype for
a combinatorial optimization problem under hard constraints. In this paper, we
develop and analyze message passing techniques, namely warning and survey
propagation, which serve as efficient heuristic algorithms for solving these
computational hard problems. We show also, how previously obtained results on
the typical-case behavior of vertex covers of random graphs can be recovered
starting from the message passing equations, and how they can be extended.Comment: 25 pages, 9 figures - version accepted for publication in PR
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
The Phase Diagram of 1-in-3 Satisfiability Problem
We study the typical case properties of the 1-in-3 satisfiability problem,
the boolean satisfaction problem where a clause is satisfied by exactly one
literal, in an enlarged random ensemble parametrized by average connectivity
and probability of negation of a variable in a clause. Random 1-in-3
Satisfiability and Exact 3-Cover are special cases of this ensemble. We
interpolate between these cases from a region where satisfiability can be
typically decided for all connectivities in polynomial time to a region where
deciding satisfiability is hard, in some interval of connectivities. We derive
several rigorous results in the first region, and develop the
one-step--replica-symmetry-breaking cavity analysis in the second one. We
discuss the prediction for the transition between the almost surely satisfiable
and the almost surely unsatisfiable phase, and other structural properties of
the phase diagram, in light of cavity method results.Comment: 30 pages, 12 figure
Integrated TiO2 resonators for visible photonics
We demonstrate waveguide-coupled titanium dioxide (TiO2) racetrack resonators
with loaded quality factors of 2x10^4 for the visible wavelengths. The
structures were fabricated in sputtered TiO2 thin films on oxidized silicon
substrates using standard top-down nanofabrication techniques, and passively
probed in transmission measurements using a tunable red laser. Devices based on
this material could serve as integrated optical elements as well as passive
platforms for coupling to visible quantum emitters.Comment: 4 pages, 3 figure
Random multi-index matching problems
The multi-index matching problem (MIMP) generalizes the well known matching
problem by going from pairs to d-uplets. We use the cavity method from
statistical physics to analyze its properties when the costs of the d-uplets
are random. At low temperatures we find for d>2 a frozen glassy phase with
vanishing entropy. We also investigate some properties of small samples by
enumerating the lowest cost matchings to compare with our theoretical
predictions.Comment: 22 pages, 16 figure
An algorithm for counting circuits: application to real-world and random graphs
We introduce an algorithm which estimates the number of circuits in a graph
as a function of their length. This approach provides analytical results for
the typical entropy of circuits in sparse random graphs. When applied to
real-world networks, it allows to estimate exponentially large numbers of
circuits in polynomial time. We illustrate the method by studying a graph of
the Internet structure.Comment: 7 pages, 3 figures, minor corrections, accepted versio
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