55,670 research outputs found
The two-dimensional random-bond Ising model, free fermions and the network model
We develop a recently-proposed mapping of the two-dimensional Ising model
with random exchange (RBIM), via the transfer matrix, to a network model for a
disordered system of non-interacting fermions. The RBIM transforms in this way
to a localisation problem belonging to one of a set of non-standard symmetry
classes, known as class D; the transition between paramagnet and ferromagnet is
equivalent to a delocalisation transition between an insulator and a quantum
Hall conductor. We establish the mapping as an exact and efficient tool for
numerical analysis: using it, the computational effort required to study a
system of width is proportional to , and not exponential in as
with conventional algorithms. We show how the approach may be used to calculate
for the RBIM: the free energy; typical correlation lengths in quasi-one
dimension for both the spin and the disorder operators; even powers of
spin-spin correlation functions and their disorder-averages. We examine in
detail the square-lattice, nearest-neighbour RBIM, in which bonds are
independently antiferromagnetic with probability , and ferromagnetic with
probability . Studying temperatures , we obtain precise
coordinates in the plane for points on the phase boundary between
ferromagnet and paramagnet, and for the multicritical (Nishimori) point. We
demonstrate scaling flow towards the pure Ising fixed point at small , and
determine critical exponents at the multicritical point.Comment: 20 pages, 25 figures, figures correcte
Correlation Decay in Random Decision Networks
We consider a decision network on an undirected graph in which each node
corresponds to a decision variable, and each node and edge of the graph is
associated with a reward function whose value depends only on the variables of
the corresponding nodes. The goal is to construct a decision vector which
maximizes the total reward. This decision problem encompasses a variety of
models, including maximum-likelihood inference in graphical models (Markov
Random Fields), combinatorial optimization on graphs, economic team theory and
statistical physics. The network is endowed with a probabilistic structure in
which costs are sampled from a distribution. Our aim is to identify sufficient
conditions to guarantee average-case polynomiality of the underlying
optimization problem. We construct a new decentralized algorithm called Cavity
Expansion and establish its theoretical performance for a variety of models.
Specifically, for certain classes of models we prove that our algorithm is able
to find near optimal solutions with high probability in a decentralized way.
The success of the algorithm is based on the network exhibiting a correlation
decay (long-range independence) property. Our results have the following
surprising implications in the area of average case complexity of algorithms.
Finding the largest independent (stable) set of a graph is a well known NP-hard
optimization problem for which no polynomial time approximation scheme is
possible even for graphs with largest connectivity equal to three, unless P=NP.
We show that the closely related maximum weighted independent set problem for
the same class of graphs admits a PTAS when the weights are i.i.d. with the
exponential distribution. Namely, randomization of the reward function turns an
NP-hard problem into a tractable one
Linear response for spiking neuronal networks with unbounded memory
We establish a general linear response relation for spiking neuronal
networks, based on chains with unbounded memory. This relation allows us to
predict the influence of a weak amplitude time-dependent external stimuli on
spatio-temporal spike correlations, from the spontaneous statistics (without
stimulus) in a general context where the memory in spike dynamics can extend
arbitrarily far in the past. Using this approach, we show how linear response
is explicitly related to neuronal dynamics with an example, the gIF model,
introduced by M. Rudolph and A. Destexhe. This example illustrates the
collective effect of the stimuli, intrinsic neuronal dynamics, and network
connectivity on spike statistics. We illustrate our results with numerical
simulations.Comment: 60 pages, 8 figure
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