4,667 research outputs found
Maximal entropy random networks with given degree distribution
Using a maximum entropy principle to assign a statistical weight to any
graph, we introduce a model of random graphs with arbitrary degree distribution
in the framework of standard statistical mechanics. We compute the free energy
and the distribution of connected components. We determine the size of the
percolation cluster above the percolation threshold. The conditional degree
distribution on the percolation cluster is also given. We briefly present the
analogous discussion for oriented graphs, giving for example the percolation
criterion.Comment: 22 pages, LateX, no figur
Dipolar SLEs
We present basic properties of Dipolar SLEs, a new version of stochastic
Loewner evolutions (SLE) in which the critical interfaces end randomly on an
interval of the boundary of a planar domain. We present a general argument
explaining why correlation functions of models of statistical mechanics are
expected to be martingales and we give a relation between dipolar SLEs and
CFTs. We compute SLE excursion and/or visiting probabilities, including the
probability for a point to be on the left/right of the SLE trace or that to be
inside the SLE hull. These functions, which turn out to be harmonic, have a
simple CFT interpretation. We also present numerical simulations of the
ferromagnetic Ising interface that confirm both the probabilistic approach and
the CFT mapping.Comment: 22 pages, 4 figure
SLE martingales and the Virasoro algebra
We present an explicit relation between representations of the Virasoro algebra and polynomial martingales in stochastic Loewner evolutions (SLE). We show that the Virasoro algebra is the spectrum generating algebra of SLE martingales. This is based on a new representation of the Virasoro algebra, inspired by the Borel-Weil construction, acting on functions depending on coordinates parametrizing conformal maps
Quantum Stochastic Processes: A Case Study
We present a detailed study of a simple quantum stochastic process, the
quantum phase space Brownian motion, which we obtain as the Markovian limit of
a simple model of open quantum system. We show that this physical description
of the process allows us to specify and to construct the dilation of the
quantum dynamical maps, including conditional quantum expectations. The quantum
phase space Brownian motion possesses many properties similar to that of the
classical Brownian motion, notably its increments are independent and
identically distributed. Possible applications to dissipative phenomena in the
quantum Hall effect are suggested.Comment: 35 pages, 1 figure
Condensation phase transition in nonlinear fitness networks
We analyze the condensation phase transitions in out-of-equilibrium complex
networks in a unifying framework which includes the nonlinear model and the
fitness model as its appropriate limits. We show a novel phase structure which
depends on both the fitness parameter and the nonlinear exponent. The
occurrence of the condensation phase transitions in the dynamical evolution of
the network is demonstrated by using Bianconi-Barabasi method. We find that the
nonlinear and the fitness preferential attachment mechanisms play important
roles in formation of an interesting phase structure.Comment: 6 pages, 5 figure
Fusion and singular vectors in A1{(1)} highest weight cyclic modules
We show how the interplay between the fusion formalism of conformal field
theory and the Knizhnik--Zamolodchikov equation leads to explicit formulae for
the singular vectors in the highest weight representations of A1{(1)}.Comment: 42 page
LERW as an example of off-critical SLEs
Two dimensional loop erased random walk (LERW) is a random curve, whose
continuum limit is known to be a Schramm-Loewner evolution (SLE) with parameter
kappa=2. In this article we study ``off-critical loop erased random walks'',
loop erasures of random walks penalized by their number of steps. On one hand
we are able to identify counterparts for some LERW observables in terms of
symplectic fermions (c=-2), thus making further steps towards a field theoretic
description of LERWs. On the other hand, we show that it is possible to
understand the Loewner driving function of the continuum limit of off-critical
LERWs, thus providing an example of application of SLE-like techniques to
models near their critical point. Such a description is bound to be quite
complicated because outside the critical point one has a finite correlation
length and therefore no conformal invariance. However, the example here shows
the question need not be intractable. We will present the results with emphasis
on general features that can be expected to be true in other off-critical
models.Comment: 45 pages, 2 figure
Calogero-Sutherland eigenfunctions with mixed boundary conditions and conformal field theory correlators
We construct certain eigenfunctions of the Calogero-Sutherland hamiltonian
for particles on a circle, with mixed boundary conditions. That is, the
behavior of the eigenfunction, as neighbouring particles collide, depend on the
pair of colliding particles. This behavior is generically a linear combination
of two types of power laws, depending on the statistics of the particles
involved. For fixed ratio of each type at each pair of neighboring particles,
there is an eigenfunction, the ground state, with lowest energy, and there is a
discrete set of eigenstates and eigenvalues, the excited states and the
energies above this ground state. We find the ground state and special excited
states along with their energies in a certain class of mixed boundary
conditions, interpreted as having pairs of neighboring bosons and other
particles being fermions. These particular eigenfunctions are characterised by
the fact that they are in direct correspondence with correlation functions in
boundary conformal field theory. We expect that they have applications to
measures on certain configurations of curves in the statistical O(n) loop
model. The derivation, although completely independent from results of
conformal field theory, uses ideas from the "Coulomb gas" formulation.Comment: 35 pages, 9 figure
Applicability of Perturbative QCD to Decays
We examine the applicability of perturbative QCD to meson decays into
mesons. We find that the perturbative QCD formalism, which includes Sudakov
effects at intermediate energy scales, is applicable to the semi-leptonic decay
, when the meson recoils fast. Following this conclusion, we
analyze the two-body non-leptonic decays and . By
comparing our predictions with experimental data, we extract the matrix element
.Comment: 18 pages in Latex, figures are available upon reques
Even-visiting random walks: exact and asymptotic results in one dimension
We reconsider the problem of even-visiting random walks in one dimension.
This problem is mapped onto a non-Hermitian Anderson model with binary
disorder. We develop very efficient numerical tools to enumerate and
characterize even-visiting walks. The number of closed walks is obtained as an
exact integer up to 1828 steps, i.e., some walks. On the analytical
side, the concepts and techniques of one-dimensional disordered systems allow
to obtain explicit asymptotic estimates for the number of closed walks of
steps up to an absolute prefactor of order unity, which is determined
numerically. All the cumulants of the maximum height reached by such walks are
shown to grow as , with exactly known prefactors. These results
illustrate the tight relationship between even-visiting walks, trapping models,
and the Lifshitz tails of disordered electron or phonon spectra.Comment: 24 pages, 4 figures. To appear in J. Phys.
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