857 research outputs found
Soft random solids and their heterogeneous elasticity
Spatial heterogeneity in the elastic properties of soft random solids is
examined via vulcanization theory. The spatial heterogeneity in the
\emph{structure} of soft random solids is a result of the fluctuations
locked-in at their synthesis, which also brings heterogeneity in their
\emph{elastic properties}. Vulcanization theory studies semi-microscopic models
of random-solid-forming systems, and applies replica field theory to deal with
their quenched disorder and thermal fluctuations. The elastic deformations of
soft random solids are argued to be described by the Goldstone sector of
fluctuations contained in vulcanization theory, associated with a subtle form
of spontaneous symmetry breaking that is associated with the
liquid-to-random-solid transition. The resulting free energy of this Goldstone
sector can be reinterpreted as arising from a phenomenological description of
an elastic medium with quenched disorder. Through this comparison, we arrive at
the statistics of the quenched disorder of the elasticity of soft random
solids, in terms of residual stress and Lam\'e-coefficient fields. In
particular, there are large residual stresses in the equilibrium reference
state, and the disorder correlators involving the residual stress are found to
be long-ranged and governed by a universal parameter that also gives the mean
shear modulus.Comment: 40 pages, 7 figure
Spectral Statistics of Erd{\H o}s-R\'enyi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues
We consider the ensemble of adjacency matrices of Erd{\H o}s-R\'enyi random
graphs, i.e.\ graphs on vertices where every edge is chosen independently
and with probability . We rescale the matrix so that its bulk
eigenvalues are of order one. Under the assumption , we prove
the universality of eigenvalue distributions both in the bulk and at the edge
of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of
the Erd{\H o}s-R\'enyi graph in the bulk of the spectrum has the same
distribution as that of the Gaussian orthogonal ensemble; and (2) that the
second largest eigenvalue of the Erd{\H o}s-R\'enyi graph has the same
distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As
an application of our method, we prove the bulk universality of generalized
Wigner matrices under the assumption that the matrix entries have at least moments
Growth of graph states in quantum networks
We propose a scheme to distribute graph states over quantum networks in the
presence of noise in the channels and in the operations. The protocol can be
implemented efficiently for large graph sates of arbitrary (complex) topology.
We benchmark our scheme with two protocols where each connected component is
prepared in a node belonging to the component and subsequently distributed via
quantum repeaters to the remaining connected nodes. We show that the fidelity
of the generated graphs can be written as the partition function of a classical
Ising-type Hamiltonian. We give exact expressions of the fidelity of the linear
cluster and results for its decay rate in random graphs with arbitrary
(uncorrelated) degree distributions.Comment: 16 pages, 7 figure
Graph realizations constrained by skeleton graphs
In 2008 Amanatidis, Green and Mihail introduced the Joint Degree Matrix (JDM)
model to capture the fundamental difference in assortativity of networks in
nature studied by the physical and life sciences and social networks studied in
the social sciences. In 2014 Czabarka proposed a direct generalization of the
JDM model, the Partition Adjacency Matrix (PAM) model. In the PAM model the
vertices have specified degrees, and the vertex set itself is partitioned into
classes. For each pair of vertex classes the number of edges between the
classes in a graph realization is prescribed. In this paper we apply the new
{\em skeleton graph} model to describe the same information as the PAM model.
Our model is more convenient for handling problems with low number of partition
classes or with special topological restrictions among the classes. We
investigate two particular cases in detail: (i) when there are only two vertex
classes and (ii) when the skeleton graph contains at most one cycle.Comment: 19 page
Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model
We consider Hermitian and symmetric random band matrices in
dimensions. The matrix elements , indexed by , are independent, uniformly distributed random variables if \abs{x-y}
is less than the band width , and zero otherwise. We prove that the time
evolution of a quantum particle subject to the Hamiltonian is diffusive on
time scales . We also show that the localization length of an
arbitrarily large majority of the eigenvectors is larger than a factor
times the band width. All results are uniform in the size
\abs{\Lambda} of the matrix.Comment: Minor corrections, Sections 4 and 11 update
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
The early evolution of the H-free process
The H-free process, for some fixed graph H, is the random graph process
defined by starting with an empty graph on n vertices and then adding edges one
at a time, chosen uniformly at random subject to the constraint that no H
subgraph is formed. Let G be the random maximal H-free graph obtained at the
end of the process. When H is strictly 2-balanced, we show that for some c>0,
with high probability as , the minimum degree in G is at least
. This gives new lower bounds for
the Tur\'an numbers of certain bipartite graphs, such as the complete bipartite
graphs with . When H is a complete graph with we show that for some C>0, with high probability the independence number of
G is at most . This gives new lower bounds
for Ramsey numbers R(s,t) for fixed and t large. We also obtain new
bounds for the independence number of G for other graphs H, including the case
when H is a cycle. Our proofs use the differential equations method for random
graph processes to analyse the evolution of the process, and give further
information about the structure of the graphs obtained, including asymptotic
formulae for a broad class of subgraph extension variables.Comment: 36 page
Network growth model with intrinsic vertex fitness
© 2013 American Physical SocietyWe study a class of network growth models with attachment rules governed by intrinsic node fitness. Both the individual node degree distribution and the degree correlation properties of the network are obtained as functions of the network growth rules. We also find analytical solutions to the inverse, design, problems of matching the growth rules to the required (e.g., power-law) node degree distribution and more generally to the required degree correlation function. We find that the design problems do not always have solutions. Among the specific conditions on the existence of solutions to the design problems is the requirement that the node degree distribution has to be broader than a certain threshold and the fact that factorizability of the correlation functions requires singular distributions of the node fitnesses. More generally, the restrictions on the input distributions and correlations that ensure solvability of the design problems are expressed in terms of the analytical properties of their generating functions
Statistics of Cycles: How Loopy is your Network?
We study the distribution of cycles of length h in large networks (of size
N>>1) and find it to be an excellent ergodic estimator, even in the extreme
inhomogeneous case of scale-free networks. The distribution is sharply peaked
around a characteristic cycle length, h* ~ N^a. Our results suggest that h* and
the exponent a might usefully characterize broad families of networks. In
addition to an exact counting of cycles in hierarchical nets, we present a
Monte-Carlo sampling algorithm for approximately locating h* and reliably
determining a. Our empirical results indicate that for small random scale-free
nets of degree exponent g, a=1/(g-1), and a grows as the nets become larger.Comment: Further work presented and conclusions revised, following referee
report
Kinetic Limit for Wave Propagation in a Random Medium
We study crystal dynamics in the harmonic approximation. The atomic masses
are weakly disordered, in the sense that their deviation from uniformity is of
order epsilon^(1/2). The dispersion relation is assumed to be a Morse function
and to suppress crossed recollisions. We then prove that in the limit epsilon
to 0 the disorder averaged Wigner function on the kinetic scale, time and space
of order epsilon^(-1), is governed by a linear Boltzmann equation.Comment: 71 pages, 3 figure
- …