2,007 research outputs found
Spectral asymptotics of the Laplacian on supercritical bond-percolation graphs
We investigate Laplacians on supercritical bond-percolation graphs with
different boundary conditions at cluster borders. The integrated density of
states of the Dirichlet Laplacian is found to exhibit a Lifshits tail at the
lower spectral edge, while that of the Neumann Laplacian shows a van Hove
asymptotics, which results from the percolating cluster. At the upper spectral
edge, the behaviour is reversed.Comment: 16 pages, typos corrected, to appear in J. Funct. Ana
The Widom-Rowlinson Model on the Delaunay Graph
We establish phase transitions for continuum Delaunay multi-type particle
systems (continuum Potts or Widom-Rowlinson models) with a repulsive
interaction between particles of different types. Our interaction potential
depends solely on the length of the Delaunay edges. We show that a phase
transition occurs for sufficiently large activities and for sufficiently large
potential parameter proving an old conjecture of Lebowitz and Lieb extended to
the Delaunay structure. Our approach involves a Delaunay random-cluster
representation analogous to the Fortuin-Kasteleyn representation of the Potts
model. The phase transition manifests itself in the mixed site-bond percolation
of the corresponding random-cluster model. Our proofs rely mainly on geometric
properties of Delaunay tessellations in and on recent studies
[DDG12] of Gibbs measures for geometry-dependent interactions. The main tool is
a uniform bound on the number of connected components in the Delaunay graph
which provides a novel approach to Delaunay Widom Rowlinson models based on
purely geometric arguments. The interaction potential ensures that shorter
Delaunay edges are more likely to be open and thus offsets the possibility of
having an unbounded number of connected components.Comment: 36 pages, 11 figure
Phase transitions in Delaunay Potts models
We establish phase transitions for classes of continuum Delaunay multi-type
particle systems (continuum Potts models) with infinite range repulsive
interaction between particles of different type. In one class of the Delaunay
Potts models studied the repulsive interaction is a triangle (multi-body)
interaction whereas in the second class the interaction is between pairs
(edges) of the Delaunay graph. The result for the edge model is an extension of
finite range results in \cite{BBD04} for the Delaunay graph and in \cite{GH96}
for continuum Potts models to an infinite range repulsion decaying with the
edge length. This is a proof of an old conjecture of Lebowitz and Lieb. The
repulsive triangle interactions have infinite range as well and depend on the
underlying geometry and thus are a first step towards studying phase
transitions for geometry-dependent multi-body systems. Our approach involves a
Delaunay random-cluster representation analogous to the Fortuin-Kasteleyn
representation of the Potts model. The phase transitions manifest themselves in
the percolation of the corresponding random-cluster model. Our proofs rely on
recent studies \cite{DDG12} of Gibbs measures for geometry-dependent
interactions
Percolation on sparse random graphs with given degree sequence
We study the two most common types of percolation process on a sparse random
graph with a given degree sequence. Namely, we examine first a bond percolation
process where the edges of the graph are retained with probability p and
afterwards we focus on site percolation where the vertices are retained with
probability p. We establish critical values for p above which a giant component
emerges in both cases. Moreover, we show that in fact these coincide. As a
special case, our results apply to power law random graphs. We obtain rigorous
proofs for formulas derived by several physicists for such graphs.Comment: 20 page
Equality of Lifshitz and van Hove exponents on amenable Cayley graphs
We study the low energy asymptotics of periodic and random Laplace operators
on Cayley graphs of amenable, finitely generated groups. For the periodic
operator the asymptotics is characterised by the van Hove exponent or zeroth
Novikov-Shubin invariant. The random model we consider is given in terms of an
adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph.
The asymptotic behaviour of the spectral distribution is exponential,
characterised by the Lifshitz exponent. We show that for the adjacency
Laplacian the two invariants/exponents coincide. The result holds also for more
general symmetric transition operators. For combinatorial Laplacians one has a
different universal behaviour of the low energy asymptotics of the spectral
distribution function, which can be actually established on quasi-transitive
graphs without an amenability assumption. The latter result holds also for long
range bond percolation models
Euler Number and Percolation Threshold on a Square Lattice with Diagonal Connection Probability and Revisiting the Island-Mainland Transition
We report some novel properties of a square lattice filled with white sites,
randomly occupied by black sites (with probability ). We consider
connections up to second nearest neighbours, according to the following rule.
Edge-sharing sites, i.e. nearest neighbours of similar type are always
considered to belong to the same cluster. A pair of black corner-sharing sites,
i.e. second nearest neighbours may form a 'cross-connection' with a pair of
white corner-sharing sites. In this case assigning connected status to both
pairs simultaneously, makes the system quasi-three dimensional, with
intertwined black and white clusters. The two-dimensional character of the
system is preserved by considering the black diagonal pair to be connected with
a probability , in which case the crossing white pair of sites are deemed
disjoint. If the black pair is disjoint, the white pair is considered
connected. In this scenario we investigate (i) the variation of the Euler
number versus graph for varying , (ii)
variation of the site percolation threshold with and (iii) size
distribution of the black clusters for varying , when . Here is
the number of black clusters and is the number of white clusters, at a
certain probability . We also discuss the earlier proposed 'Island-Mainland'
transition (Khatun, T., Dutta, T. & Tarafdar, S. Eur. Phys. J. B (2017) 90:
213) and show mathematically that the proposed transition is not, in fact, a
critical phase transition and does not survive finite size scaling. It is also
explained mathematically why clusters of size 1 are always the most numerous
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