92,605 research outputs found
Processes on Unimodular Random Networks
We investigate unimodular random networks. Our motivations include their
characterization via reversibility of an associated random walk and their
similarities to unimodular quasi-transitive graphs. We extend various theorems
concerning random walks, percolation, spanning forests, and amenability from
the known context of unimodular quasi-transitive graphs to the more general
context of unimodular random networks. We give properties of a trace associated
to unimodular random networks with applications to stochastic comparison of
continuous-time random walk.Comment: 66 pages; 3rd version corrects formula (4.4) -- the published version
is incorrect --, as well as a minor error in the proof of Proposition 4.10;
4th version corrects proof of Proposition 7.1; 5th version corrects proof of
Theorem 5.1; 6th version makes a few more minor correction
Quasi-randomness and algorithmic regularity for graphs with general degree distributions
We deal with two intimately related subjects: quasi-randomness and regular partitions. The purpose of the concept of quasi-randomness is to express how much a given graph āresemblesā a random one. Moreover, a regular partition approximates a given graph by a bounded number of quasi-random graphs. Regarding quasi-randomness, we present a new spectral characterization of low discrepancy, which extends to sparse graphs. Concerning regular partitions, we introduce a concept of regularity that takes into account vertex weights, and show that if satisfies a certain boundedness condition, then admits a regular partition. In addition, building on the work of Alon and Naor [Proceedings of the 36th ACM Symposium on Theory of Computing (STOC), Chicago, IL, ACM, New York, 2004, pp. 72ā80], we provide an algorithm that computes a regular partition of a given (possibly sparse) graph in polynomial time. As an application, we present a polynomial time approximation scheme for MAX CUT on (sparse) graphs without ādense spots.
Quasi-random oriented graphs
We show that a number of conditions on oriented graphs, all of which are
satisfied with high probability by randomly oriented graphs, are equivalent.
These equivalences are similar to those given by Chung, Graham and Wilson in
the case of unoriented graphs, and by Chung and Graham in the case of
tournaments. Indeed, our main theorem extends to the case of a general
underlying graph G the main result of Chung and Graham which corresponds to the
case that G is complete.
One interesting aspect of these results is that exactly two of the four
orientations of a four-cycle can be used for a quasi-randomness condition,
i.e., if the number of appearances they make in D is close to the expected
number in a random orientation of the same underlying graph, then the same is
true for every small oriented graph HComment: 11 page
Central limit theorem for exponentially quasi-local statistics of spin models on Cayley graphs
Central limit theorems for linear statistics of lattice random fields
(including spin models) are usually proven under suitable mixing conditions or
quasi-associativity. Many interesting examples of spin models do not satisfy
mixing conditions, and on the other hand, it does not seem easy to show central
limit theorem for local statistics via quasi-associativity. In this work, we
prove general central limit theorems for local statistics and exponentially
quasi-local statistics of spin models on discrete Cayley graphs with polynomial
growth. Further, we supplement these results by proving similar central limit
theorems for random fields on discrete Cayley graphs and taking values in a
countable space but under the stronger assumptions of {\alpha}-mixing (for
local statistics) and exponential {\alpha}-mixing (for exponentially
quasi-local statistics). All our central limit theorems assume a suitable
variance lower bound like many others in the literature. We illustrate our
general central limit theorem with specific examples of lattice spin models and
statistics arising in computational topology, statistical physics and random
networks. Examples of clustering spin models include quasi-associated spin
models with fast decaying covariances like the off-critical Ising model, level
sets of Gaussian random fields with fast decaying covariances like the massive
Gaussian free field and determinantal point processes with fast decaying
kernels. Examples of local statistics include intrinsic volumes, face counts,
component counts of random cubical complexes while exponentially quasi-local
statistics include nearest neighbour distances in spin models and Betti numbers
of sub-critical random cubical complexes.Comment: Minor changes incorporated based on suggestions by referee
The asymptotics of strongly regular graphs
A strongly regular graph is called trivial if it or its complement is a union
of disjoint cliques. We prove that every infinite family of nontrivial strongly
regular graphs is quasi-random in the sense of Chung, Graham and Wilson
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