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Processes on Unimodular Random Networks
David Aldous,Russell Lyons +1 more
TL;DR: In this article, the authors investigate unimodular random networks and their properties via reversibility of an associated random walk and their similarities to unimmodular quasi-transitive graphs, and extend various theorems concerning random walks, percolation, spanning forests, and amenability.
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Abstract: 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.
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Random Graphs and Complex Networks
Remco van der Hofstad
- 01 Jan 2017
TL;DR: This chapter explains why many real-world networks are small worlds and have large fluctuations in their degrees, and why Probability theory offers a highly effective way to deal with the complexity of networks, and leads us to consider random graphs.
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Probability on Trees and Networks
Russell Lyons,Yuval Peres +1 more
- 20 Jan 2017
TL;DR: In this article, the authors present a state-of-the-art account of probability on networks, including percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks.
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Ising models on locally tree-like graphs
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TL;DR: In this article, it was shown that the Boltzmannian prediction for the limiting free energy per spin is correct for any positive temperature and external field, and local marginals can be approximated by iterating a set of mean field (cavity) equations.
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Proof of the Satisfiability Conjecture for Large k
Jian Ding,Allan Sly,Nike Sun +2 more
- 14 Jun 2015
TL;DR: The satisfiability threshold αs is established, given explicitly by the one-step replica symmetry breaking (1SRB) prediction from statistical physics, and it is believed that the methods may apply to a range of random constraint satisfaction problems in the 1RSB class.
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Around the circular law
Charles Bordenave,Djalil Chafaï +1 more
TL;DR: In this paper, it was shown that the empirical spectral distribution of a nxn random matrix with iid entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension $n$ tends to infinity, which is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices.
201
References
An Introduction To Probability Theory And Its Applications
Feller William
- 01 Jan 1950
TL;DR: A First Course in Probability (8th ed.) by S. Ross is a lively text that covers the basic ideas of probability theory including those needed in statistics.
10.2K
Self-organized criticality
TL;DR: In this article, the authors show that certain extended dissipative dynamical systems naturally evolve into a critical state, with no characteristic time or length scales, and the temporal fingerprint of the self-organized critical state is the presence of flicker noise or 1/f noise; its spatial signature is the emergence of scale-invariant (fractal) structure.
4.7K
•Book
Fundamentals of the Theory of Operator Algebras
Richard V. Kadison,John R. Ringrose +1 more
- 01 Jan 1983
TL;DR: In this article, the authors compare normal states and unitary equivalence of von Neumann algebras, including the trace and the trace trace of the trace of a projection.
3.1K
A critical point for random graphs with a given degree sequence
Michael Molloy,Bruce Reed +1 more
TL;DR: It is shown that if Σ i(i - 2)λi > 0, then such graphs almost surely have a giant component, while if λ0, λ1… which sum to 1, then almost surely all components in such graphs are small.
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