scholarly journals Random subgraphs of finite graphs. II. The lace expansion and the triangle condition

2005 ◽  
Vol 33 (5) ◽  
pp. 1886-1944 ◽  
Author(s):  
Christian Borgs ◽  
Jennifer T. Chayes ◽  
Remco van der Hofstad ◽  
Gordon Slade ◽  
Joel Spencer
Keyword(s):  
Lace Expansion ◽  
Finite Graphs ◽  
Triangle Condition ◽  
Random Subgraphs

scholarly journals Random subgraphs of finite graphs: I. The scaling window under the triangle condition

2005 ◽  
Vol 27 (2) ◽  
pp. 137-184 ◽  
Author(s):  
Christian Borgs ◽  
Jennifer T. Chayes ◽  
Remco van der Hofstad ◽  
Gordon Slade ◽  
Joel Spencer
Keyword(s):  
Finite Graphs ◽  
Triangle Condition ◽  
Scaling Window ◽  
Random Subgraphs

scholarly journals Critical Site Percolation in High Dimension

2020 ◽  
Vol 181 (3) ◽  
pp. 816-853
Author(s):  
Markus Heydenreich ◽  
Kilian Matzke
Keyword(s):  
High Dimension ◽  
Critical Exponents ◽  
Mean Field ◽  
High Dimensions ◽  
Lace Expansion ◽  
Site Percolation ◽  
Triangle Condition ◽  
Hypercubic Lattice ◽  
Infra Red ◽  
Critical Site

Abstract We use the lace expansion to prove an infra-red bound for site percolation on the hypercubic lattice in high dimension. This implies the triangle condition and allows us to derive several critical exponents that characterize mean-field behavior in high dimensions.

scholarly journals Random Subgraphs Of Finite Graphs: III. The Phase Transition For The n-Cube

COMBINATORICA ◽  
2006 ◽  
Vol 26 (4) ◽  
pp. 395-410 ◽  
Author(s):  
Christian Borgs ◽  
Jennifer T. Chayes ◽  
Remco van der Hofstad ◽  
Gordon Slade ◽  
Joel Spencer
Keyword(s):  
Phase Transition ◽  
Finite Graphs ◽  
Random Subgraphs

scholarly journals Mass-conserving diffusion-based dynamics on graphs

2021 ◽  
pp. 1-49
Author(s):  
J.M BUDD ◽  
Y. VAN GENNIP
Keyword(s):  
Multiscale Modeling ◽  
Classification Problems ◽  
Theoretical Justification ◽  
Finite Graphs ◽  
Discrete Scheme ◽  
Cahn Equation ◽  
Separating Flows ◽  
Special Case ◽  
Appl Math ◽  
Convergence Of The Scheme

An emerging technique in image segmentation, semi-supervised learning and general classification problems concerns the use of phase-separating flows defined on finite graphs. This technique was pioneered in Bertozzi and Flenner (2012, Multiscale Modeling and Simulation10(3), 1090–1118), which used the Allen–Cahn flow on a graph, and was then extended in Merkurjev et al. (2013, SIAM J. Imaging Sci.6(4), 1903–1930) using instead the Merriman–Bence–Osher (MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip (2020, SIAM J. Math. Anal.52(5), 4101–4139), we gave a theoretical justification for this use of the MBO scheme in place of Allen–Cahn flow, showing that the MBO scheme is a special case of a ‘semi-discrete’ numerical scheme for Allen–Cahn flow. In this paper, we extend this earlier work, showing that this link via the semi-discrete scheme is robust to passing to the mass-conserving case. Inspired by Rubinstein and Sternberg (1992, IMA J. Appl. Math.48, 249–264), we define a mass-conserving Allen–Cahn equation on a graph. Then, with the help of the tools of convex optimisation, we show that our earlier machinery can be applied to derive the mass-conserving MBO scheme on a graph as a special case of a semi-discrete scheme for mass-conserving Allen–Cahn. We give a theoretical analysis of this flow and scheme, proving various desired properties like existence and uniqueness of the flow and convergence of the scheme, and also show that the semi-discrete scheme yields a choice function for solutions to the mass-conserving MBO scheme.

Self-avoiding walk enumeration via the lace expansion

2007 ◽  
Vol 40 (36) ◽  
pp. 10973-11017 ◽  
Author(s):  
Nathan Clisby ◽  
Richard Liang ◽  
Gordon Slade
Keyword(s):  
Lace Expansion ◽  
Self Avoiding Walk

scholarly journals Liftings in Finite Graphs and Linkages in Infinite Graphs with Prescribed Edge-Connectivity

2016 ◽  
Vol 32 (6) ◽  
pp. 2575-2589
Author(s):  
Seongmin Ok ◽  
R. Bruce Richter ◽  
Carsten Thomassen
Keyword(s):  
Infinite Graphs ◽  
Edge Connectivity ◽  
Finite Graphs

scholarly journals The edmonds—Gallai decomposition for matchings in locally finite graphs

COMBINATORICA ◽  
1982 ◽  
Vol 2 (3) ◽  
pp. 229-235 ◽  
Author(s):  
François Bry ◽  
Michel Las Vergnas
Keyword(s):  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs

scholarly journals Approximating Cayley Diagrams Versus Cayley Graphs

2012 ◽  
Vol 21 (4) ◽  
pp. 635-641
Author(s):  
ÁDÁM TIMÁR
Keyword(s):  
Cayley Graph ◽  
Hamiltonian Cycle ◽  
Cayley Graphs ◽  
The Other ◽  
Finite Graphs ◽  
Similar Construction ◽  
Graph Property

We construct a sequence of finite graphs that weakly converge to a Cayley graph, but there is no labelling of the edges that would converge to the corresponding Cayley diagram. A similar construction is used to give graph sequences that converge to the same limit, and such that a Hamiltonian cycle in one of them has a limit that is not approximable by any subgraph of the other. We give an example where this holds, but convergence is meant in a stronger sense. This is related to whether having a Hamiltonian cycle is a testable graph property.

scholarly journals L p -distortion and p -spectral gap of finite graphs

2014 ◽  
Vol 46 (2) ◽  
pp. 329-341 ◽  
Author(s):  
Pierre-Nicolas Jolissaint ◽  
Alain Valette
Keyword(s):  
Spectral Gap ◽  
Finite Graphs
Sign in / Sign up

Export Citation Format

Share Document