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Abstract A coupling of random walkers on the same finite graph, who take turns sequentially, is said to be anavoidance couplingif the walkers never collide. Previous studies of these processes have focused almost exclusively on complete graphs, in particular how many walkers an avoidance coupling can include. For other graphs, apart from special cases, it has been unsettled whether even two noncolliding simple random walkers can be coupled. In this article, we construct such a coupling on (i) anyd‐regular graph avoiding a fixed subgraph depending ond; and (ii) any square‐free graph with minimum degree at least three. A corollary of the first result is that a uniformly random regular graph onnvertices admits an avoidance coupling with high probability.
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Spectrum of random d ‐regular graphs up to the edge
Abstract Consider the normalized adjacency matrices of randomd‐regular graphs onNvertices with fixed degree . We prove that, with probability for any , the following two properties hold as provided that : (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten–McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound inN, that is, . (ii) All eigenvectors of randomd‐regular graphs are completely delocalized.
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- Award ID(s):
- 2153335
- PAR ID:
- 10486371
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Communications on Pure and Applied Mathematics
- Volume:
- 77
- Issue:
- 3
- ISSN:
- 0010-3640
- Format(s):
- Medium: X Size: p. 1635-1723
- Size(s):
- p. 1635-1723
- Sponsoring Org:
- National Science Foundation
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