Size biased couplings and the spectral gap for random regular graphs
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[1] Stanford University
Stanford University
Estados Unidos
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[2] University of Southtern California (USA)
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 46, Nº. 1, 2018, págs. 72-125
- Idioma: inglés
- Enlaces
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Resumen
Let λλ be the second largest eigenvalue in absolute value of a uniform random dd-regular graph on nn vertices. It was famously conjectured by Alon and proved by Friedman that if dd is fixed independent of nn, then λ=2d−1−−−−√+o(1)λ=2d−1+o(1) with high probability. In the present work, we show that λ=O(d−−√)λ=O(d) continues to hold with high probability as long as d=O(n2/3)d=O(n2/3), making progress toward a conjecture of Vu that the bound holds for all 1≤d≤n/21≤d≤n/2. Prior to this work the best result was obtained by Broder, Frieze, Suen and Upfal (1999) using the configuration model, which hits a barrier at d=o(n1/2)d=o(n1/2). We are able to go beyond this barrier by proving concentration of measure results directly for the uniform distribution on dd-regular graphs. These come as consequences of advances we make in the theory of concentration by size biased couplings. Specifically, we obtain Bennett-type tail estimates for random variables admitting certain unbounded size biased couplings.
