Academic Journal
Components, large and small, are as they should be I: Supercritical percolation on regular graphs of growing degree.
| Τίτλος: | Components, large and small, are as they should be I: Supercritical percolation on regular graphs of growing degree. |
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| Συγγραφείς: | Diskin, Sahar1 (AUTHOR), Krivelevich, Michael1 (AUTHOR) |
| Πηγή: | Transactions of the American Mathematical Society. Oct2026, Vol. 379 Issue 10, p7251-7270. 20p. |
| Θεματικοί όροι: | *Regular graphs, *Percolation theory, *Mathematical expansion, *Graph connectivity, *Branching processes |
| Περίληψη: | We provide sufficient conditions for a regular graph G of growing degree d, guaranteeing a phase transition in its random subgraph G_p similar to that of G(n,p) when p\cdot d\approx 1. These conditions capture several well-studied graphs, such as (percolation on) the complete graph K_n, the binary hypercube Q^d, d-regular expanders, and random d-regular graphs. In particular, this serves as a unified proof for these (and other) cases. Suppose that G is a d-regular graph on n vertices, with d=\omega (1). Let \epsilon >0 be a small constant, and let p=\frac {1+\epsilon }{d}. Let y(\epsilon) be the survival probability of a Galton-Watson tree with offspring distribution Po(1+\epsilon). We show that if G satisfies a (very) mild edge expansion requirement, and if one has fairly good control on the expansion of small sets in G, then typically the percolated random subgraph G_p contains a unique giant component of asymptotic order y(\epsilon)n, and all the other components in G_p are of order O(\log n/\epsilon ^2). We also show that this result is tight, in the sense that if one asks for a slightly weaker control on the expansion of small sets in G, then there are d-regular graphs G on n vertices, where typically the second largest component is of order \Omega (d\log (n/d))=\omega (\log n). This is the first of a two-part sequence of papers. In the subsequent work, we consider supercritical percolation on regular graphs of constant degree, and establish similar sufficient (and essentially tight) conditions in that setting. [ABSTRACT FROM AUTHOR] |
| Βάση Δεδομένων: | Academic Search Index |
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