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Components, large and small, are as they should be I: Supercritical percolation on regular graphs of growing degree.

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Τίτλος: Components, large and small, are as they should be I: Supercritical percolation on regular graphs of growing degree.
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]
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  Label: Title
  Group: Ti
  Data: Components, large and small, are as they should be I: Supercritical percolation on regular graphs of growing degree.
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  Data: <searchLink fieldCode="AR" term="%22Diskin%2C+Sahar%22">Diskin, Sahar</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Krivelevich%2C+Michael%22">Krivelevich, Michael</searchLink><relatesTo>1</relatesTo> (AUTHOR)
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  Data: <searchLink fieldCode="JN" term="%22Transactions+of+the+American+Mathematical+Society%22">Transactions of the American Mathematical Society</searchLink>. Oct2026, Vol. 379 Issue 10, p7251-7270. 20p.
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  Data: *<searchLink fieldCode="DE" term="%22Regular+graphs%22">Regular graphs</searchLink><br />*<searchLink fieldCode="DE" term="%22Percolation+theory%22">Percolation theory</searchLink><br />*<searchLink fieldCode="DE" term="%22Mathematical+expansion%22">Mathematical expansion</searchLink><br />*<searchLink fieldCode="DE" term="%22Graph+connectivity%22">Graph connectivity</searchLink><br />*<searchLink fieldCode="DE" term="%22Branching+processes%22">Branching processes</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: 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]
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RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1090/tran/9684
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 20
        StartPage: 7251
    Subjects:
      – SubjectFull: Regular graphs
        Type: general
      – SubjectFull: Percolation theory
        Type: general
      – SubjectFull: Mathematical expansion
        Type: general
      – SubjectFull: Graph connectivity
        Type: general
      – SubjectFull: Branching processes
        Type: general
    Titles:
      – TitleFull: Components, large and small, are as they should be I: Supercritical percolation on regular graphs of growing degree.
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            NameFull: Diskin, Sahar
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            NameFull: Krivelevich, Michael
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            – D: 01
              M: 10
              Text: Oct2026
              Type: published
              Y: 2026
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              Value: 379
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            – TitleFull: Transactions of the American Mathematical Society
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