We determine the critical probability for Bernoulli bond percolation on essentially any finite graph. Namely, letting $\lambda(G)$ denote the spectral radius (maximum eigenvalue) of $G$, we prove that the critical probability is at $1/\lambda(G)$: above this probability there is typically a component of order $\Omega(\lambda(G))$, whereas below it all components are of order at most $O(\sqrt{|G|})$. These results in particular confirm a conjecture of Krivelevich and Samotij about percolation on graphs of a given average degree, and vastly extend theorems of Bollob\'as, Borgs, Chayes, and Riordan, who proved analogous results but only for dense graphs. Our theorems are optimal in many regimes, and also demonstrate that percolation has an unexpectedly subtle behaviour on graphs whose spectral radius is roughly the square root of their maximum degree.
Many important results in extremal graph theory can be roughly summarized as “if a triangle‐free graph G$$ G $$ has certain properties, then it has a homomorphism to a triangle‐free graph Γ$$ \Gamma $$ of bounded size.” For example, bounds on homomorphism thresholds give such a statement if G$$ G $$ has sufficiently high minimum degree, and the approximate homomorphism theorem gives such a statement for all G$$ G $$ if one weakens the notion of homomorphism appropriately. In this paper, we study asymmetric versions of these results, where the assumptions on G$$ G $$ and Γ$$ \Gamma $$ need not match. For example, we prove that if G$$ G $$ is a graph with odd girth at least 9 and minimum degree at least δ|G|$$ \delta \mid G\mid $$ , then G$$ G $$ is homomorphic to a triangle‐free graph whose size depends only on δ$$ \delta $$ . Moreover, the odd girth assumption can be weakened to odd girth at least 7 if G$$ G $$ has bounded VC dimension or bounded domination number. This gives a new and improved proof of a result of Huang, Liu, Rong, and Xu. We also prove that in the asymmetric approximate homomorphism theorem, the bounds exhibit a rather surprising “double phase transition”: the bounds are super‐exponential if G$$ G $$ is only assumed to be triangle‐free, they become exponential if G$$ G $$ is assumed to have odd girth 7 or 9, and become linear if G$$ G $$ has odd girth at least 11. Our proofs use a wide variety of techniques, including entropy arguments, the Frieze–Kannan weak regularity lemma, properties of the generalized Mycielskian construction, and recent work on abundance and the asymmetric removal lemma.
Lior Gishboliner, Eoin Hurley, Yuval Wigderson· Random Structures & Algo...· 0 citations
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