is captured by the local limit of local limit is red. General framework Decay of correlation = This local view of . Develop a machinery to calculate the probability that the root of the Let G n be a graph sequence satisfying | G n | → ∞ . We wish to calculate the asymptotics of ι ( G n ) . We approximate E [ ι ( G n )] = P ( ρ n ∈ I ( G n )) for ρ n chosen uniformly. We hope that this is determined by a small neighbourhood of ρ n . ⇒ ι ( G n ) ∼ E [ ι ( G n )] a.a.s.
local limit is red. General framework Decay of correlation = Develop a machinery to calculate the probability that the root of the Let G n be a graph sequence satisfying | G n | → ∞ . We wish to calculate the asymptotics of ι ( G n ) . We approximate E [ ι ( G n )] = P ( ρ n ∈ I ( G n )) for ρ n chosen uniformly. We hope that this is determined by a small neighbourhood of ρ n . ⇒ ι ( G n ) ∼ E [ ι ( G n )] a.a.s. This local view of ρ n is captured by the local limit of G n .
General framework Decay of correlation = Develop a machinery to calculate the probability that the root of the local limit is red. Let G n be a graph sequence satisfying | G n | → ∞ . We wish to calculate the asymptotics of ι ( G n ) . We approximate E [ ι ( G n )] = P ( ρ n ∈ I ( G n )) for ρ n chosen uniformly. We hope that this is determined by a small neighbourhood of ρ n . ⇒ ι ( G n ) ∼ E [ ι ( G n )] a.a.s. This local view of ρ n is captured by the local limit of G n .
Local limits (a.k.a. Benjamini–Schramm Limits) We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n .
Local limits (a.k.a. Benjamini–Schramm Limits) We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n .
Local limits (a.k.a. Benjamini–Schramm Limits) We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n .
Local limits (a.k.a. Benjamini–Schramm Limits) We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n .
Local limits (a.k.a. Benjamini–Schramm Limits) We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n .
Local limits (a.k.a. Benjamini–Schramm Limits) We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n . · · ·
Local limits (a.k.a. Benjamini–Schramm Limits) We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n . · · · � ➪ ➪
Local limits (a.k.a. Benjamini–Schramm Limits) We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n . · · · � ➪ ➪ � ➪ ➪
Local limits (a.k.a. Benjamini–Schramm Limits) loc Examples loc loc We say that a (random) graph sequence G n converges locally to a (random) rooted graph ( U, ρ ) if for every r ≥ 0 the ball B G n ( ρ n , r ) converges in distribution to B U ( ρ, r ) , where ρ n is a uniform vertex of G n . P n , C n − → Z [ n ] d loc → Z d − G ( n, d / n ) loc − → T d , a Galton–Watson Pois ( d ) tree − → the d -regular tree G n,d → ˆ Uniform random tree T n − T 1 , a size-biased GW Pois (1) tree − → the canopy tree Finite d -ary balanced tree loc
Convergence of the greedy independence ratio from a typical vertex is subfactorial in their length. (bounded degree subfactorial path growth) Theorem (Krivelevich, Mészáros, M., Shikhelman ’20) Suppose has subfactorial path growth. If loc then a.a.s. is red Say that G n has subfactorial path growth if the expected number of paths
Convergence of the greedy independence ratio from a typical vertex is subfactorial in their length. Theorem (Krivelevich, Mészáros, M., Shikhelman ’20) Suppose has subfactorial path growth. If loc then a.a.s. is red Say that G n has subfactorial path growth if the expected number of paths (bounded degree ⊊ subfactorial path growth)
Convergence of the greedy independence ratio from a typical vertex is subfactorial in their length. Theorem (Krivelevich, Mészáros, M., Shikhelman ’20) loc is red Say that G n has subfactorial path growth if the expected number of paths (bounded degree ⊊ subfactorial path growth) Suppose G n has subfactorial path growth. If G n − → ( U, ρ ) then ι ( G n ) → ι ( U, ρ ) a.a.s.
Convergence of the greedy independence ratio from a typical vertex is subfactorial in their length. Theorem (Krivelevich, Mészáros, M., Shikhelman ’20) loc Say that G n has subfactorial path growth if the expected number of paths (bounded degree ⊊ subfactorial path growth) Suppose G n has subfactorial path growth. If G n − → ( U, ρ ) then ι ( G n ) → ι ( U, ρ ) a.a.s. P ( ρ is red )
Exploration algorithms / decay of correlation
Exploration algorithms / decay of correlation
Exploration algorithms / decay of correlation
Exploration algorithms / decay of correlation
Exploration algorithms / decay of correlation
Exploration algorithms / decay of correlation
Exploration algorithms / decay of correlation
Exploration algorithms / decay of correlation
Exploration algorithms / decay of correlation
Exploration algorithms / decay of correlation
Exploration algorithms / decay of correlation
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