Local and global convergence

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Dedicated to the Memory of Oded Schramm
Local and global convergence in bounded degree
graphs
László Lovász Eötvös Loránd University, Budapest
Joint work with Christian Borgs, Jennifer Chayes
and Jeff Kahn
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The Benjamini-Schramm limit
G simple graph with all degrees D
BG(v,r) nodes at distance r from node v
Ar simple rooted graphs with all degrees D
and radius r
v random uniform node ? BG(v,r) random graph in
Ar
PG(A) P(BG(v,r)A)
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The Benjamini-Schramm limit
A1

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The Benjamini-Schramm limit
? maximal paths from rooted
countable graphs with degrees D
?A maximal paths through A
A ?-algebra generated by the ?A
P probability measure on (?,A)
P has some special properties
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Other limit constructions
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Other limit constructions
?
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Other limit constructions
Measure preserving graph G(0,1,E) (a) all
degrees D (b) X?0,1 Borel ? N(X) is Borel
R.Kleinberg L
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Other limit constructions
Graphing G(0,1,E)
Elek
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Homomorphism functions
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Homomorphism functions
Examples
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Homomorphism functions
We know
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Homomorphism functions
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Left and right convergence
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Left and right convergence
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Examples
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Examples
Feketes Lemma ? convergence
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Examples
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Examples
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Examples
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Examples
Construct auxiliary graph G
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Examples
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Left and right convergence
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Analogy the dense case
Left-convergence (homomorphisms from small
graphs)
Right-convergence (homomorphisms into small
graphs)
Distance of two graphs (optimal overlay
convergent?Cauchy)
Limit objects (2-variable functions)
Approximation by bounded-size graphs (Szemerédi
Lemma, sampling)
Parameters continuous at infinity (parameter
testing)
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Limit objects
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Limit objects
LS
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Amenable (hyperfinite) limits
Small cut decomposition
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Amenable (hyperfinite) limits
G1,G2, amenable (hyperfinite)
Can be decomposed into bounded pieces by small
cut decomposition.
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Amenable graphs and hyperfinite limits
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Regularity Lemma?
?-homogeneous ? small cut decomposition, each
piece H satisfies
Every sufficiently large graph of bounded degree
can be decomposed into quasi-homogeneous
pieces by small cuts.
Elek Lippner Angel - Szegedy
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Regularity Lemma?
Easy observation
Alon
A construction for H? Effective bound on q?