Convergent sequences of sparse graphs (status report) László Lovász Eötvös University, Budapest

For dense graphs: Left-convergence (homomorphisms from “small” graphs) Right-convergence (homomorphisms into “small” graphs) Distance of two graphs (optimal overlay; convergent  Cauchy) Limit of a convergent sequence (2-variable functions, reflection positive graph parameters, ergodic measures on countable graphs) Approximation by bounded-size graphs (Szemerédi Lemma, sampling) For sparse graphs? Parameters “continuous at infinity” (parameter testing, spectrum)

Left-convergence Right-convergence Distance of two graphs Limit of a convergent sequence Approximation by bounded-size graphs Parameters “continuous at infinity”

Weighted version:

G n : sequence of graphs with degrees  D G n is left-convergent if converges  connected F

Equivalent definition: G  ( ) G n is left-convergent if converges for all r All possible neighborhoods with radius r

Left-convergence Right-convergence Distance of two graphs Limit of a convergent sequence Approximation by bounded-size graphs Parameters “continuous at infinity”

G n is right-convergent if is convergent  q  1  H in a neighborhood of J q J q : complete graph K q with loops

Left-convergent   q  2 D is convergent. Right-convergent  left-convergent Borgs-Chayes-Kahn-L Number of q -colorings

G n : n  n discrete torus is convergent if H is connected nonbipartite. Long-range interaction between colors

Key to the proof: Mayer expansion where infinite sum!

: Dobrushin Lemma The expansions are convergent if H-J q is small enough

Mayer expansion: where

: Erdős-L-Spencer Let F 1,…F N be all connected graphs on at most q nodes. Then the matrix is nonsingular. Sample Lemma:

Left-convergence Right-convergence Distance of two graphs Limit of a convergent sequence Approximation by bounded-size graphs Parameters “continuous at infinity”

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Left-convergence Right-convergence Distance of two graphs Limit of a convergent sequence Approximation by bounded-size graphs Parameters “continuous at infinity”

The limit object

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Benjamini – Schramm: probability distribution on rooted countable graphs with degrees  D with “unimodularity” condition

All possible neighborhoods with radius 0: radius 1: radius 2: radius 3: x0x0 x 11 x 12 x 13 x 14 x 121 x 122 x 123 x 124 x 1241 x 1242 x 1243 x further equations frequency of this neighborhood

The limit object Benjamini – Schramm: probability distribution on rooted countable graphs with degrees  D with “unimodularity” condition R.Kleinberg – L: bounded degree graph on [0,1] with “measure-preserving” condition

The limit object complete binary trees

The limit object Benjamini – Schramm: probability distribution on rooted countable graphs with degrees  D with “unimodularity” condition R.Kleinberg – L: bounded degree graph on [0,1] with “measure-preserving” condition Elek: “graphing”: measure-preserving involution

Open problem: are all these limit objects?

expander same subgraph densities This notion of limit (or convergence) is not enough... Does not see global structure

This notion of limit (or convergence) is not enough... Girth of G n tends to   G n tends to union of trees Bollobás et al. “ W -random” with probabilities W(x,y)/n tends to union of trees Does not distinguish graphs with large girth

This notion of limit (or convergence) is not enough... have the same limit Does not see the geometry or topology of the graphs

Left-convergence Right-convergence Distance of two graphs Limit of a convergent sequence Approximation by bounded-size graphs Parameters “continuous at infinity”

Easy observation: For every r,D  1 and  0 there is a q(r, ,D) such that for every graph G with degrees  D there is a graph H with degrees  D and with  q nodes such that for all for all connected graphs F with  r nodes Thanks to Noga, Nati,... A construction for H ? Effective bound on q ?

Left-convergence Right-convergence Distance of two graphs Limit of a convergent sequence Approximation by bounded-size graphs Parameters “continuous at infinity”

Other parameters “continuous at infinity” T(G): number of spanning trees If (G n ) is a convergent sequence of connected graphs, then is convergent. Lyons

Perfect matching If (G n ) is a convergent sequence of bipartite graphs with perfect matchings  Limit graphing has a measurable perfect matching G. Kun If (G n ) is a convergent sequence of bipartite graphs with maximum matching < (1-  )|V(G n )| / 2  Limit graphing has no measurable perfect matching What about non-measurable?