1. Local and global convergence in bounded degree graphs László Lovász Eötvös Loránd University, Budapest Joint work with Christian Borgs, Jennifer Chayes and Jeff Kahn December 20091 Dedicated to the Memory of Oded Schramm

2. December 20092 The Benjamini-Schramm limit G: simple graph with all degrees ≤ D B G (v,r)= {nodes at distance ≤ r from node v} v random uniform node  B G (v,r) random graph in A r P G (A)= P(B G (v,r)≈A) A r = {simple rooted graphs with all degrees ≤ D and radius ≤r } (G 1,G 2,…) convergent: is convergent for all A

3. December 20093 The Benjamini-Schramm limit A1A1 A2A2 A3A3 …

4. December 20094 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…

5. December 20095 Other limit constructions

6. December 20096 Other limit constructions ?

7. December 20097 Other limit constructions Measure preserving graph: G=([0,1],E) (a) all degrees ≤D (b) X  [0,1] Borel  N(X) is Borel (c) X,Y  [0,1] Borel  R.Kleinberg – L

8. December 20098 Other limit constructions Graphing: G=([0,1],E) Elek

9. December 20099 Homomorphism functions Weighted version: Probability that random map V(G)  V(H) is a hom

10. December 200910 Homomorphism functions Examples: hom(G, ) = # of independent sets in G

11. December 200911 Homomorphism functions We know  we know

12. December 200912 Homomorphism functions

13. December 200913 Left and right convergence very large graph counting edges, triangles,... spectra,... counting colorations, stable sets,... statistical physics,... maximum cut,...

14. December 200914 Left and right convergence ?

15. December 200915 Examples

16. December 200916 Examples Fekete’s Lemma  convergence

17. December 200917 Examples

18. December 200918 Examples

19. December 200919 Examples

20. December 200920 Examples Construct auxiliary graph G : H connected nonbipartite  G connected nonbipartite 

21. December 200921 Examples

22. December 200922 Left and right convergence

23. December 200923 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)

24. December 2009 Limit objects 24 Borgs, Chayes,L,Sós,Vesztergombi

25. For every convergent graph sequence (G n ) there is a graphon such that December 200925 Limit objects LS Conversely, for every graphon W there is a graph sequence (G n ) such that LS W is essentially unique (up to measure-preserving transformation). BCL

26. December 200926 Amenable (hyperfinite) limits o(n) edges  (n) nodes Small cut decomposition:

27. December 200927 Amenable (hyperfinite) limits {G 1,G 2,…} amenable (hyperfinite): Can be decomposed into bounded pieces by small cut decomposition.

28. December 200928 Amenable graphs and hyperfinite limits For a convergent graph sequence, hyperfiniteness is reflected by the limit. Schramm Every minor-closed property is testable for graphs with bounded degree. Benjamini-Schramm-Shapira

29. December 200929 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

30. December 200930 Regularity Lemma? 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 Alon A construction for H ? Effective bound on q ?