Graph homomorphisms, statistical physics, and quasirandom graphs László Lovász Microsoft Research lovasz@microsoft.com Joint work with: Christian Borgs, Jennifer Chayes, Mike Freedman, Jeff Kahn, Lex Schrijver, Vera T. Sós, Balázs Szegedy, Kati Vesztergombi

Homomomorphism: adjacency-preserving map coloring independent set triangles

Weighted version:

G connected 

L1966

probability that random map Homomorphism density: probability that random map is a homomorphism every node in G weighted by 1/|V(G)| Homomorphism entropy:

Examples: if G has no loops

3 3 -1 1/4 1/4 -1 -1 -1 -1 1/4 1/4 -1 3 3 H 1 2 H

Hom functions and statistical physics atoms are in states (e.g. up or down): interaction only between neighboring atoms: graph G energy of interaction: energy of state: partition function:

H=Kq, all weights are positive  soft-core model sparse G bounded degree partition function: All weights in H are 1  hard-core model H=Kq, all weights are positive  soft-core model dense G

Recall: : set of connected graphs Erdős – Lovász – Spencer

Kruskal-Katona 1 Goodman 1/2 2/3 3/4 1 Bollobás Lovász-Simonovits

small probe (subgraph) small template (model) large graph

Turán’s Theorem for triangles: Kruskal-Katona Theorem for triangles: Erdős’s Theorem on quadrilaterals:

k-labeled graph: k nodes labeled 1,...,k Connection matrices k-labeled graph: k nodes labeled 1,...,k Connection matrix (for target graph G):

... k=2: ...

Main Lemma: is positive semidefinite reflection positivity has rank

Proof of Kruskal-Katona

How much does the positive semidefinite property capture? ...almost everything!

Connection matrix of a parameter: graph parameter is positive semidefinite has rank reflection positivity equality holds in “generic” case (H has no automorphism)

k-labeled quantum graph: finite sum is a commutative algebra with unit element ... Inner product: positive semidefinite suppose =

Distance of graphs: Converse???

(Gn: n=1,2,...) is quasirandom, if d(Gn, G(n,p))  0 a.s. Quasirandom graphs Thomason Chung – Graham – Wilson (Gn: n=1,2,...) is quasirandom, if d(Gn, G(n,p))  0 a.s. Example: Paley graphs p: prime 1 mod 4 How to see that these graphs are quasirandom?

For a sequence (Gn: n=1,2,...), the following are equivalent: (Gn) is quasirandom;  simple graph F; for F=K2 and C4. Chung – Graham – Wilson Converse if G’ is a random graph.

Suppose that Want: k=1: ... ... 1 p p2 pk pk+1

k=2: ... ... 1 p2 p4 p2k p2k+2

k=deg(v) 1 pk p2k p|E(G’)| p|E(G)| ... ... ... ... ... ... ... ... ...

Generalized (quasi)random graphs 0.1 0.1n 0.2n 0.3n 0.4n density 0.2 0.5 0.7 0.2 0.3 0.2 0.4 0.5 0.35 0.3 density 0.35 For a sequence (Gn: n=1,2,...), the following are equivalent: d(Gn, G(n,H))  0;  simple graph F;

(Gn) left-convergent: Recall: (Gn) left-convergent: (Gn) right-convergent:

(C2n) is right-convergent Example: (C2n) is right-convergent But... (Cn) is not convergent for bipartite H

Any connection between left and right convergence?

(Gn) left-convergent: Graphs with bounded degree D (Gn) left-convergent: e.g. H=Kq, q>8D