DIMACS 20 th Birthday Celebration, 20 November 2009

1. DIMACS 20th Birthday Celebration, 20 November 2009 The Combinatorial Side of Statistical Physics Peter Winkler, Dartmouth

2. Phase I: The Parties Meet

3. We begin with a checkerboard on which checkers are placed uniformly at random subject to the condition that no two are orthogonally adjacent. Combinatorics--- or Statistical Physics?

4. To see better what’s going on, we color the even occupied squares blue and the odd ones red. Notice the tendency to cluster… Discrete hard-core

5. Actually, that was just one corner of this picture (generated by me and Peter Shor using “coupling from the past.”) Now, let’s raise the stakes by rewarding larger independent sets with a factor for each extra occupied site. The big picture

6. This is what it looks like when we set = 3.787. The plot thickens

7. At = 3.792, one of the colors “breaks symmetry” and takes over the picture. We suddenly get “ordered phase,” “long-range correlation” and “slow mixing.” Take-over

8. self-avoiding random walks hard-core model random independent sets monomer-dimer random matchings branched polymers random lattice trees Potts model random colorings linear polymers percolation random subgraphs Statistical physics  Combinatorics

9. Physics techniques (e.g., “cavity method”) have helped to make major progress in understanding satisfiability. [Mezard, Parisi and Virasoro ’85] x (x z v) (x y u) w z (x y z) y (w z t) (y z w) CS theory’s favorite hard-constraint model

10. On the Bethe lattice, where things are nice: [Brightwell & W. ’99] Which graphs cause a phase transition?

11. Phase II: DIMACS Makes a Match

12. In 3 dimensions, the “critical activity” for the discrete hard-core model drops from about 3.8 to about 2.2. What happens when the dimension gets very high? Along came [David Galvin and Jeff Kahn ’04] (with ideas from Sapozhenko) to show the critical activity goes to zero. Even a certain well-known married couple at Microsoft Research couldn’t agree. Combinatorialists settle a controversy

13. Phase III: MSR Leads the Charge

14. The physicists’ scaling methods are quite powerful. E.g., [Borgs, Chayes, Kesten & Spencer ’01] find the scaling window and critical exponent for the Erdos-Renyi giant component. For example: edges of a large empty graph are created independently with probability p. When do you get a giant connected component? Physicists call this game percolation and usually play it on a grid, asking: when is there an infinite connected component? Take your favorite graph G, and let its vertices (or its edges) live or die at random. What happens? Percolation

15. Colour the points of a Poisson process green (with probability p ) or red. Now draw in the Voronoi cells; do the green cells percolate? [Bollobas and Riordan ’07] proved that the critical probability is ½. Read their new book on percolation! Voronoi percolation

16. Coordinate Percolation coordinate percolation: independent percolation: ? ? ? a vertex lives or dies based on an independent event associated with the vertex. a vertex lives or dies based on independent events associated with the vertex’s coordinates. Motivation: water seeping through a porous material. Motivation: scheduling!

17. Here, each row and each column has been randomly assigned a number from {1,2,3,4}. A site is killed if it gets the same number from both coordinates. This type of dependent percolation came up in the study of a self-stabilizing token management protocol. Coordinate Percolation

18. .4 .7 .3 .6 .3 .3 .8 1.1 .7 1.0 .7 .7 .2 .5 .1 .4 .1 .1 .5 .8 .4 .7 .4 .4 .2 .5 .1 .4 .1 .1 .1 .4 0 .3 0 Let be the probability of escape from (0,0) when the threshold is t. t An easy variant of coordinate percolation Random reals (say, uniform in [0,1]) are assigned to the coordinates; each grid point inherits the sum of its coordinate reals; and any vertex whose sum exceeds some threshold t is deleted. (t=.75 in figure.) t Hmmm… does it matter if we’re allowed to move left or down, as well?

19. Theta-functions, Independent vs. Coordinate Percolation 1 p p 1 c p independent 1 t 2 2 1 t coordinate Unknown: behavior of theta just above the critical point, e.g.: what is the critical exponent? Known: a precise closed expression for the probability of percolation!

20. Back to the continuum? Sometimes, paradoxically, you get better combinatorics by not moving to the grid. Example: branched polymers. Physicists have been studying these on the grid. But…

21. 1/2 [Brydges and Imbrie ’03], using equivariant cohomology, proved a deep connection between branched polymers in dimension D+2 and the hard-core model in dimension D. They get an exact formula for the volume of the space of branched polymers in dimensions 2 and 3. [Kenyon and W. ’09] use elementary calculus and combinatorics to duplicate and extend some of these results, i.e. showing that branched polymers of n balls in 3-space have diameter ~n . Appearing in the work are random permutations, Cayley’s Theorem, Euler numbers, Tutte polynomial . . . Branched polymers in dimensions 2 and 3

22. From this work we also get a method for generating perfectly random polymers. Generating random polymers

23. Phase IV: DIMACS and StatisticalPhysics Face a Brilliant Future

24. “Boundary Influence” DIMACS DIMACS DIMACS DIMACS DIMACS Mezard Dyer Galvin Schramm DIMACS Sorkin Van den Berg DIMACS Randall Tetali Riordan Propp Vigoda Kannan Lebowitz Propp Montenari Spencer Kenyon Reimer DIMACS Bowen Bollobas Peres DIMACS Borgs Sinclair Sidoravicius Kahn Kesten Gacs Lyons Brightwell Steif Martinelli Holroyd Haggstrom DIMACS Chayes Radin DIMACS Jerrum Lovasz DIMACS DIMACS DIMACS DIMACS DIMACS

25. Happy 20th Birthday!! DIMACS