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

Slide 2 Phase I: The Parties Meet

Slide 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?

Slide 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

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

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

Slide 7 Take-over 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.”

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

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

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

Slide 11 Phase II: DIMACS Makes a Match

Slide 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? Combinatorialists settle a controversy Even a certain well-known married couple at Microsoft Research couldn’t agree. Along came [David Galvin and Jeff Kahn ’04] (with ideas from Sapozhenko) to show the critical activity goes to zero.

Slide 13 Phase III: MSR Leads the Charge

Slide 14 Take your favorite graph G, and let its vertices (or its edges) live or die at random. What happens? Percolation 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? 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.

Slide 15 Colour the points of a Poisson process green (with probability p ) or red. Voronoi percolation 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!

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

Slide 17 This type of dependent percolation came up in the study of a self-stabilizing token management protocol. Coordinate Percolation 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.

Slide 18 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 Let be the probability of escape from (0,0) when the threshold is t. t Hmmm… does it matter if we’re allowed to move left or down, as well?

Slide 19 Theta-functions, Independent vs. Coordinate Percolation independent p 1 1 p p c 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! 2 t t coordinate

Slide 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…

Slide 21 Branched polymers in dimensions 2 and 3 [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. Appearing in the work are random permutations, Cayley’s Theorem, Euler numbers, Tutte polynomial... [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. 1/2

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

Slide 23 Phase IV: DIMACS and Statistical Physics Face a Brilliant Future

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

Slide 25 DIMACS Happy 20 th Birthday!!