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

Slide 1

DIMACS 20th Birthday Celebration, 20 November 2009

The Combinatori

al 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 = 3.787.

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 modelrandom independent

sets

monomer-dimer random matchings

branched 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

understandingsatisfiability. [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 arecreated independently with probability p. When

do you get a giant connected component?

Physicists call this game percolation and usuallyplay 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.)

0.1 .4 .30.1

.4

.1

.7

.3

.1 .4 .1.5.2

.5 .8 .4 .7 .4

.2 .5 .1 .4 .1

.8 1.1 .7 1.0 .7

.4 .7 .3 .6 .3

t

Let be the probability of escape from (0,0) when the

threshold is t.

tHmmm… 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

1p

pc

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

1

1 2tcoordinate

Slide 20

Back to the continuum?

Sometimes, paradoxically, you get better combinatoricsby 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 spaceof 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 andcombinatorics to duplicate and extend some of

these results, i.e. showing that branched polymersof 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

randompolymers

Slide 23

Phase IV: DIMACS and StatisticalPhysics Face a Brilliant Future

Slide 24

“Boundary Influence”

Riordan

DIMACS DIMACS DIMACS DIMACS DIMACS

DIMACS DIMACS DIMACS DIMACS DIMACS

DIM

AC

S

DIM

AC

S

DIM

AC

S

DIM

AC

S D

IMA

CS

D

IMA

CS

Bollobas

LebowitzBowen

Chayes

BorgsLyons

RadinSteif

Van den Berg

SpencerVigoda

KestenSidoravicius

Martinelli

Schramm

Lovasz

Kannan

PeresSinclair

Jerrum

DyerRandall

HolroydBrightwell

Kenyon

MezardTetali

Montenari

Haggstrom

ProppPropp

SorkinGalvin

Kahn Gacs

Reimer

Slide 25

DIMACS

Happy 20th Birthday!!