DIMACS 20th Birthday Celebration, 20 November 2009
The
Combinatorial
Side of
Statistical
Physics
Peter Winkler, Dartmouth
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Phase I: The Parties Meet
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Combinatorics--We begin with a
checkerboard on
which checkers are
placed uniformly at
random subject to
the condition that
no two are
orthogonally
adjacent.
or Statistical
Physics?
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Discrete hard-core
To see better what’s
going on, we color the
even occupied squares
blue and the odd ones
red. Notice the
tendency to cluster…
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The big picture
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.
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The plot thickens
This is what
it looks like
when we set
= 3.787.
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Take-over
At = 3.792,
one of the
colors “breaks
symmetry” and
takes over the
picture.
We suddenly
get “ordered
phase,” “longrange
correlation” and
“slow mixing.”
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Statistical physics  Combinatorics
hard-core model
monomer-dimer
Potts model
percolation
linear polymers
branched polymers
random independent sets
random matchings
random colorings
random subgraphs
self-avoiding
random walks
random lattice trees
Slide ‹#›
CS theory’s favorite hard-constraint model
x
(x z v)
(x y u)
y
w
(x y z)
z
(w z t)
(y z w)
Physics techniques (e.g., “cavity method”) have
helped to make major progress in understanding
satisfiability. [Mezard, Parisi and Virasoro ’85]
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Which graphs cause a phase transition?
On the Bethe
lattice, where
things are nice:
[Brightwell
& W. ’99]
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Phase II: DIMACS Makes a Match
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Combinatorialists settle a controversy
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?
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.
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Phase III: MSR Leads the Charge
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Percolation
Take your favorite graph G, and let its vertices
(or its edges) live or die at random. What happens?
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.
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Voronoi percolation
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!
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Coordinate Percolation
independent percolation:
?
a vertex lives or dies based on
an independent event
associated with the vertex.
Motivation: water seeping
through a porous material.
coordinate percolation:
?
?
a vertex lives or dies based on
independent events associated
with the vertex’s coordinates.
Motivation: scheduling!
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Coordinate Percolation
This type of
dependent percolation
came up in the study
of a self-stabilizing
token management
protocol.
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.
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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.)
Let t be the probability
of escape from (0,0) when
the threshold is t.
.3
.7
.1
.4
.1
.4
.7
.3
.6
.3
.8
1.1
.7
1.0
.7
.2
.5
.1
.4
.1
.5
.8
.4
.7
.4
.2
.5
.1
.4
.1
.1
.4
0
.3
0
Hmmm…t does it matter if we’re
allowed to move left or down, as well?
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Theta-functions, Independent vs. Coordinate Percolation
Unknown: behavior of theta
just above the critical point,
e.g.: what is the critical
exponent?
1
p
p
p
c
1
Known: a precise
closed expression for
the probability of
percolation!
independent
1
t
t
1
2
2
coordinate
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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…
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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.
[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 .
Appearing in the work are random permutations,
Cayley’s Theorem, Euler numbers, Tutte polynomial . . .
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Generating
random
polymers
From this work
we also get a
method for
generating
perfectly
random
polymers.
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Phase IV: DIMACS and Statistical
Physics Face a Brilliant Future
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“Boundary Influence”
DIMACS DIMACS DIMACS
DIMACS DIMACS DIMACS
DIMACS DIMACS DIMACS DIMACS DIMACS
DIMACS DIMACS DIMACS DIMACS DIMACS
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th
20
Happy
Birthday!!
DIMACS
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