Combinatorics and Statistical Physics:

a story of hopping particles

Lauren K. Williams

Harvard University and MSRI

1

Combinatorics and statistical physics: a story of hopping particles

Program

1. Background on a model from statistical mechanics: the

asymmetric exclusion process (ASEP). Motivation from traffic flow

and biology.

2. Some combinatorial objects coming from geometry:

Γ

-diagrams

and permutation tableaux

3. A surprising relationship between 1 and 2.

4. Applications and connections to other things ...

Lauren K. Williams 2

Combinatorics and statistical physics: a story of hopping particles

Understanding traffic

Consider a one-way road with cars entering at one end and exiting

at the other end. We’d like to understand the behavior of traffic in

this simple situation, addressing questions such as the following:

• Current: how many cars move past one spot in unit time?

• Density: what is the average number of cars present?

• Distribution: what is the probability that we see a particular

configuration of cars?

• Lines: a line of cars can form at either end of the road if cars

enter too fast or exit too slowly.

• Shocks: this is a distinct transition at some point on the road

from low density to high density traffic. This is when one needs to

suddenly slow down because of the buildup of cars ahead.

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Combinatorics and statistical physics: a story of hopping particles

Traffic flow

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http://ops.fhwa.dot.gov/opssecurity/dev-mx/images/fig09.jpg

Combinatorics and statistical physics: a story of hopping particles

A model for traffic: the asymmetric exclusion process

We’d like a model simple enough to mathematically analyze, yet

rich enough to exhibit various traffic phenomena.

Fix a one-dimensional lattice of n sites, and represent cars by

particles which occupy the sites. Choose a parameter q (0 ≤ q ≤ 1).

• New particles can enter the lattice from the left at rate α, and

particles can exit from the right at rate β.

• The model is asymmetric in the sense that the probability of a

particle jumping left is q times the probability of jumping right.

• Exclusion: at most one particle on each site

We’ll depict particles as • or 1 and empty sites as ◦ or 0.

Lauren K. Williams 5

Combinatorics and statistical physics: a story of hopping particles

The asymmetric exclusion process

• Introduced by biologists (MacDonald, Gibbs, Pipkin) in 1968,

and independently by a mathematician (Spitzer) in 1970. Much

more work on the ASEP by Liggett, Derrida, Spohn, Sasamoto,

Lebowitz, Ferrari, etc. (More than 300 papers on arXiv!)

Let Bn be the set of all 2n words of length n on letters {◦, •}.

The ASEP is the Markov chain on Bn with transition probabilities:

• If X = A•◦B and Y = A◦•B then PX,Y = 1n+1 and PY,X = q

n+1 .

• If X = ◦B and Y = •B then PX,Y = αn+1 .

• If X = B• and Y = B◦ then PX,Y = βn+1 .

• Otherwise PX,Y = 0 for Y #= X and PX,X = 1 −∑

X !=Y PX,Y .

Lauren K. Williams 6

Combinatorics and statistical physics: a story of hopping particles

ASEP model

The state diagram of the ASEP model for n = 2.

1/3q/3

α/3 β/3

α/3β/3

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Combinatorics and statistical physics: a story of hopping particles

Some features of the ASEP

Hydrodynamics: particle density in ASEP evolves according to

Burger’s equation.

The ASEP exhibits boundary-induced phase transitions. (Here,

q = 0.)

12

12

1

1

HIGH DENSITY

MAXIMALCURRENTDENSITY

LOW

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Combinatorics and statistical physics: a story of hopping particles

(a) α = 0.2, β = 1 (b) α = 1, β = 0.2 (c) α = β = 1

Lauren K. Williams 9 http://front.math.ucdavis.edu/9910.0270

Combinatorics and statistical physics: a story of hopping particles

ASEP in the context of biology

• Sequence alignment can be mapped onto the asymmetric

exclusion process: Drasdo, Hwa, Lassig, Bundschuh, etc.

• The ASEP models the nuclear pore complex, a multiprotein

machine that manages the transport of material into and out of the

nucleus (through a single-file pore represented by the 1D lattice):

Colvin, etc.

• The ASEP models the process of translation in protein synthesis:

Macdonald, Gibbs, Pipkin, etc.

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Combinatorics and statistical physics: a story of hopping particles

The ASEP as a model for protein synthesis

In the translation step, ribo-

somes “read” the codons of mes-

senger RNA (mRNA) as the ri-

bosomes move along an mRNA

chain. Three steps: initiation,

where ribosomes attach them-

selves one at a time at the “start”

end of the mRNA; elongation,

where ribosomes move down the

chain in a series of steps; termi-

nation, where they detach them-

selves at the “stop” codon.

Lauren K. Williams 11

http://www.accessexcellence.org/RC/VL/GG/images/protein_synthesis.gif

Combinatorics and statistical physics: a story of hopping particles

Stationary Distribution of the ASEP model

The ASEP has a unique stationary distribution – that is, it has a

unique left eigenvector of the transition matrix associated with

eigenvalue 1. This is called the steady state.

1/3q/3

α/3 β/3

α/3β/3

(Solve for prob.’s, say when α = β = 1.)

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Combinatorics and statistical physics: a story of hopping particles

Some combinatorics:

Γ

-diagrams and permutation tableaux

Definition: A

Γ

-diagram is a partition λ = (λ1, . . . , λn) (where

λi ≥ 0) together with a filling with 0’s and 1’s such that:

• There is no 0 which has a 1 above it in the same column and a 1

to its left in the same row.

0 0 11 1 1 10 0 1 01 1 0 0

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Combinatorics and statistical physics: a story of hopping particles

Γ

-diagrams and permutation tableaux

0 0 11 1 1 10 0 1 01 1 0 0

•

Γ

-diagrams were introduced by Postnikov and shown to

correspond to cells in a cell decomposition of the totally

nonnegative part of the Grassmannian Gr+kn (subset of the real

Grassmannian with all Plucker coordinates positive).

• Gr+kn is really coming from representation theory. More generally,

Lusztig has introduced the totally non-negative part of any real

flag variety – deep connections to Lusztig’s canonical basis, and to

the cluster algebras of Fomin and Zelevinsky.

• Surprisingly, this combinatorics is also related to the asymmetric

exclusion process.

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Combinatorics and statistical physics: a story of hopping particles

Permutation tableaux

Definition: We say that a

Γ

-diagram is a permutation tableau if:

• Each column of the rectangle contains at least one 1.

There is a nice bijection from perm-tableaux to permutations which

carries statistics on tableaux to statistics on permutations

(Steingrimsson-W.).

0 0 11 1 1 10 0 1 01 1 0 0

.. . . .

.. .

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Combinatorics and statistical physics: a story of hopping particles

Enumeration: q-Eulerian numbers

Theorem(W.) There is an explicit rank-generating function for cells

in Gr+kn. As a consequence, one gets an enumeration formula for

permutation tableaux:

The weight wt(T ) of a permutation tableau T is the number of 1’s

minus the number of columns.

Let Ek,n(q) =∑

T qwt(T ), summing over all perm-tableaux T with

k rows and n − k columns.

Theorem(W.): Let [i] := 1 + q + q2 + · · · + qi−1.

Ek,n(q) = qk−k2k−1∑

i=0

(−1)i[k − i]qnqki−k

((

n

i

)

qk−i +

(

n

i − 1

))

.

Additionally, Ek,n(q) specializes at q = −1, 0, 1 to binomial

numbers, Narayana numbers, and Eulerian numbers.

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Combinatorics and statistical physics: a story of hopping particles

Corteel’s result

Theorem (Corteel): Let α = β = 1. In the steady state, the

probability that the ASEP model with n sites is in a configuration

with precisely k particles is:

Ek+1,n+1(q)

Zn

Here, Zn is the partition function for the model – the sum of the

probabilities of all possible states. So Zn =∑n

k=0 Ek+1,n+1(q).

Question: Corteel’s result doesn’t say anything about the location

of the particles. How can we refine this result?

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Combinatorics and statistical physics: a story of hopping particles

Refinement

There is a easy bijection between words τ in {0, 1}n and partitions

of semiperimeter n + 1 (where each column has length at least one):

1

0 0 0 1 1 0 1 00011

0

This associates the partition λ(τ) to τ .

Theorem (Corteel, W). In the steady state, the probability that the

ASEP is in configuration τ is:∑

T qwt(T )

Zn

where the sum is over all permutation tableaux of shape λ(τ).

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Combinatorics and statistical physics: a story of hopping particles

Example

(q+2)/(q+5)

1

1 1

11

1100

q =1

q =1

q =1

q +2

0

0

0

State ProbabilityPartition Perm Tableaux Weight

1/(q+5)

1/(q+5)

1/(q+5)

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Combinatorics and statistical physics: a story of hopping particles

Further Refinement

Now want α and β to be general. Two more definitions ...

Given a permutation tableaux T , let f(T ) be the number of 1’s in

the first row of T .

We say that a 0 of T is restricted if it lies below some 1. And we

say that a row is unrestricted if it does not contain a restricted 0.

Let u(T ) be the number of unrestricted rows of T minus 1.

0 0 11 1 1 10 0 1 01 1 0 0

Above, f(T ) = 2 and u(T ) = 1.

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Combinatorics and statistical physics: a story of hopping particles

Refined Theorem

Theorem (Corteel, W). In the steady state, the probability that the

ASEP is in configuration τ is∑

T qwt(T )α−f(T )β−u(T )

Zn,

where the sum ranges over all permutation tableaux T of shape λ.

This is an exact combinatorial formula for the probability of being

in each state.

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Combinatorics and statistical physics: a story of hopping particles

Idea of the proof

We gave two proofs of this result.

• First: algebraic proof, based on the matrix ansatz of Derrida et al.

• Second: we construct a bigger Markov chain (the PT chain) on

the set of all permutation tableaux, which “projects” to the ASEP

in a very strong sense. We prove that in the PT chain, the steady

state probability of being at a particular tableau T is

qwt(T )α−f(T )β−u(T )

Zn.

The PT chain on perm-tableaux can be viewed as a Markov chain

on certain configurations of totally positive vectors.

Lauren K. Williams 22

Combinatorics and statistical physics: a story of hopping particles

The “PT chain”

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Combinatorics and statistical physics: a story of hopping particles

The “PT chain” on permutations

4123

1423

4321

4213

3421

4312

24133124

3142

4132

21433214

1234

4231

1432

1243

2431

1342

3241

2314

1324

2134

2341

3412

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Combinatorics and statistical physics: a story of hopping particles

Transitions in the “PT chain”: enter and exit

1

0 0 0 0 0

0001

Lauren K. Williams 25

Combinatorics and statistical physics: a story of hopping particles

Transitions in the “PT chain”: hop right

1

1

1

00001

000001

0 0 0 0 0

0 0 0 0

Lauren K. Williams 26

Combinatorics and statistical physics: a story of hopping particles

Transitions in the “PT chain”: hop left

1

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Combinatorics and statistical physics: a story of hopping particles

Application: exact formula for probabilities (α = β = 1)

What is the steady state probability of being in state • ◦ ◦ ◦ • ◦ ◦?

This corresponds to perm-tableaux of this shape:

Count columns of each height and add 1: get (3, 4, 1).

The number of perm-tableaux of this shape is

Num := q−3[3]3[2]4[1]1 − q−3[2]7[1]1 − q−2[2]3[1]5 + q−2[1]8.

So probability is Num /Z7. When q = 1, get 297/7! = 0.058928 . . . .

Theorem (Novelli, Thibon, W.): There is a connection between the

steady-state probabilities and combinatorial Hopf algebras.

Corollary: this example can be generalized for any state of ASEP.

Lauren K. Williams 28

Combinatorics and statistical physics: a story of hopping particles

Application: easy proofs of recurrences for ASEP ...

Theorem (Brak, Corteel, Rechnitzer, Essam): Steady state

probabilities of the ASEP obey the following recurrence:

fn(τ1, τ2, . . . ,τj−1, •, ◦, τj+2, . . . , τn) =

fn−1(τ1, τ2, . . . , τj−1, •, τj+2, . . . , τn)+

qfn(τ1, τ2, . . . , τj−1, ◦, •, τj+2, . . . , τn)+

fn−1(τ1, . . . , τj−1, ◦, τj+2, . . . , τn).

We get a new and much simpler picture proof of this result:

* * 1

00

or00 0

***

1* *

***or0

Lauren K. Williams 29

Combinatorics and statistical physics: a story of hopping particles

Application: monotonicity results (α = β = 1)...

q +3q+32q +3q+32

q +4q +7q+53 2q +4q +6q+43 2

6 5q +5q +14q +26q +33q +26q+104 3 2

5 4q +5q +13q +21q +20q+9

1

q+2

4 3q +5q +11q +13q+74 3 q +4q +9q +11q+62

3 2

2

Lauren K. Williams 30

Combinatorics and statistical physics: a story of hopping particles

Applications: monotonicity results (α = β = 1)...

(Steingrimsson, W.)

=

=

= 1

92

2

2

3

3

4

4

5

5

6 7 820+71q+128q +153q +135q +92q +49q +20q +6q +q

6+15q+20q +15q +6q +q3 4 5

15+50q+85q +94q +75q +45q +20q +6q +q6 7 8

Lauren K. Williams 31

Combinatorics and statistical physics: a story of hopping particles

Applications: monotonicity results ...

Def: Let τ, τ ′ ∈ {0, 1}n be two states of the ASEP which contain

exactly k particles. We define the partial order ≺ by τ ≺ τ ′ if and

only if λ(τ) ⊂ λ(τ ′).

Proposition (Corteel, W). Let α = β = 1. Suppose that τ ≺ τ ′, and

let d := |λ(τ ′)|− |λ(τ)|. Then fn(τ ′) − fn(τ) is a non-negative

polynomial. In other words, as one moves up the partial order ≺,

the coefficients of fn(τ) monotonically increase.

Proposition (Steingrimsson, W). Let α = β = 1. If d < n/2, then

fn(•d+1◦n−d−1) − fn(•d◦n−d) is a non-negative polynomial. As a

corollary, the most probable state of the ASEP is •n/2◦n/2.

Lauren K. Williams 32

Combinatorics and statistical physics: a story of hopping particles

Thank you!

Preprints available at:

• http://www.math.harvard.edu/∼lauren

The main references for this talk are:

Tableaux combinatorics for the asymmetric exclusion process (with

Sylvie Corteel), Advances in Applied Math, Vol. 39, Sept. 2007.

A Markov chain on permutations which projects to the PASEP

(with Sylvie Corteel), International Mathematics Research Notices,

2007.

Lauren K. Williams 33