# TASEP: LDP via MPA

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The Semi-Infinite TASEP: Large Deviations viaMatrix Products

H. G. Duhart, P. Morters, J. Zimmer

University of Bath

21 November 2014

The TASEP

The totally asymmetric simple exclusion process is one of thesimplest interacting particle systems. It was introduced by Liggettin 1975.

Create particles in site 1 at rate (0, 1).Particles jump to the right with rate 1.

At most one particle per site.

1

. . .

Horacio Gonzalez Duhart TASEP: LDP via MPA

The TASEP

The totally asymmetric simple exclusion process is one of thesimplest interacting particle systems. It was introduced by Liggettin 1975.

Create particles in site 1 at rate (0, 1).

Particles jump to the right with rate 1.

At most one particle per site.

1

. . .

Horacio Gonzalez Duhart TASEP: LDP via MPA

The TASEP

The totally asymmetric simple exclusion process is one of thesimplest interacting particle systems. It was introduced by Liggettin 1975.

Create particles in site 1 at rate (0, 1).Particles jump to the right with rate 1.

At most one particle per site.

1

. . .

Horacio Gonzalez Duhart TASEP: LDP via MPA

The TASEP

Create particles in site 1 at rate (0, 1).Particles jump to the right with rate 1.

At most one particle per site.

1

. . .

Horacio Gonzalez Duhart TASEP: LDP via MPA

The TASEP

Create particles in site 1 at rate (0, 1).Particles jump to the right with rate 1.

At most one particle per site.

1

. . .

Horacio Gonzalez Duhart TASEP: LDP via MPA

The TASEP

Formally, the state space is {0, 1}N and its generator:

Gf () = (1 1)(f (1) f ())

+kN

k(1 k+1)(

f (k,k+1) f ())

Horacio Gonzalez Duhart TASEP: LDP via MPA

The TASEP

Theorem (Liggett 1975)

Let be a product measure on {0, 1}N for which := lim

k{ : k = 1} exists. Then there exist probability

measures % defined if either 12 and % > 1 , or > 12 and12 % 1, which are asymptotically product with density %, suchthat

if 12

then limtS(t) =

{ if 1 if > 1 ,

and if >1

2then lim

tS(t) =

1/2 if

1

2

if >1

2.

Horacio Gonzalez Duhart TASEP: LDP via MPA

Main result

We will assume that our process {(t)}t0 starts with noparticles. That is

P[k(0) = 0 k N] = 1

Working under the reached invariant measure, we will find alarge deviation principle for the sequence of random variables{Xn}nN of the empirical density of the first n sites.

Xn =1

n

nk=1

k

Horacio Gonzalez Duhart TASEP: LDP via MPA

Main result

We will assume that our process {(t)}t0 starts with noparticles. That is

P[k(0) = 0 k N] = 1

Working under the reached invariant measure, we will find alarge deviation principle for the sequence of random variables{Xn}nN of the empirical density of the first n sites.

Xn =1

n

nk=1

k

Horacio Gonzalez Duhart TASEP: LDP via MPA

Main result

Theorem

Let Xn be the empirical density of a semi-infinite TASEP withinjection rate (0, 1) starting with an empty lattice. Then,under the invariant probability measure given by Theorem 1,{Xn}nN satisfies a large deviation principle with convex ratefunction I : [0, 1] [0,] given as follows:(a) If 1

2, then I (x) = x log

x

+ (1 x) log 1 x

1 .

(b) If >1

2, then

I (x) =

x log

x

+ (1 x) log 1 x

1 + log (4(1 )) if 0 x 1 ,

2 [x log x + (1 x) log(1 x) + log 2] if 1 < x 12,

x log x + (1 x) log(1 x) + log 2 if 12< x 1.

Horacio Gonzalez Duhart TASEP: LDP via MPA

Main result

Horacio Gonzalez Duhart TASEP: LDP via MPA

The MPA

Grokinsky (2004), based on the work of Derrida et. al. (1993),found a way to completely characterise the measure of Theorem 1via a matrix representation.

Theorem (Grokinsky)

Suppose there exist (possibly infinite) non-negative matrices D, Eand vectors w and v, fulfilling the algebraic relations

DE = D + E , wTE = wT , c(D + E )v = v

for some c > 0. Then the probability measure c defined by

c { : 1 = 1, . . . , n = n} =wT (

nk=1 kD + (1 k)E ) v

wT (D + E )nv

is invariant for the semi-infinite TASEP.

Horacio Gonzalez Duhart TASEP: LDP via MPA

The MPA

The invariant measure given by Liggett is the same as the onegiven by Grokinsky.

We will focus on the case when we start with an empty lattice.

When 12 , sites behave like iid Bernoulli random variableswith parameter .

We can find explicit solutions for the matrices and vectors inthe case > 12 .

D =

1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1

. . ....

......

.... . .

, E =

1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1

. . ....

......

.... . .

, v =

123...

,

and wT =

(1,

1

1,

(1

1)2

, )

Horacio Gonzalez Duhart TASEP: LDP via MPA

The MPA

The invariant measure given by Liggett is the same as the onegiven by Grokinsky.

We will focus on the case when we start with an empty lattice.

When 12 , sites behave like iid Bernoulli random variableswith parameter .

We can find explicit solutions for the matrices and vectors inthe case > 12 .

D =

1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1

. . ....

......

.... . .

, E =

1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1

. . ....

......

.... . .

, v =

123...

,

and wT =

(1,

1

1,

(1

1)2

, )

Horacio Gonzalez Duhart TASEP: LDP via MPA

The MPA

The invariant measure given by Liggett is the same as the onegiven by Grokinsky.

We will focus on the case when we start with an empty lattice.

When 12 , sites behave like iid Bernoulli random variableswith parameter .

We can find explicit solutions for the matrices and vectors inthe case > 12 .

D =

1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1

. . ....

......

.... . .

, E =

1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1

. . ....

......

.... . .

, v =

123...

,

and wT =

(1,

1

1,

(1

1)2

, )

Horacio Gonzalez Duhart TASEP: LDP via MPA

The MPA

The invariant measure given by Liggett is the same as the onegiven by Grokinsky.

We will focus on the case when we start with an empty lattice.

When 12 , sites behave like iid Bernoulli random variableswith parameter .

We can find explicit solutions for the matrices and vectors inthe case > 12 .

D =

1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1

. . ....

......

.... . .

, E =

1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1

. . ....

......

.... . .

, v =

123...

,

and wT =

(1,

1

1,

(1

1)2

, )

Horacio Gonzalez Duhart TASEP: LDP via MPA

The MPA

The invariant measure given by Liggett is the same as the onegiven by Grokinsky.

We will focus on the case when we start with an empty lattice.

When 12 , sites behave like iid Bernoulli random variableswith parameter .

We can find explicit solutions for the matrices and vectors inthe case > 12 .

D =

1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1

. . ....

......

.... . .

, E =

1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1

. . ....

......

.... . .

, v =

123...

,

and wT =

(1,

1

1,

(1

1)2

, )

Horacio Gonzalez Duhart TASEP: LDP via MPA

Large deviations

We now want to find a large deviation principle.

Simply put, a sequence of random variables {Xn}nN satisfiesa LDP with rate function I if

P[Xn x ] exp{nI (x)}

for some non-negative function I : R [0,]

Horacio Gonzalez Duhart TASEP: LDP via MPA

Large deviations

We now want to find a large deviation principle.

Simply put, a sequence of random variables {Xn}nN satisfiesa LDP with rate function I if

P[Xn x ] exp{nI (x)}

for some non-negative function I : R [0,]

Horacio Gonzalez Duhart TASEP: LDP via MPA

Large deviations

Formally,

Definition (Large deviation principle)

Let X be a Polish space. Let {Pn}nN be a sequence of probabilityof measures on X . We say {Pn}nN satisfies a large deviationprinciple with rate function I if the following three conditions meet:

i) I is a rate function (non-negative and lsc).

ii) lim supn

1

nlogPn[F ] inf

xFI (x) F X closed

iii) lim infn

1

nlogPn[G ] inf

xGI (x) G X open.

Horacio Gonzalez Duhart TASEP: LDP via MPA

Large deviations

The study of large deviations has been developed sinceVaradhan unified the theory in 1966.

The result we will use for our proof the Gartner-Ellis Theorem.

Theorem (Gartner-Ellis)

Let {Xn}nN be a sequence of random variables on a probabilityspace (,A,P), where is a nonempty subset of R. If

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