TASEP: LDP via MPA

43
The Semi-Infinite TASEP: Large Deviations via Matrix Products H. G. Duhart, P. M¨ orters, J. Zimmer University of Bath 21 November 2014

Transcript of TASEP: LDP via MPA

Page 1: TASEP: LDP via MPA

The Semi-Infinite TASEP: Large Deviations viaMatrix Products

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

University of Bath

21 November 2014

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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 α ∈ (0, 1).

Particles jump to the right with rate 1.

At most one particle per site.

α 1

. . .

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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 α ∈ (0, 1).

Particles jump to the right with rate 1.

At most one particle per site.

α 1

. . .

Horacio Gonzalez Duhart TASEP: LDP via MPA

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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 α ∈ (0, 1).

Particles jump to the right with rate 1.

At most one particle per site.

α 1

. . .

Horacio Gonzalez Duhart TASEP: LDP via MPA

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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 α ∈ (0, 1).

Particles jump to the right with rate 1.

At most one particle per site.

α 1

. . .

Horacio Gonzalez Duhart TASEP: LDP via MPA

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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 α ∈ (0, 1).

Particles jump to the right with rate 1.

At most one particle per site.

α 1

. . .

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The TASEP

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

Gf (η) = α(1− η1)(f (η1)− f (η)

)+∑k∈N

ηk(1− ηk+1)(

f (ηk,k+1)− f (η))

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The TASEP

Theorem (Liggett 1975)

Let µ be a product measure on 0, 1N for whichρ := lim

k→∞µη : ηk = 1 exists. Then there exist probability

measures µα% defined if either α ≤ 12 and % > 1− α, or α > 1

2 and12 ≤ % ≤ 1, which are asymptotically product with density %, suchthat

if α ≤ 1

2then lim

t→∞µS(t) =

να if ρ ≤ 1− α

µαρ if ρ > 1− α,

and if α >1

2then lim

t→∞µS(t) =

µα1/2 if ρ ≤ 1

2

µαρ if ρ >1

2.

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Main result

We will assume that our process ξ(t)t≥0 starts with noparticles. That is

P[ξk(0) = 0 ∀k ∈ N] = 1

Working under the reached invariant measure, we will find alarge deviation principle for the sequence of random variablesXnn∈N of the empirical density of the first n sites.

Xn =1

n

n∑k=1

ξk

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Main result

We will assume that our process ξ(t)t≥0 starts with noparticles. That is

P[ξk(0) = 0 ∀k ∈ N] = 1

Working under the reached invariant measure, we will find alarge deviation principle for the sequence of random variablesXnn∈N of the empirical density of the first n sites.

Xn =1

n

n∑k=1

ξk

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Main result

Theorem

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

(a) If α ≤ 1

2, then I (x) = x log

x

α+ (1− x) log

1− x

1− α.

(b) If α >1

2, then

I (x) =

x log

x

α+ (1 − x) log

1 − x

1 − α+ log (4α(1 − α)) if 0 ≤ x ≤ 1 − α,

2 [x log x + (1 − x) log(1 − x) + log 2] if 1 − α < x ≤ 1

2,

x log x + (1 − x) log(1 − x) + log 2 if1

2< x ≤ 1.

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Main result

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The MPA

Großkinsky (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 (Großkinsky)

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

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

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

ναc ζ : ζ1 = η1, . . . , ζn = ηn =wT (

∏nk=1 ηkD + (1− ηk)E ) v

wT (D + E )nv

is invariant for the semi-infinite TASEP.

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The MPA

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

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

When α ≤ 12 , sites behave like iid Bernoulli random variables

with parameter α.

We can find explicit solutions for the matrices and vectors inthe case α > 1

2 .

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

α− 1,

(1

α− 1

)2

, · · ·)

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The MPA

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

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

When α ≤ 12 , sites behave like iid Bernoulli random variables

with parameter α.

We can find explicit solutions for the matrices and vectors inthe case α > 1

2 .

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

α− 1,

(1

α− 1

)2

, · · ·)

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The MPA

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

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

When α ≤ 12 , sites behave like iid Bernoulli random variables

with parameter α.

We can find explicit solutions for the matrices and vectors inthe case α > 1

2 .

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

α− 1,

(1

α− 1

)2

, · · ·)

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The MPA

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

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

When α ≤ 12 , sites behave like iid Bernoulli random variables

with parameter α.

We can find explicit solutions for the matrices and vectors inthe case α > 1

2 .

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

α− 1,

(1

α− 1

)2

, · · ·)

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The MPA

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

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

When α ≤ 12 , sites behave like iid Bernoulli random variables

with parameter α.

We can find explicit solutions for the matrices and vectors inthe case α > 1

2 .

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

α− 1,

(1

α− 1

)2

, · · ·)

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Large deviations

We now want to find a large deviation principle.

Simply put, a sequence of random variables Xnn∈N satisfiesa LDP with rate function I if

P[Xn ≈ x ] ≈ exp−nI (x)

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

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Large deviations

We now want to find a large deviation principle.

Simply put, a sequence of random variables Xnn∈N satisfiesa LDP with rate function I if

P[Xn ≈ x ] ≈ exp−nI (x)

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

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Large deviations

Formally,

Definition (Large deviation principle)

Let X be a Polish space. Let Pnn∈N be a sequence of probabilityof measures on X . We say Pnn∈N 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

x∈FI (x) ∀F ⊂ X closed

iii) lim infn→∞

1

nlogPn[G ] ≥ − inf

x∈GI (x) ∀G ⊂ X open.

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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 Xnn∈N be a sequence of random variables on a probabilityspace (Ω,A,P), where Ω is a nonempty subset of R. If the limitcumulant generating function Λ: R→ R defined by

Λ(θ) = limn→∞

1n logE[enθXn ]

exists and is differentiable on all R, then Xnn∈N satisfies a largedeviation principle with rate function I : Ω→ [−∞,∞] defined by

I (x) = supθ∈Rxθ − Λ(θ).

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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 Xnn∈N be a sequence of random variables on a probabilityspace (Ω,A,P), where Ω is a nonempty subset of R. If the limitcumulant generating function Λ: R→ R defined by

Λ(θ) = limn→∞

1n logE[enθXn ]

exists and is differentiable on all R, then Xnn∈N satisfies a largedeviation principle with rate function I : Ω→ [−∞,∞] defined by

I (x) = supθ∈Rxθ − Λ(θ).

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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 Xnn∈N be a sequence of random variables on a probabilityspace (Ω,A,P), where Ω is a nonempty subset of R. If the limitcumulant generating function Λ: R→ R defined by

Λ(θ) = limn→∞

1n logE[enθXn ]

exists and is differentiable on all R, then Xnn∈N satisfies a largedeviation principle with rate function I : Ω→ [−∞,∞] defined by

I (x) = supθ∈Rxθ − Λ(θ).

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Idea of the proof of the main result

Now the idea is to use the the MPA in calculating thefunction in Gartner-Ellis Theorem.

Λ(θ) = limn→∞

1

nlogE

[enθXn

]= lim

n→∞

1

nlogE

[exp

n∑k=1

ξk)]

= limn→∞

1

nlog

∑η∈0,1n

να1/4ξ : ξk = ηk for k ≤ n exp

n∑k=1

ηk

)

= limn→∞

1

nlog

wT (eθD + E )nv

wT (D + E )nv

= limn→∞

1n log wT (eθD + E )nv − 2 log 2.

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Idea of the proof of the main result

Now the idea is to use the the MPA in calculating thefunction in Gartner-Ellis Theorem.

Λ(θ) = limn→∞

1

nlogE

[enθXn

]= lim

n→∞

1

nlogE

[exp

n∑k=1

ξk)]

= limn→∞

1

nlog

∑η∈0,1n

να1/4ξ : ξk = ηk for k ≤ n exp

n∑k=1

ηk

)

= limn→∞

1

nlog

wT (eθD + E )nv

wT (D + E )nv

= limn→∞

1n log wT (eθD + E )nv − 2 log 2.

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Idea of the proof of the main result

Now the idea is to use the the MPA in calculating thefunction in Gartner-Ellis Theorem.

Λ(θ) = limn→∞

1

nlogE

[enθXn

]= lim

n→∞

1

nlogE

[exp

n∑k=1

ξk)]

= limn→∞

1

nlog

∑η∈0,1n

να1/4ξ : ξk = ηk for k ≤ n exp

n∑k=1

ηk

)

= limn→∞

1

nlog

wT (eθD + E )nv

wT (D + E )nv

= limn→∞

1n log wT (eθD + E )nv − 2 log 2.

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Idea of the proof of the main result

Now the idea is to use the the MPA in calculating thefunction in Gartner-Ellis Theorem.

Λ(θ) = limn→∞

1

nlogE

[enθXn

]= lim

n→∞

1

nlogE

[exp

n∑k=1

ξk)]

= limn→∞

1

nlog

∑η∈0,1n

να1/4ξ : ξk = ηk for k ≤ n exp

n∑k=1

ηk

)

= limn→∞

1

nlog

wT (eθD + E )nv

wT (D + E )nv

= limn→∞

1n log wT (eθD + E )nv − 2 log 2.

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Idea of the proof of the main result

Now the idea is to use the the MPA in calculating thefunction in Gartner-Ellis Theorem.

Λ(θ) = limn→∞

1

nlogE

[enθXn

]= lim

n→∞

1

nlogE

[exp

n∑k=1

ξk)]

= limn→∞

1

nlog

∑η∈0,1n

να1/4ξ : ξk = ηk for k ≤ n exp

n∑k=1

ηk

)

= limn→∞

1

nlog

wT (eθD + E )nv

wT (D + E )nv

= limn→∞

1n log wT (eθD + E )nv − 2 log 2.

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Idea of the proof of the main result

Having the equation

Λ(θ) = limn→∞

1n log wT (eθD + E )nv − 2 log 2

we would like to simplify it and use Gartner-Ellis Theorem.

However, even when we have a explicit form of D, E , v , andw , the term wT (eθD + E )nv is not easy to handle, and so wesplit into a lower bound and an upper bound.

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Idea of the proof of the main result

Having the equation

Λ(θ) = limn→∞

1n log wT (eθD + E )nv − 2 log 2

we would like to simplify it and use Gartner-Ellis Theorem.

However, even when we have a explicit form of D, E , v , andw , the term wT (eθD + E )nv is not easy to handle, and so wesplit into a lower bound and an upper bound.

Horacio Gonzalez Duhart TASEP: LDP via MPA

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Upper bound

The upper bound comes from noticing that (eθD + E ) is aToeplitz operator (constant diagonals in the matrix), and vand w live on a family of weighted spaces `2

s and its dual,respectively.

We then use Cauchy-Schwarz inequality and optimise over theparameter s of permissible weights.

wT (eθD + E )nv ≤ |w |`2?s||(eθD + E )n|B(`2

s )|v |`2s

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Upper bound

The upper bound comes from noticing that (eθD + E ) is aToeplitz operator (constant diagonals in the matrix), and vand w live on a family of weighted spaces `2

s and its dual,respectively.

We then use Cauchy-Schwarz inequality and optimise over theparameter s of permissible weights.

wT (eθD + E )nv ≤ |w |`2?s||(eθD + E )n|B(`2

s )|v |`2s

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Lower bound

The lower bound comes from expanding the term

wT (eθD + E )nv =n∑

p=1

p∑j=0

f np,j(θ)wTEp−jD jv

where f np,j(θ) are polynomials on eθ with non-negative

coefficients.

We then find a lower bound when j = 0:

wT (eθD + E )nv ≥n∑

p=1

f np,0(θ)wTEpv .

And another when j = p:

wT (eθD + E )nv ≥n∑

p=1

f np,p(θ)wTDpv .

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Lower bound

The lower bound comes from expanding the term

wT (eθD + E )nv =n∑

p=1

p∑j=0

f np,j(θ)wTEp−jD jv

where f np,j(θ) are polynomials on eθ with non-negative

coefficients.

We then find a lower bound when j = 0:

wT (eθD + E )nv ≥n∑

p=1

f np,0(θ)wTEpv .

And another when j = p:

wT (eθD + E )nv ≥n∑

p=1

f np,p(θ)wTDpv .

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Lower bound

The lower bound comes from expanding the term

wT (eθD + E )nv =n∑

p=1

p∑j=0

f np,j(θ)wTEp−jD jv

where f np,j(θ) are polynomials on eθ with non-negative

coefficients.

We then find a lower bound when j = 0:

wT (eθD + E )nv ≥n∑

p=1

f np,0(θ)wTEpv .

And another when j = p:

wT (eθD + E )nv ≥n∑

p=1

f np,p(θ)wTDpv .

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Summarising. . .

TASEP: Create particles at rate α. Move them to the right at rate 1.One particle per site.

MPA: Find matrices and vectors satisfying certain condition to find theinvariant measure.

LDP: Find the exponential rate of convergence to 0 of unlikely events.

Our result: Find an LDP of the empirical density of the TASEP via theMPA.

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Summarising. . .

TASEP: Create particles at rate α. Move them to the right at rate 1.One particle per site.

MPA: Find matrices and vectors satisfying certain condition to find theinvariant measure.

LDP: Find the exponential rate of convergence to 0 of unlikely events.

Our result: Find an LDP of the empirical density of the TASEP via theMPA.

Horacio Gonzalez Duhart TASEP: LDP via MPA

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Summarising. . .

TASEP: Create particles at rate α. Move them to the right at rate 1.One particle per site.

MPA: Find matrices and vectors satisfying certain condition to find theinvariant measure.

LDP: Find the exponential rate of convergence to 0 of unlikely events.

Our result: Find an LDP of the empirical density of the TASEP via theMPA.

Horacio Gonzalez Duhart TASEP: LDP via MPA

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Summarising. . .

TASEP: Create particles at rate α. Move them to the right at rate 1.One particle per site.

MPA: Find matrices and vectors satisfying certain condition to find theinvariant measure.

LDP: Find the exponential rate of convergence to 0 of unlikely events.

Our result: Find an LDP of the empirical density of the TASEP via theMPA.

Horacio Gonzalez Duhart TASEP: LDP via MPA

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Summarising. . .

TASEP: Create particles at rate α. Move them to the right at rate 1.One particle per site.

MPA: Find matrices and vectors satisfying certain condition to find theinvariant measure.

LDP: Find the exponential rate of convergence to 0 of unlikely events.

Our result: Find an LDP of the empirical density of the TASEP via theMPA.

Horacio Gonzalez Duhart TASEP: LDP via MPA

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References

F. den Hollander. Large Deviations, volume 14 of Fields InstituteMonographs. American Mathematical Society, Providence, RI, 2000.

B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier. Exact solutionof a 1D asymmetric exclusion model using a matrix formulation. J.Phys. A, 26(7):1493,1993.

S. Großkinsky. Phase transitions in nonequilibrium stochasticparticle systems with local conservation laws. PhD thesis, TUMunich, 2004.

T. M. Liggett. Ergodic theorems for the asymmetric simpleexclusion process. Trans. Amer. Math. Soc., 213:237-261,1975.

H. G. Duhart, P. Morters, and J. Zimmer. The Semi-InfiniteAsymmetric Exclusion Process: Large Deviations via MatrixProducts. ArXiv e-prints, arXiv:1411.3270v1, November 2014.

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Functional Materials Far From Equilibrium

21 November 2014, University of Bristol