TASEP: LDP via MPA
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Transcript of 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
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
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
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
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
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
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))
Horacio Gonzalez Duhart TASEP: LDP via MPA
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.
Horacio Gonzalez Duhart TASEP: LDP via MPA
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
Horacio Gonzalez Duhart TASEP: LDP via MPA
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
Horacio Gonzalez Duhart TASEP: LDP via MPA
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.
Horacio Gonzalez Duhart TASEP: LDP via MPA
Main result
Horacio Gonzalez Duhart TASEP: LDP via MPA
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.
Horacio Gonzalez Duhart TASEP: LDP via MPA
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
, · · ·)
Horacio Gonzalez Duhart TASEP: LDP via MPA
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
, · · ·)
Horacio Gonzalez Duhart TASEP: LDP via MPA
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
, · · ·)
Horacio Gonzalez Duhart TASEP: LDP via MPA
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
, · · ·)
Horacio Gonzalez Duhart TASEP: LDP via MPA
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
, · · ·)
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 Xnn∈N 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 Xnn∈N 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 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.
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 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).
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 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).
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 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).
Horacio Gonzalez Duhart TASEP: LDP via MPA
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
(θ
n∑k=1
ξk)]
= limn→∞
1
nlog
∑η∈0,1n
να1/4ξ : ξk = ηk for k ≤ n exp
(θ
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.
Horacio Gonzalez Duhart TASEP: LDP via MPA
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
(θ
n∑k=1
ξk)]
= limn→∞
1
nlog
∑η∈0,1n
να1/4ξ : ξk = ηk for k ≤ n exp
(θ
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.
Horacio Gonzalez Duhart TASEP: LDP via MPA
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
(θ
n∑k=1
ξk)]
= limn→∞
1
nlog
∑η∈0,1n
να1/4ξ : ξk = ηk for k ≤ n exp
(θ
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.
Horacio Gonzalez Duhart TASEP: LDP via MPA
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
(θ
n∑k=1
ξk)]
= limn→∞
1
nlog
∑η∈0,1n
να1/4ξ : ξk = ηk for k ≤ n exp
(θ
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.
Horacio Gonzalez Duhart TASEP: LDP via MPA
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
(θ
n∑k=1
ξk)]
= limn→∞
1
nlog
∑η∈0,1n
να1/4ξ : ξk = ηk for k ≤ n exp
(θ
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.
Horacio Gonzalez Duhart TASEP: LDP via MPA
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
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
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
Horacio Gonzalez Duhart TASEP: LDP via MPA
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
Horacio Gonzalez Duhart TASEP: LDP via MPA
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 .
Horacio Gonzalez Duhart TASEP: LDP via MPA
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 .
Horacio Gonzalez Duhart TASEP: LDP via MPA
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 .
Horacio Gonzalez Duhart TASEP: LDP via MPA
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
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
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
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
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
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.
Horacio Gonzalez Duhart TASEP: LDP via MPA
Functional Materials Far From Equilibrium
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