The document discusses the totally asymmetric simple exclusion process (TASEP), which is one of the simplest interacting particle systems. It introduces the TASEP model, which involves particles jumping to the right on a one-dimensional lattice at rate 1, with particles being injected into the first site at rate α. The main result establishes a large deviation principle for the empirical density of particles in the TASEP under its invariant measure. The proof uses a matrix product ansatz to represent the invariant measure and characterize the limiting cumulant generating function, allowing application of Gärtner-Ellis theorem to obtain the large deviation principle.

1. The Semi-Infinite TASEP: Large Deviations via
Matrix Products
H. G. Duhart, P. M¨orters, J. Zimmer
University of Bath
11 November 2015

2. The TASEP
The totally asymmetric simple exclusion process is one of the
simplest interacting particle systems. It was introduced by Liggett
in 1975.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

3. The TASEP
The totally asymmetric simple exclusion process is one of the
simplest interacting particle systems. It was introduced by Liggett
in 1975.
Create particles in site 1 at rate α ∈ (0, 1).
Horacio Gonz´alez Duhart TASEP: LDP via MPA

4. The TASEP
The totally asymmetric simple exclusion process is one of the
simplest interacting particle systems. It was introduced by Liggett
in 1975.
Create particles in site 1 at rate α ∈ (0, 1).
Particles jump to the right with rate 1.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

5. The TASEP
The totally asymmetric simple exclusion process is one of the
simplest interacting particle systems. It was introduced by Liggett
in 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.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

6. The TASEP
The totally asymmetric simple exclusion process is one of the
simplest interacting particle systems. It was introduced by Liggett
in 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 Gonz´alez Duhart TASEP: LDP via MPA

7. The TASEP
Formally, the state space is {0, 1}N
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 Gonz´alez Duhart TASEP: LDP via MPA

8. The TASEP
Theorem (Liggett 1975)
Let µ be a product measure on {0, 1}N
for which
ρ := lim
k→∞
µ{η : ηk = 1} exists. Then there exist probability
measures µα
defined if either α ≤ 1
2 and > 1 − α, or α > 1
2 and
1
2 ≤ ≤ 1, which are asymptotically product with density , such
that
if α ≤
1
2
then lim
t→∞
µS(t) =
να if ρ ≤ 1 − α
µα
ρ if ρ > 1 − α,
and if α >
1
2
then lim
t→∞
µS(t) =



µα
1/2 if ρ ≤
1
2
µα
ρ if ρ >
1
2
.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

9. Main result
We will assume that our process {ξ(t)}t≥0 starts with no
particles. That is
P[ξk(0) = 0 ∀k ∈ N] = 1
Horacio Gonz´alez Duhart TASEP: LDP via MPA

10. Main result
We will assume that our process {ξ(t)}t≥0 starts with no
particles. That is
P[ξk(0) = 0 ∀k ∈ N] = 1
Working under the reached invariant measure, we will find a
large deviation principle for the sequence of random variables
{Xn}n∈N of the empirical density of the first n sites.
Xn =
1
n
n
k=1
ξk
Horacio Gonz´alez Duhart TASEP: LDP via MPA

11. Main result
Theorem
Let Xn be the empirical density of a semi-infinite TASEP with
injection rate α ∈ (0, 1) starting with an empty lattice. Then,
under the invariant probability measure given by Theorem 1,
{Xn}n∈N satisfies a large deviation principle with convex rate
function 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 if
1
2
< x ≤ 1.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

12. Main result
Horacio Gonz´alez Duhart TASEP: LDP via MPA

13. The MPA
Großkinsky (2004), based on the work of Derrida et. al. (1993),
found a way to completely characterise the measure of Theorem 1
via a matrix representation.
Theorem (Großkinsky)
Suppose there exist (possibly infinite) non-negative matrices D, E
and vectors w and v, fulfilling the algebraic relations
DE = D + E, αwT
E = wT
, c(D + E)v = v
for some c > 0. Then the probability measure ¯να
c defined by
¯να
c {ζ : ζ1 = η1, . . . , ζn = ηn} =
wT ( n
k=1 ηkD + (1 − ηk)E) v
wT (D + E)nv
is invariant for the semi-infinite TASEP.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

14. The MPA
The invariant measure given by Liggett is the same as the one
given by Großkinsky.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

15. The MPA
The invariant measure given by Liggett is the same as the one
given by Großkinsky.
We will focus on the case when we start with an empty lattice.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

16. The MPA
The invariant measure given by Liggett is the same as the one
given by Großkinsky.
We will focus on the case when we start with an empty lattice.
When α ≤ 1
2, sites behave like iid Bernoulli random variables
with parameter α.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

17. The MPA
The invariant measure given by Liggett is the same as the one
given by Großkinsky.
We will focus on the case when we start with an empty lattice.
When α ≤ 1
2, sites behave like iid Bernoulli random variables
with parameter α.
We can find explicit solutions for the matrices and vectors in
the case α > 1
2.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

18. The MPA
The invariant measure given by Liggett is the same as the one
given by Großkinsky.
We will focus on the case when we start with an empty lattice.
When α ≤ 1
2, sites behave like iid Bernoulli random variables
with parameter α.
We can find explicit solutions for the matrices and vectors in
the 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 =





1
2
3
...





,
and wT
= 1,
1
α
− 1,
1
α
− 1
2
, · · ·
Horacio Gonz´alez Duhart TASEP: LDP via MPA

19. Large deviations
We now want to find a large deviation principle.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

20. Large deviations
We now want to find a large deviation principle.
Simply put, a sequence of random variables {Xn}n∈N satisfies
a LDP with rate function I if
P[Xn ≈ x] ≈ exp{−nI(x)}
for some non-negative function I : R → [0, ∞]
Horacio Gonz´alez Duhart TASEP: LDP via MPA

21. Large deviations
Formally,
Definition (Large deviation principle)
Let X be a Polish space. Let {Pn}n∈N be a sequence of probability
of measures on X. We say {Pn}n∈N satisfies a large deviation
principle with rate function I if the following three conditions meet:
i) I is a rate function (non-negative and lsc).
ii) lim sup
n→∞
1
n
log Pn[F] ≤ − inf
x∈F
I(x) ∀F ⊂ X closed
iii) lim inf
n→∞
1
n
log Pn[G] ≥ − inf
x∈G
I(x) ∀G ⊂ X open.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

22. Large deviations
The study of large deviations has been developed since
Varadhan unified the theory in 1966.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

23. Large deviations
The study of large deviations has been developed since
Varadhan unified the theory in 1966.
The result we will use for our proof the G¨artner-Ellis Theorem.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

24. Large deviations
The study of large deviations has been developed since
Varadhan unified the theory in 1966.
The result we will use for our proof the G¨artner-Ellis Theorem.
Theorem (G¨artner-Ellis)
Let {Xn}n∈N be a sequence of random variables on a probability
space (Ω, A, P), where Ω is a nonempty subset of R. If the limit
cumulant generating function Λ: R → R defined by
Λ(θ) = lim
n→∞
1
n log E[enθXn
]
exists and is differentiable on all R, then {Xn}n∈N satisfies a large
deviation principle with rate function I : Ω → [−∞, ∞] defined by
I(x) = sup
θ∈R
{xθ − Λ(θ)}.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

25. Idea of the proof of the main result
Now the idea is to use the MPA in calculating the function in
G¨artner-Ellis Theorem.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

26. Idea of the proof of the main result
Now the idea is to use the MPA in calculating the function in
G¨artner-Ellis Theorem.
Λ(θ) = lim
n→∞
1
n
log E enθXn
= lim
n→∞
1
n
log E exp θ
n
k=1
ξk
Horacio Gonz´alez Duhart TASEP: LDP via MPA

27. Idea of the proof of the main result
Now the idea is to use the MPA in calculating the function in
G¨artner-Ellis Theorem.
Λ(θ) = lim
n→∞
1
n
log E enθXn
= lim
n→∞
1
n
log E exp θ
n
k=1
ξk
= lim
n→∞
1
n
log
η∈{0,1}n
¯να
1/4{ξ : ξk = ηk for k ≤ n} exp θ
n
k=1
ηk
Horacio Gonz´alez Duhart TASEP: LDP via MPA

28. Idea of the proof of the main result
Now the idea is to use the MPA in calculating the function in
G¨artner-Ellis Theorem.
Λ(θ) = lim
n→∞
1
n
log E enθXn
= lim
n→∞
1
n
log E exp θ
n
k=1
ξk
= lim
n→∞
1
n
log
η∈{0,1}n
¯να
1/4{ξ : ξk = ηk for k ≤ n} exp θ
n
k=1
ηk
= lim
n→∞
1
n
log
wT
(eθ
D + E)n
v
wT (D + E)nv
Horacio Gonz´alez Duhart TASEP: LDP via MPA

29. Idea of the proof of the main result
Now the idea is to use the MPA in calculating the function in
G¨artner-Ellis Theorem.
Λ(θ) = lim
n→∞
1
n
log E enθXn
= lim
n→∞
1
n
log E exp θ
n
k=1
ξk
= lim
n→∞
1
n
log
η∈{0,1}n
¯να
1/4{ξ : ξk = ηk for k ≤ n} exp θ
n
k=1
ηk
= lim
n→∞
1
n
log
wT
(eθ
D + E)n
v
wT (D + E)nv
= lim
n→∞
1
n log wT
(eθ
D + E)n
v − 2 log 2.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

30. Idea of the proof of the main result
Having the equation
Λ(θ) = lim
n→∞
1
n log wT
(eθ
D + E)n
v − 2 log 2
we would like to simplify it and use G¨artner-Ellis Theorem.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

31. Idea of the proof of the main result
Having the equation
Λ(θ) = lim
n→∞
1
n log wT
(eθ
D + E)n
v − 2 log 2
we would like to simplify it and use G¨artner-Ellis Theorem.
However, even when we have a explicit form of D, E, v, and
w, the term wT
(eθ
D + E)n
v is not easy to handle, and so we
split into a lower bound and an upper bound.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

32. Upper bound
The upper bound comes from noticing that (eθ
D + E) is a
Toeplitz operator (constant diagonals in the matrix), and v
and w live on a family of weighted spaces 2
s and its dual,
respectively.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

33. Upper bound
The upper bound comes from noticing that (eθ
D + E) is a
Toeplitz operator (constant diagonals in the matrix), and v
and w live on a family of weighted spaces 2
s and its dual,
respectively.
We then use Cauchy-Schwarz inequality and optimise over the
parameter s of permissible weights.
wT
(eθ
D + E)n
v ≤ |w| 2
s
||(eθ
D + E)n
|B( 2
s )|v| 2
s
Horacio Gonz´alez Duhart TASEP: LDP via MPA

34. Lower bound
The lower bound comes from expanding the term
wT
(eθ
D + E)n
v =
n
p=1
p
j=0
f n
p,j (θ)wT
Ep−j
Dj
v
where f n
p,j (θ) are polynomials on eθ
with non-negative
coefficients.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

35. Lower bound
The lower bound comes from expanding the term
wT
(eθ
D + E)n
v =
n
p=1
p
j=0
f n
p,j (θ)wT
Ep−j
Dj
v
where f n
p,j (θ) are polynomials on eθ
with non-negative
coefficients.
We then find a lower bound when j = 0:
wT
(eθ
D + E)n
v ≥
n
p=1
f n
p,0(θ)wT
Ep
v.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

36. Lower bound
The lower bound comes from expanding the term
wT
(eθ
D + E)n
v =
n
p=1
p
j=0
f n
p,j (θ)wT
Ep−j
Dj
v
where f n
p,j (θ) are polynomials on eθ
with non-negative
coefficients.
We then find a lower bound when j = 0:
wT
(eθ
D + E)n
v ≥
n
p=1
f n
p,0(θ)wT
Ep
v.
And another when j = p:
wT
(eθ
D + E)n
v ≥
n
p=1
f n
p,p(θ)wT
Dp
v.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

37. Summarising. . .
TASEP: Create particles at rate α. Move them to the right at rate 1.
One particle per site.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

38. 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 the
invariant measure.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

39. 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 the
invariant measure.
LDP: Find the exponential rate of convergence to 0 of unlikely events.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

40. 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 the
invariant 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 the
MPA.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

41. 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 the
invariant 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 the
MPA.
Horacio Gonz´alez Duhart TASEP: LDP via MPA

42. References
F. den Hollander. Large Deviations, volume 14 of Fields Institute
Monographs. American Mathematical Society, Providence, RI, 2000.
B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier. Exact solution
of a 1D asymmetric exclusion model using a matrix formulation. J.
Phys. A, 26(7):1493,1993.
S. Großkinsky. Phase transitions in nonequilibrium stochastic
particle systems with local conservation laws. PhD thesis, TU
Munich, 2004.
T. M. Liggett. Ergodic theorems for the asymmetric simple
exclusion process. Trans. Amer. Math. Soc., 213:237-261,1975.
H. G. Duhart, P. M¨orters, and J. Zimmer. The Semi-Infinite
Asymmetric Exclusion Process: Large Deviations via Matrix
Products. ArXiv e-prints, arXiv:1411.3270v1, November 2014.
Go play with the applet!
Horacio Gonz´alez Duhart TASEP: LDP via MPA

43. Bath, UNAM and CIMAT Workshop Series
11 November 2015, CIMAT