TASEP: LDP via MPA

download TASEP: LDP via MPA

of 43

  • date post

    11-Jul-2015
  • Category

    Science

  • view

    83
  • download

    1

Embed Size (px)

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 (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

    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

    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