Frank Redig Joint work with G. Carinci C. Giardin a M. Jara · Some news on the symmetric inclusion...

Some news on the symmetric inclusion process

Frank RedigJoint work with

G. CarinciC. Giardina

M. Jara

January 10, 2017

Frank Redig Joint work with G. Carinci C. Giardina M. Jara Some news on the symmetric inclusion process

BackgroundThe SIP(κ) on Z is the process on Ω = NZ with formal generator

Lf (η) =∑i

ηi (κ+ ηi+1)(f (ηi ,i+1)− f (η)

)+

∑i

ηi+1(κ+ ηi )(f (ηi+1,i )− f (η)

)More generally, for S a countable set, and p : S × S → [0,∞) anirreducible symmetric function the SIP(κ, p) on S is the process onNS with formal generator

Lf (η) =∑i ,j

p(i , j)ηi (κ+ ηj)(f (ηi ,j)− f (η)

)REMARK

If S is infinite, some conditions on p are required in order to ensureexistence. E.g. S = Zd then symmetric translation invariant finiterange p is ok.


Basic properties1. Invariant (reversible) product measures with marginals

νλ(n) =1

Zκ

λn

n!

Γ(κ+ n)

Γ(κ)

with Zκ = (1− λ)−κ. For κ integer, these are negativebinomials.

2. Self-duality. Define for ξ, η ∈ Ω, ξ finite

D(ξ, η) =∏i

d(ξi , ηi )

with

d(k, n) =n!

(n − k)!

Γ(κ)

Γ(κ+ k)

These polynomials (indexed by k) are naturally related to thereversible product measures via∑

n

νλ(n)d(k , n) =

(λ

1− λ

)k

=

(∑n

νλ(n)d(1, n)

)k


With these polynomials, we have the self-duality relation, i.e., forall ξ, η, with ξ finite

EηD(ξ, η(t)) = EξD(ξ(t), η)

Two examples:

a) One dual particle

Eη(ηi (t)) = κEηD(δi , ηt)

= κEδiD(ξ(t), η)

= κErandom walki

(ηX (t)

κ

)=

∑j

pt(i , j)ηj


b) Two dual particles

D(δx + δy , η) =

ηxηyκ2 if x 6= yηx (ηx−1)κ(κ+1) if x = y

and so we have

EηD(δx + δy , η(t)) = Etwo sip particlesx ,y D(δx(t) + δy(t), η)

So in order to have information on second moments, oneneeds two dual particles.


c) One dual particle, continued.Start the SIP from a product initial measure ν with localdensity

∫ηidν = ρ(i). Then at time t > 0 we have

ρt(i) =

∫Eη(ηi (t))dν =

∫ ∑j

pt(i , j)ηjdν =∑j

pt(i , j)ρ(j)

If we now choose a slowly varying density ρ(N)(i) = ρ(i/N)(where ρ : R→ R), and speed up time diffusively

ρ(N)tN2(i) =

∑j

ptN2(i , j)ρ(j/N)→ Ebrownian motionx ρ(Bt)

provided i/N → x ∈ R. Hence the density profile converges tothe solution of the diffusion equation

∂ρ(x , t)

∂t− κ∂

2ρ(x , t)

∂x2= 0


d) Two dual particles continued.Start the SIP from a product initial measure ν with localdensity

∫ηidν = ρ(i). Define the time-dependent local

density covariance

Ξ(t, x , y ; ν) =

∫Eη(ηx(t)− ρt(x))(ηy (t)− ρt(y))ν(dη)

where we remind

ρt(x) =∑y

pt(x , y)ρ(y)

with ρ(y) =∫ηydν. Then we have the following formula,

using duality with ≤ 2 dual particles.


Ξ(t, x , y ; ν)

= (1 + (c∗ − 1)δx ,y )(ECx ,y ;x ,y

(ρ(Xt)ρ(Yt)− ρ(X ′t)ρ(Y ′t )

))+ Ex ,y

(I (Xt = Yt)

(1

c∗χ(Xt)− ρ(Xt)

2

))+ δx ,y ((c∗ − 1)ρt(x)2 + ρt(x)) (1)

with c∗ = 1 + 12κ , and χ(x) =

∫ηx(ηx − 1)dν


If we are in local equilibrium, i.e., if the marginals are the reversibleproduct measure marginals, then the term

Ex ,y

(I (Xt = Yt)

(1

c∗χ(Xt)− ρ(Xt)

2

))drops because in that case

1

c∗χ(x)− ρ(x)2

is zero for all x . So the second term in Ξ(t, x , y ; ν) can be viewedas a contribution to the time-dependent correlations due to thedeviation from local equilibrium.


We can use this formula to compute the variance of the densityfluctuation field, i.e., we start from a product measure νN with∫

ηxdνN = ρ( xN )

1

c∗

∫ηx(ηx − 1) = ρ2( x

N )

K2( xN ) = ρ2( x

N )− ρ( xN )2 (2)

REMARK

K2 is zero when the measures νN have the reversible productmarginals, so it measures (in second order) the deviation from localequilibrium.

We define the density fluctuation field

XN(Φ, η, t) =1√N

∑i

Φ( iN )(ηi (tN

2)− ρtN2( iN ))

where Φ is a test function.Frank Redig Joint work with G. Carinci C. Giardina M. Jara Some news on the symmetric inclusion process

Theorem

limN→∞

EνN(XN(Φ, η, t)2

)=

∫Φ(x)2

((c∗ − 1)ρt(x)2 + ρt(x)

)+

∫ ∫ ∫Φ(x)Φ(y)K2(u)πt(x , u)πt(y , u) dxdydu (3)

where πt denotes the transition density of the limiting Brownianmotion, i.e. Bκt .


REMARK

I The first term is what we expect when we are in localequilibrium.

I The second term is only present when we do not start in localequilibrium. Simplest example is when K2 is constant, we findthe contribution

K2

∫Φ(x)Φ(y)π2t(x , y)dxdy = K2

∫Φ(k)2e−k

2κtdk

which converges slowly (as 1/√t) to zero when t →∞.


The proof is based on a coupling between two sip particles and twoindependent particles such that we have the following twoproperties

a) Unbiased:

E((Xt ,Yt)− (X ′t ,Y

′t )|(X ′s ,Y ′s ) : 0 ≤ s ≤ t

)= (0, 0)

b) Not too far from each other:

1√t‖(Xt ,Yt)− (X ′t ,Y

′t )‖ → 0, as t →∞

The existence of such a coupling is plausible if one realizes that theSIP particles only interact when they are at neighboring positionsand then create discrepancies which have no preferred sign, due tothe symmetry. This coupling takes care of the first term in (1) (thecontribution of this term vanishes, despite the fact it looks“threatening”, i.e., not of the “correct order”).


Interesting: the sign of correlations in the course of time

1) I It is well-known that for the SEP, when starting from aproduct measure, the measure at time t > 0 has negativecorrelations. This follows from Liggett’s comparison inequalitybetween independent walkers and SEP walkers.

I In that (SEP) context there is no distinction between productmeasure and local equilibrium product measures, as all productmeasures are products of Bernoulli distributions (i.e., are localequilibrium product measures).


2) I For SIP we showed that when starting from a product localequilibrium measure, the measure at time t > 0 has positivecorrelations. This follows from a similar comparison inequality,in the other sense.

I Notice that the SIP is not monotone, so this is not related toFKG, rather to bosonic versus fermionic interaction.

I The SIP is monotone in a generalized sense, namely under theweaker order ν ≤ ν′ when

∫D(ξ, η)ν(dη) ≤

∫D(ξ, η)ν′(dη)

for all ξ.


3) I Now we can check what happens for a general productmeasure, from the formula (1). We see that the measure attime t can have positive or negative correlations, depending onthe second term (the first term is positive for x 6= y).

I So the fact that we have to start from from a product localequilibrium measure, in order for the measure at time t > 0 tohave positive correlations is “sharp”.

I This is even true for independent random walkers: thecorrelations at time t > 0 can be positive or negative, and arezero when started from a product of Poisson measures (localequilibrium).

I It is an interesting open problem to characterize the measureswhich under evolution of SIP produce positive correlations attime t > 0 i.e., develop the analogue of Branden-Liggett’stheorem for SIP.


Exact formula for two interacting particlesConsider a particle system with generator of the form

Lf (η) =∑i

c+(ηi , ηi+1)(f (ηi ,i+1)− f (η)

)+

∑i

c−(ηi+1, ηi )(f (ηi+1,i )− f (η)

)We assume

1. Linearity of rates

c+(2, 0) + c−(2, 0) = 2(c+(1, 0) + c−(1, 0)) = α

2. Constant asymmetry

c+(k ,m)

c−(k ,m)= C

If there is asymmetry it has to be produced by a constantfactor in front of the corresponding symmetric term. We thendefine q = c+(1, 0)/α, p = 1− p, and γ = c+(1, 1) + c−(1, 1),and finally θ = γ

α − 1.


Define x1(t), x2(t) the positions of two particles evolving accordingto the above defined generator, and define

w(t) = |x2(t)− x1(t)|, u(t) = x1(t) + x2(t)

Then w is Markov and u is “subjected” to w , i.e., given therealization of w the process u evolves according to

L f (u|w) = 2α(1 + θ

2 1w=1

)[q(f (u + 1)− f (u))

+ p(f (u − 1)− f (u))] (4)

which is an (a)symmetric (when p 6= q) translation-invariantrandom walk with time-dependent (via w) total jump rate. Theprocess w , in turn is behaving like a reflected random walk movingat different rates when at 0, at 1, or > 1.

G (w ,w ′; k , λ) =∑u

e−iku∫ ∞

0Pt((0,w); (u,w ′))e−λtdt


Theorem

G (w ,w ′;κ, λ)

=f θλ,κ(w ,w ′)

Z(0)λ,κ

ζ |w ′−w |−1λ,κ + ζw

′+w−1λ,κ ·

2Z

(0)λ,κ

Zλ,κ− 1

with νκ = cos(κ)− i(q − p) sin(κ),

ζλ,κ := ζ (xλ,κ) = xλ,κ −√x2λ,κ − 1, xλ,κ :=

1

νκ

(1 +

λ

2α

)

Zλ,κ = Z(0)λ,κ + 2θα(xλ,κ − νκ), Z

(0)λ,κ := ανκ(ζ−2

λ,κ − 1)

and

f θλ,κ(w ,w ′) = 1 + 1w ′=0 ·1

2θ − 1 + θ(ν−1

κ ζλ,κ − 1)1w=0


A first application is the behavior of two SIP particles in thecondensation regime, i.e., with κ = ε→ 0. Denote their positionsby X ε(t),Y ε(t) and putW ε(t) = |X ε(t)− Y ε(t)|,Uε(t) = X ε(t) + Y ε(t). Then we havefor all λ > 0, k ,m ∈ R

limε→0

∫ ∞0

E0,0e−ikUε(ε−3t)−imW ε(ε−3t)e−λtdt

=

√k2 + λ√

k2 + λ+ k2 + λ2

Ψ(k ,m, λ)

+1

2

1√k2 + λ+ k2 + λ

2

(5)


where

Ψ(k ,m, λ) =

∫ ∞0

E0,0e−i(k(xt+yt)+m|xt−yt |)e−λtdt

where xt , yt are independent Brownian motions.


A second application is the second-order Boltzmann Gibbs principlewhich is the essential ingredient in the proof of the convergence ofthe density fluctuations for WASIP to the stochastic Burgersequation. From theorem (??) we obtain, by explicit asymptoticanalysis around k ,m→ 0, λ→ 0 the so-called (strong) one-blockestimate.


TheoremLet Φ be a test function

F (n, η,Φ) =∑x

Φ′( xn )

((ηx − ρ)(ηx+1 − ρ)− 1

ln

ln∑r=1

(ηx − ρ)(ηx+r − ρ)

)

where ln/n→ 0, ln →∞. Then we have the estimate

lim supn→∞

1

lnn

∣∣∣∣∫ ∞0

Eνλ(ρ)(F (n, η,Φ)F (n, η(t),Φ)dt

∣∣∣∣ <∞More informally,

1

n2

∣∣∣∣∫ ∞0

Eνλ(ρ)(F (n, η,Φ)F (n, η(t),Φ)dt

∣∣∣∣ ≤ Clnn


This estimate allows to perform the “replacement” of∫ t

0

∑x

Φ′(x/n)(ηx(s))(ηx+1(s))

by ∫ t

0

∑x

Φ′(x/n)(ηlnx (s))2

with

ηlnx =1

ln

ln∑j=1

ηx+j

which gives then a closed equation for the density fluctuation field.


Frank Redig Joint work with G. Carinci C. Giardin a M. Jara · Some news on the symmetric inclusion...

Documents

Transcript of Frank Redig Joint work with G. Carinci C. Giardin a M. Jara · Some news on the symmetric inclusion...