Invariant measures of discrete interacting particles systems: - - PowerPoint PPT Presentation

Invariant measures of discrete interacting particles systems: algebraic aspects Luis Fredes

(joint work with J.F. Marckert).

École d’été St. Flour 2018

Luis Fredes Invariant measures of discrete IPS 1 / 23

Particle system Define a set of κ colors Eκ := {0, 1, . . . , κ − 1} for κ ∈ {∞, 2, 3, . . . }. An interacting particle system (IPS) is a stochastic process (ηt)t∈R+ embedded on a graph G = (V , E) with configuration space in SV. We will work with S = Eκ and with G = Z, Z/nZ. .

Luis Fredes Invariant measures of discrete IPS 2 / 23

Particle system Define a set of κ colors Eκ := {0, 1, . . . , κ − 1} for κ ∈ {∞, 2, 3, . . . }. An interacting particle system (IPS) is a stochastic process (ηt)t∈R+ embedded on a graph G = (V , E) with configuration space in SV. We will work with S = Eκ and with G = Z, Z/nZ. .

Luis Fredes Invariant measures of discrete IPS 2 / 23

TASEP t t + ∆t

Luis Fredes Invariant measures of discrete IPS 3 / 23

TASEP t t + ∆t

∆t ∼ exp(1)

Luis Fredes Invariant measures of discrete IPS 3 / 23

Contact process t t + ∆t t + ∆t

∆t ∼ exp(1)

Luis Fredes Invariant measures of discrete IPS 4 / 23

Contact process t t + ∆t t + ∆t

∆t ∼ exp(1) ∆t ∼ exp(2λ)

Luis Fredes Invariant measures of discrete IPS 4 / 23

General case t t + ∆t

Luis Fredes Invariant measures of discrete IPS 5 / 23

General case t t + ∆t L

Luis Fredes Invariant measures of discrete IPS 5 / 23

General case t t + ∆t L

∆t ∼ exp(T[ ]) |

Luis Fredes Invariant measures of discrete IPS 5 / 23

Invariant measure of particle system

Definition

A distribution µ on E V

κ is said to be invariant if ηt ∼ µ for

any t ≥ 0, when η0 ∼ µ.

Luis Fredes Invariant measures of discrete IPS 6 / 23

Invariant measure of particle system

Definition

A distribution µ on E V

κ is said to be invariant if ηt ∼ µ for

any t ≥ 0, when η0 ∼ µ. Usual questions in the topic:

Luis Fredes Invariant measures of discrete IPS 6 / 23

Invariant measure of particle system

Definition

A distribution µ on E V

κ is said to be invariant if ηt ∼ µ for

any t ≥ 0, when η0 ∼ µ. Usual questions in the topic: Existence?

Luis Fredes Invariant measures of discrete IPS 6 / 23

Invariant measure of particle system

Definition

A distribution µ on E V

κ is said to be invariant if ηt ∼ µ for

any t ≥ 0, when η0 ∼ µ. Usual questions in the topic: Existence? Uniqueness?

Luis Fredes Invariant measures of discrete IPS 6 / 23

Invariant measure of particle system

Definition

A distribution µ on E V

κ is said to be invariant if ηt ∼ µ for

any t ≥ 0, when η0 ∼ µ. Usual questions in the topic: Existence? Uniqueness? Convergence?

Luis Fredes Invariant measures of discrete IPS 6 / 23

Invariant measure of particle system

Definition

A distribution µ on E V

κ is said to be invariant if ηt ∼ µ for

any t ≥ 0, when η0 ∼ µ. Usual questions in the topic: Existence? Uniqueness? Convergence? Rate of convergence?

Luis Fredes Invariant measures of discrete IPS 6 / 23

Invariant measure of particle system

Definition

A distribution µ on E V

κ is said to be invariant if ηt ∼ µ for

any t ≥ 0, when η0 ∼ µ. Usual questions in the topic: Existence? Uniqueness? Convergence? Rate of convergence? Simple representation?

Luis Fredes Invariant measures of discrete IPS 6 / 23

Invariant measure of particle system

Definition

A distribution µ on E V

κ is said to be invariant if ηt ∼ µ for

any t ≥ 0, when η0 ∼ µ. Usual questions in the topic: Existence? Uniqueness? Convergence? Rate of convergence? Simple representation? (Integrability)

Luis Fredes Invariant measures of discrete IPS 6 / 23

Some things (not much) are known about I.I.D. random invariant distributions of IPS.

Luis Fredes Invariant measures of discrete IPS 7 / 23

Some things (not much) are known about I.I.D. random invariant distributions of IPS. [ Andjel ’82,

Ferrari ’93, Balazs–Rassoul-Agha–Seppalainen–Sethuraman ’07, Borodin–Corwin ’11, Fajfrová–Gobron–Saada ’16...]

Luis Fredes Invariant measures of discrete IPS 7 / 23

Some things (not much) are known about I.I.D. random invariant distributions of IPS. [ Andjel ’82,

Ferrari ’93, Balazs–Rassoul-Agha–Seppalainen–Sethuraman ’07, Borodin–Corwin ’11, Fajfrová–Gobron–Saada ’16...]

What about another type of distribution?

Luis Fredes Invariant measures of discrete IPS 7 / 23

Some things (not much) are known about I.I.D. random invariant distributions of IPS. [ Andjel ’82,

Ferrari ’93, Balazs–Rassoul-Agha–Seppalainen–Sethuraman ’07, Borodin–Corwin ’11, Fajfrová–Gobron–Saada ’16...]

What about another type of distribution? MARKOV!!!!!!

Luis Fredes Invariant measures of discrete IPS 7 / 23

Consider a Markov distribution (MD) (ρ, M), with Markov Kernel (MK) M of memory m = 1 and ρ the invariant measure of M, i.e. for any x ∈ E Ja,bK

κ

P(XJa, bK = x) = ρxa

b−1

Y

j=a

Mxj,xj+1 .

Luis Fredes Invariant measures of discrete IPS 8 / 23

Consider a Markov distribution (MD) (ρ, M), with Markov Kernel (MK) M of memory m = 1 and ρ the invariant measure of M, i.e. for any x ∈ E Ja,bK

κ

P(XJa, bK = x) = ρxa

b−1

Y

j=a

Mxj,xj+1=: γ(x).

Luis Fredes Invariant measures of discrete IPS 8 / 23

Denote by µt the measure of the process on E Z

κ at time

t ≥ 0. t = 0

X1 X2 X3 X4 X5 X6 X7

X ∼ µ0 = γ t > 0

Y1 Y2 Y3 Y4 Y5 Y6 Y7

Y ∼ µt = γ

Evolution under T

Luis Fredes Invariant measures of discrete IPS 9 / 23

Definition

A process (Xk, k ∈ Z/nZ) taking its values in E Z/nZ

κ

is said to have a Gibbs distribution G(M) characterized by a MK M, if for any x ∈ E J0,n−1K

κ

, P(XJ0, n − 1K = x) = Qn−1

j=0 Mxj,xj+1 mod n

Trace(Mn) .

Luis Fredes Invariant measures of discrete IPS 10 / 23

Definition

A process (Xk, k ∈ Z/nZ) taking its values in E Z/nZ

κ

is said to have a Gibbs distribution G(M) characterized by a MK M, if for any x ∈ E J0,n−1K

κ

, P(XJ0, n − 1K = x) = Qn−1

j=0 Mxj,xj+1 mod n

Trace(Mn) =: ν(x).

Luis Fredes Invariant measures of discrete IPS 10 / 23

Y6 X6 Y7 X7 Y8 X8 Y9 X9 Y10 X10 Y5 X5 Y1 X1 Y2 X2 Y3 X3 Y4 X4

t = 0 t > 0 X ∼ µ0 = ν Y ∼ µt = ν

Evolution under T

Luis Fredes Invariant measures of discrete IPS 11 / 23

Theorem 1 (F- Marckert ’17)

Let Eκ be finite, L = 2, m = 1. If M > 0 then the following statements are equivalent for the couple (T, M):

1 (ρ, M) is invariant by T on Z. 2 G(M) is invariant by T on Z/nZ, for all n ≥ 3 3 G(M) is invariant by T on Z/7Z 4 A finite system of equations of degree 7 in M and linear

in T.

Luis Fredes Invariant measures of discrete IPS 12 / 23

Suppose µt is described with a MD. For any x ∈ E J1,nK

κ

we define

LineM,T

n

(x) := ∂ ∂t µt

J1,nK(x)

where w k differs from x in w kJk, k + 1K = (u, v).

Luis Fredes Invariant measures of discrete IPS 13 / 23

Suppose µt is described with a MD. For any x ∈ E J1,nK

κ

we define

LineM,T

n

(x) := ∂ ∂t µt

J1,nK(x)

where w k differs from x in w kJk, k + 1K = (u, v).

Definition

A (ρ, M) MD under its invariant distribution is said to be AI by T on the line when Linen ≡ 0, for all n ∈ N.

Luis Fredes Invariant measures of discrete IPS 13 / 23

But you are CHEATING!!!

Luis Fredes Invariant measures of discrete IPS 13 / 23

Suppose µt is described with a MD. For any x ∈ E J1,nK

κ

we define

LineM,T

n

(x) := ∂ ∂t µt

J1,nK(x)

where w k differs from x in w kJk, k + 1K = (u, v).

Definition

A (ρ, M) MD under its invariant distribution is said to be AI by T on the line when Linen ≡ 0, for all n ∈ N.

Luis Fredes Invariant measures of discrete IPS 13 / 23

Suppose µt is described with a MD. For any x ∈ E J1,nK

κ

we define

LineM,T

n

(x) := ∂ ∂t µt

J1,nK(x)

=

Mass creation rate of x

− Mass destruction rate of x

where w k differs from x in w kJk, k + 1K = (u, v).

Definition

A (ρ, M) MD under its invariant distribution is said to be AI by T on the line when Linen ≡ 0, for all n ∈ N.

Luis Fredes Invariant measures of discrete IPS 13 / 23

Suppose µt is described with a MD. For any x ∈ E J1,nK

κ

we define

LineM,T

n

(x) := ∂ ∂t µt

J1,nK(x)

= lim

h→0

X

w∈E Z

κ

P(ηt+hJ1, nK = x|ηt = w) − Mass destruction rate of x

where w k differs from x in w kJk, k + 1K = (u, v).

Definition

A (ρ, M) MD under its invariant distribution is said to be AI by T on the line when Linen ≡ 0, for all n ∈ N.

Luis Fredes Invariant measures of discrete IPS 13 / 23

Suppose µt is described with a MD. For any x ∈ E J1,nK

κ

we define

LineM,T

n

(x) := ∂ ∂t µt

J1,nK(x)

= lim

h→0

X

w∈E Z

κ

P(ηt+hJ1, nK = x|ηt = w) − lim

h→0

X

w∈E Z

κ

P(ηt+h = w|ηtJ1, nK = x)

where w k differs from x in w kJk, k + 1K = (u, v).

Definition

A (ρ, M) MD under its invariant distribution is said to be AI by T on the line when Linen ≡ 0, for all n ∈ N.

Luis Fredes Invariant measures of discrete IPS 13 / 23

Suppose µt is described with a MD. For any x ∈ E J1,nK

κ

we define

LineM,T

n

(x) := ∂ ∂t µt

J1,nK(x)

= lim

h→0

X

w∈E Z

κ

P(ηt+hJ1, nK = x|ηt = w) − X

x1,x0, xn+1,xn+22Eκ

n

X

j=0

γ(xJ−1, n + 2K) X

(u,v)∈E 2

κ

T[xj,xj+1|u,v]

where w k differs from x in w kJk, k + 1K = (u, v).

Definition

A (ρ, M) MD under its invariant distribution is said to be AI by T on the line when Linen ≡ 0, for all n ∈ N.

Luis Fredes Invariant measures of discrete IPS 13 / 23

Suppose µt is described with a MD. For any x ∈ E J1,nK

κ

we define

LineM,T

n

(x) := ∂ ∂t µt

J1,nK(x)

= X

x1,x0, xn+1,xn+22Eκ

n

X

j=0

X

(u,v)∈E 2

κ

γ(w jJ−1, n + 2K)T[u,v|xj,xj+1] − X

x1,x0, xn+1,xn+22Eκ

n

X

j=0

γ(xJ−1, n + 2K) X

(u,v)∈E 2

κ

T[xj,xj+1|u,v]

where w k differs from x in w kJk, k + 1K = (u, v).

Definition

A (ρ, M) MD under its invariant distribution is said to be AI by T on the line when Linen ≡ 0, for all n ∈ N.

Luis Fredes Invariant measures of discrete IPS 13 / 23

Suppose µt is described with a MD. For any x ∈ E J1,nK

κ

we define

LineM,T

n

(x) := ∂ ∂t µt

J1,nK(x)

= X

x1,x0, xn+1,xn+22Eκ

n

X

j=0

B @ X

(u,v)∈E 2

κ

γ(w jJ−1, n + 2K)T[u,v|xj,xj+1] −γ(xJ−1, n + 2K) X

(u,v)∈E 2

κ

T[xj,xj+1|u,v] 1 A

where w k differs from x in w kJk, k + 1K = (u, v).

Definition

A (ρ, M) MD under its invariant distribution is said to be AI by T on the line when Linen ≡ 0, for all n ∈ N.

Luis Fredes Invariant measures of discrete IPS 13 / 23

Suppose µt is described with a MD. For any x ∈ E J1,nK

κ

we define

LineM,T

n

(x) := ∂ ∂t µt

J1,nK(x)

= X

x1,x0, xn+1,xn+22Eκ

n

X

j=0

B @ X

(u,v)∈E 2

κ

⇣ ρx1 Y

1kn+1 k62{j1,j,j+1}

Mxk,xk+1 ⌘ Mxj1,uMu,vMv,xj+2T[u,v|xj,xj+1] − ⇣ ρx1

n+1

Y

k=−1

Mxk,xk+1 ⌘ Tout

[xj,xj+1]

1 C A

where w k differs from x in w kJk, k + 1K = (u, v).

Definition

A (ρ, M) MD is said to be invariant by T on the line when Linen ≡ 0, for all n ∈ N.

Luis Fredes Invariant measures of discrete IPS 14 / 23

Suppose µt is described with a MD. For any x ∈ E J1,nK

κ

we define

LineM,T

n

(x) := ∂ ∂t µt

J1,nK(x)

= X

x1,x0, xn+1,xn+22Eκ

n

X

j=0

ρx1

n+1

Y

k=−1

Mxk,xk+1 ! × @ @ X

(u,v)∈E 2

κ

T[u,v|xj,xj+1] Mxj1,uMu,vMv,xj+2 Mxj1,xjMxj,xj+1Mxj+1,xj+2 1 A − Tout

[xj,xj+1]

1 A

where w k differs from x in w kJk, k + 1K = (u, v).

Definition

A (ρ, M) MD is said to be invariant by T on the line when Linen ≡ 0, for all n ∈ N.

Luis Fredes Invariant measures of discrete IPS 14 / 23

Suppose µt is described with a MD. For any x ∈ E J1,nK

κ

we define

LineM,T

n

(x) := ∂ ∂t µt

J1,nK(x)

= X

x1,x0, xn+1,xn+22Eκ

n

X

j=0

ρx1

n+1

Y

k=−1

Mxk,xk+1 ! × @ @ X

(u,v)∈E 2

κ

T[u,v|xj,xj+1] Mxj1,uMu,vMv,xj+2 Mxj1,xjMxj,xj+1Mxj+1,xj+2 1 A − Tout

[xj,xj+1]

1 A | {z }

Zxj1,xj ,xj+1,xj+2

Definition

A (ρ, M) MD is said to be invariant by T on the line when Linen ≡ 0, for all n ∈ N.

Luis Fredes Invariant measures of discrete IPS 14 / 23

Definitions

• Define for every a, b, c, d ∈ Eκ

Z M,T

a,b,c,d :=

@ X

(u,v)∈E 2

κ

T[u,v|b,c] Ma,uMu,vMv,d Ma,bMb,cMc,d 1 A − Tout

[b,c].

Luis Fredes Invariant measures of discrete IPS 15 / 23

Definition

A Gibbs measure with kernel M is said to be invariant by T

• n Z/nZ when Cyclen ≡ 0, where

Cyclen(x) :=

n−1

X

j=0

X

u,v∈Eκ

⇣ ν(w j)T[u,v|xj,xj+1 mod n] − ν(x)Tout

[xj,xj+1 mod n]

⌘ where w k differs from x in w kJk, k + 1 mod nK = (u, v).

Luis Fredes Invariant measures of discrete IPS 16 / 23

Definition

A Gibbs measure with kernel M is said to be invariant by T

• n Z/nZ when Cyclen ≡ 0, where

Cyclen(x) := ν(x) ×

n−1

X

j=0

Zxj−1,xj,xj+1,xj+2 where w k differs from x in w kJk, k + 1 mod nK = (u, v).

Luis Fredes Invariant measures of discrete IPS 16 / 23

Extensions

Luis Fredes Invariant measures of discrete IPS 17 / 23

Memory and amplitude

Theorem 1 (F- Marckert)

Let Eκ be finite, L = 2, m = 1. If M > 0 then the following statements are equivalent:

1 (ρ, M) is invariant by T on Z. 2 G(M) is invariant by T on Z/nZ, for all n ≥ 3 3 G(M) is invariant by T on Z/7Z 4 A finite system of equations of degree 7 in M and linear

in T.

Luis Fredes Invariant measures of discrete IPS 18 / 23

Memory and amplitude

Theorem 1- Strongest form (F- Marckert ’17)

Let Eκ be finite, L ≥ 2, m ∈ N. If M > 0 then the following statements are equivalent:

1 (ρ, M) is invariant by T on Z. 2 G(M) is invariant by T on Z/nZ, for all n ≥ m + L 3 G(M) is invariant by T on Z/hZ 4 A finite system of equations of degree h in M and linear

in T. h := 4m + 2L − 1

Luis Fredes Invariant measures of discrete IPS 18 / 23

Other extensions: Theorem 1 when κ = ∞.

Luis Fredes Invariant measures of discrete IPS 19 / 23

Other extensions: Theorem 1 when κ = ∞. I.I.D. invariant measures on Zd. 11 00 − − − − − − − − − − → 01 10

Luis Fredes Invariant measures of discrete IPS 19 / 23

Other extensions: Theorem 1 when κ = ∞. I.I.D. invariant measures on Zd. 11 00

T 1 1 0 0 0 1 1 0

− − − − − − − − − − → 01 10

Luis Fredes Invariant measures of discrete IPS 19 / 23

Other extensions: Theorem 1 when κ = ∞. I.I.D. invariant measures on Zd. 11 00

T 1 1 0 0 0 1 1 0

− − − − − − − − − − → 01 10 We link our results with the TASEP’s matrix ansatz.

Luis Fredes Invariant measures of discrete IPS 19 / 23

Other extensions: Theorem 1 when κ = ∞. I.I.D. invariant measures on Zd. 11 00

T 1 1 0 0 0 1 1 0

− − − − − − − − − − → 01 10 We link our results with the TASEP’s matrix ansatz. Problem: MK with zero entries.

Luis Fredes Invariant measures of discrete IPS 19 / 23

Applications

Luis Fredes Invariant measures of discrete IPS 20 / 23

Theorem 3 (F.- Marckert ’17)

Consider κ < ∞. Consider an IRM T with amplitude L, which is not identically 0. If for infinitely many integers n the IPS with IRM T possesses an absorbing subset Sn of E Z/nZ

κ

, with ? ( Sn ( E Z/nZ

κ

. Then, there does not exist any MD with any memory m with full support, invariant by T on the line.

Luis Fredes Invariant measures of discrete IPS 21 / 23

Theorem 3 (F.- Marckert ’17)

Consider κ < ∞. Consider an IRM T with amplitude L, which is not identically 0. If for infinitely many integers n the IPS with IRM T possesses an absorbing subset Sn of E Z/nZ

κ

, with ? ( Sn ( E Z/nZ

κ

. Then, there does not exist any MD with any memory m with full support, invariant by T on the line.

Corollary

The contact process do not have a MD of any memory m ≥ 0 as invariant distribution.

Luis Fredes Invariant measures of discrete IPS 21 / 23

Summary of other applications The case κ = 2, m = 1 and L = 2 is totally explicitly solved.

Luis Fredes Invariant measures of discrete IPS 22 / 23

Summary of other applications The case κ = 2, m = 1 and L = 2 is totally explicitly solved. For κ < ∞, L = 2 and m = 1 we have an algorithm to find the set of all possible M MK which are invariant for a given T.

Luis Fredes Invariant measures of discrete IPS 22 / 23

Summary of other applications The case κ = 2, m = 1 and L = 2 is totally explicitly solved. For κ < ∞, L = 2 and m = 1 we have an algorithm to find the set of all possible M MK which are invariant for a given T. Examples of I.I.D. invariant measures on Z2.

Luis Fredes Invariant measures of discrete IPS 22 / 23

Summary of other applications The case κ = 2, m = 1 and L = 2 is totally explicitly solved. For κ < ∞, L = 2 and m = 1 we have an algorithm to find the set of all possible M MK which are invariant for a given T. Examples of I.I.D. invariant measures on Z2. Zero range, voter model, etc. Also when we make mild changes on these models we have some results.

Luis Fredes Invariant measures of discrete IPS 22 / 23

Summary of other applications The case κ = 2, m = 1 and L = 2 is totally explicitly solved. For κ < ∞, L = 2 and m = 1 we have an algorithm to find the set of all possible M MK which are invariant for a given T. Examples of I.I.D. invariant measures on Z2. Zero range, voter model, etc. Also when we make mild changes on these models we have some results. We find an IRM T which possesses some hidden Markov chain as invariant distributions. It is done using a projection from E3 to E2.

Luis Fredes Invariant measures of discrete IPS 22 / 23

Thank you!

Luis Fredes Invariant measures of discrete IPS 23 / 23