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 E κ := { 0 , 1 , . . . , κ − 1 } for κ ∈ { ∞ , 2 , 3 , . . . } . An interacting particle system (IPS) is a stochastic process ( η t ) t ∈ R + embedded on a graph G = ( V , E ) with configuration space in S V . We will work with S = E κ and with G = Z , Z / n Z . . Luis Fredes Invariant measures of discrete IPS 2 / 23

Particle system Define a set of κ colors E κ := { 0 , 1 , . . . , κ − 1 } for κ ∈ { ∞ , 2 , 3 , . . . } . An interacting particle system (IPS) is a stochastic process ( η t ) t ∈ R + embedded on a graph G = ( V , E ) with configuration space in S V . We will work with S = E κ and with G = Z , Z / n Z . . 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 ∼ exp ( 1 ) t Luis Fredes Invariant measures of discrete IPS 3 / 23

Contact process t + ∆ t t + ∆ t ∆ t ∼ exp ( 1 ) t Luis Fredes Invariant measures of discrete IPS 4 / 23

Contact process t + ∆ t t + ∆ t ∆ t ∼ exp ( 1 ) ∆ t ∼ exp ( 2 λ ) t 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 L t Luis Fredes Invariant measures of discrete IPS 5 / 23

General case t + ∆ t L ∆ t ∼ exp ( T [ | ]) t Luis Fredes Invariant measures of discrete IPS 5 / 23

Invariant measure of particle system Definition κ is said to be invariant if η t ∼ µ for A distribution µ on E V any t ≥ 0, when η 0 ∼ µ . Luis Fredes Invariant measures of discrete IPS 6 / 23

Invariant measure of particle system Definition κ is said to be invariant if η t ∼ µ for A distribution µ on E V any t ≥ 0, when η 0 ∼ µ . Usual questions in the topic: Luis Fredes Invariant measures of discrete IPS 6 / 23

Invariant measure of particle system Definition κ is said to be invariant if η t ∼ µ for A distribution µ on E V any t ≥ 0, when η 0 ∼ µ . Usual questions in the topic: Existence? Luis Fredes Invariant measures of discrete IPS 6 / 23

Invariant measure of particle system Definition κ is said to be invariant if η t ∼ µ for A distribution µ on E V any t ≥ 0, when η 0 ∼ µ . Usual questions in the topic: Existence? Uniqueness? Luis Fredes Invariant measures of discrete IPS 6 / 23

Invariant measure of particle system Definition κ is said to be invariant if η t ∼ µ for A distribution µ on E V any t ≥ 0, when η 0 ∼ µ . Usual questions in the topic: Existence? Uniqueness? Convergence? Luis Fredes Invariant measures of discrete IPS 6 / 23

Invariant measure of particle system Definition κ is said to be invariant if η t ∼ µ for A distribution µ on E V any t ≥ 0, when η 0 ∼ 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 κ is said to be invariant if η t ∼ µ for A distribution µ on E V any t ≥ 0, when η 0 ∼ 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 κ is said to be invariant if η t ∼ µ for A distribution µ on E V any t ≥ 0, when η 0 ∼ 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 J a , b K κ b − 1 Y P ( X J a , b K = x ) = ρ x a . M x j , x j + 1 j = a 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 J a , b K κ b − 1 Y P ( X J a , b K = x ) = ρ x a M x j , x j + 1 =: γ ( x ) . j = a Luis Fredes Invariant measures of discrete IPS 8 / 23

Denote by µ t the measure of the process on E Z κ at time t ≥ 0. Y ∼ µ t = γ t > 0 Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Evolution under T X ∼ µ 0 = γ t = 0 X 1 X 2 X 3 X 4 X 5 X 6 X 7 Luis Fredes Invariant measures of discrete IPS 9 / 23

Definition A process ( X k , k ∈ Z / n Z ) taking its values in E Z / n Z is said κ to have a Gibbs distribution G ( M ) characterized by a MK M , if for any x ∈ E J 0 , n − 1 K , κ Q n − 1 j = 0 M x j , x j + 1 mod n P ( X J 0 , n − 1 K = x ) = . Trace ( M n ) Luis Fredes Invariant measures of discrete IPS 10 / 23

Definition A process ( X k , k ∈ Z / n Z ) taking its values in E Z / n Z is said κ to have a Gibbs distribution G ( M ) characterized by a MK M , if for any x ∈ E J 0 , n − 1 K , κ Q n − 1 j = 0 M x j , x j + 1 mod n P ( X J 0 , n − 1 K = x ) = =: ν ( x ) . Trace ( M n ) Luis Fredes Invariant measures of discrete IPS 10 / 23

Y 8 Y 7 Y 9 Y 6 Y ∼ µ t = ν t > 0 Y 10 Y 5 Y 1 Y 4 Y 2 Y 3 Evolution under T X 8 X 7 X 9 X 6 X ∼ µ 0 = ν t = 0 X 10 X 5 X 1 X 4 X 2 X 3 Luis Fredes Invariant measures of discrete IPS 11 / 23

Theorem 1 (F- Marckert ’17) Let E κ 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 / n Z , for all n ≥ 3 3 G ( M ) is invariant by T on Z / 7 Z 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 µ t is described with a MD. For any x ∈ E J 1 , n K we κ define ( x ) := ∂ Line M , T ∂ t µ t J 1 , n K ( x ) n where w k di ff ers from x in w k J k , k + 1 K = ( u , v ) . Luis Fredes Invariant measures of discrete IPS 13 / 23

Suppose µ t is described with a MD. For any x ∈ E J 1 , n K we κ define ( x ) := ∂ Line M , T ∂ t µ t J 1 , n K ( x ) n where w k di ff ers from x in w k J k , k + 1 K = ( u , v ) . Definition A ( ρ , M ) MD under its invariant distribution is said to be AI by T on the line when Line n ≡ 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 µ t is described with a MD. For any x ∈ E J 1 , n K we κ define ( x ) := ∂ Line M , T ∂ t µ t J 1 , n K ( x ) n where w k di ff ers from x in w k J k , k + 1 K = ( u , v ) . Definition A ( ρ , M ) MD under its invariant distribution is said to be AI by T on the line when Line n ≡ 0, for all n ∈ N . Luis Fredes Invariant measures of discrete IPS 13 / 23

Suppose µ t is described with a MD. For any x ∈ E J 1 , n K we κ define ( x ) := ∂ Line M , T ∂ t µ t J 1 , n K ( x ) n Mass creation rate of x = − Mass destruction rate of x where w k di ff ers from x in w k J k , k + 1 K = ( u , v ) . Definition A ( ρ , M ) MD under its invariant distribution is said to be AI by T on the line when Line n ≡ 0, for all n ∈ N . Luis Fredes Invariant measures of discrete IPS 13 / 23

Suppose µ t is described with a MD. For any x ∈ E J 1 , n K we κ define ( x ) := ∂ Line M , T ∂ t µ t J 1 , n K ( x ) n X P ( η t + h J 1 , n K = x | η t = w ) = lim h → 0 w ∈ E Z κ − Mass destruction rate of x where w k di ff ers from x in w k J k , k + 1 K = ( u , v ) . Definition A ( ρ , M ) MD under its invariant distribution is said to be AI by T on the line when Line n ≡ 0, for all n ∈ N . Luis Fredes Invariant measures of discrete IPS 13 / 23