Some models at the interface of probability theory and combinatorics: particle systems and maps. Luis Fredes PhD defense. Under the supervision of J.F. Marckert. Invariant measures of discrete interacting particles systems: algebraic aspects. 1 Survival and coexistence for spatial population models with forest fire epidemics. 2 Tree-decorated planar maps: combinatorial results. 3 Tree-decorated planar maps: local limits. 4 Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 1 / 39
Invariant measures of discrete interacting particles systems: algebraic aspects. with J.F. Marckert. Example: TASEP η t + dt η t Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 2 / 39
Invariant measures of discrete interacting particles systems: algebraic aspects. with J.F. Marckert. Example: TASEP η t + dt Exp ( 1 ) η t Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 2 / 39
Particle systems of our interests η t + dt L | Exp ( T [ ]) η t Our setting: model depends on 4 parameters: Graph G = ( V , E ) belonging to Z d , Z / n Z . Set of κ ∈ N ∪ {∞} colors E κ = { 0 , 1 , . . . , κ − 1 } . Dependence neighborhood L ≥ 2. Jump rate matrix T = [ T [ u | w ] ] { u , w ∈ E L κ } . Notation: η t ( v ) → color of vertex v at time t . Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 3 / 39
Particle systems of our interests η t + dt ∼ µ t + dt L | Exp ( T [ ]) η t ∼ µ t Our setting: model depends on 4 parameters: Graph G = ( V , E ) belonging to Z d , Z / n Z . Set of κ ∈ N ∪ {∞} colors E κ = { 0 , 1 , . . . , κ − 1 } . Dependence neighborhood L ≥ 2. Jump rate matrix T = [ T [ u | w ] ] { u , w ∈ E L κ } . Notation: η t ( v ) → color of vertex v at time t . Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 3 / 39
Invariant measure Definition Given T, a distribution µ is said to be invariant if η t ∼ µ for any t ≥ 0, when η 0 ∼ µ . η t + dt ∼ µ L Exp ( T [ | ]) η t ∼ µ Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 4 / 39
Some references: Well definition of PS given T: κ < ∞ , always well defined. [Liggett ’85] . 1 κ = ∞ , not always. Some techniques: 2 Graphic method [Harris 72’] . Functional analysis [Liggett ’73] . Stochastic domination [Andjel ’82] . Existence of invariant measures for specific PS . [Andjel ’82] . Computation of invariant measures for specific PS. [Derrida et al ’93, Blythe & Evans ’07...] . Uniqueness / ergodicity? Convergence to the invariant measure for specific PS? Rate of convergence? [Benjamini et al ’05, Labbé & Lacoin ’16...] . Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 5 / 39
Our main question: Given a (class of) measure, is it possible to characterize the T ’s for which this measure is invariant? Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 6 / 39
A classical sufficient condition for invariance of product measures Detailed balance equations The product measure ρ Z is invariant by T on Z if ρ a ρ b T [ a , b | u , v ] = ρ u ρ v T [ u , v | a , b ] ∀ a , b , u , v ∈ E κ ρ u ρ v u v T [ a , b | u , v ] T [ u , v | a , b ] a b ρ a ρ b The product measure case is partially known. [Fajfrová et al ’16] . Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 7 / 39
Our main question: Is it possible to characterize the T for which the distribution of a Markov chain is invariant ?" Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 8 / 39
Markov Distribution P ( X � 0 , 2 � = x ) = A process X has a Markov distribution M x 0 , x 1 M x 1 , x 2 ρ x 0 ( ρ, M ) , with Markov Kernel (MK) M of memory m = 1 and initial distribution ρ , if for any x ∈ E n + 1 x 0 x 1 x 2 κ n − 1 � P ( X � 0 , n � = x ) = ρ x 0 M x j , x j + 1 . j = 0 Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 9 / 39
Markov Distribution P ( X � 0 , 2 � = x ) = A process X has a Markov distribution M x 0 , x 1 M x 1 , x 2 ρ x 0 ( ρ, M ) , with Markov Kernel (MK) M of memory m = 1 and initial distribution ρ , if for any x ∈ E n + 1 x 0 x 1 x 2 κ n − 1 � P ( X � 0 , n � = x ) = ρ x 0 M x j , x j + 1 . j = 0 P ( X � 0 , 2 � = x ) ∝ Gibbs Distribution · M x 2 , x 0 A vector ( X k , k ∈ Z / n Z ) is said to have X 0 a Gibbs distribution G ( M ) characterized · M x 0 , x 1 X 2 by a MK M , if for any x ∈ E n κ , X 1 � n − 1 j = 0 M x j , x j + 1 · M x 1 , x 2 mod n P ( X � 0 , n − 1 � = x ) = . Trace( M n ) Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 9 / 39
Invariance schemes Y 8 Y 7 Y 9 Y 6 Y 0 Y 5 Y 0 Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 1 Y 4 Y 2 Y 3 Y := η t ∼ G ( M ) t > 0 Y := η t ∼ ( ρ, M ) t > 0 Evolution under T Evolution under T X 8 X 7 X 9 X 6 X 0 X 5 X 0 X 1 X 2 X 3 X 4 X 5 X 6 X 1 X 4 X 2 X 3 X := η 0 ∼ ( ρ, M ) X := η 0 ∼ G ( M ) t = 0 t = 0 Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 10 / 39
Invariance schemes Y 8 Y 7 Y 9 Y 6 Y 0 Y 5 Y 0 Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 1 Y 4 Y 2 Y 3 Y := η t ∼ G ( M ) t > 0 Y := η t ∼ ( ρ, M ) t > 0 Evolution under T Evolution under T X 8 X 7 X 9 X 6 X 0 X 5 X 0 X 1 X 2 X 3 X 4 X 5 X 6 X 1 X 4 X 2 X 3 X := η 0 ∼ ( ρ, M ) X := η 0 ∼ G ( M ) t = 0 t = 0 Main Theorem (F. & Marckert ’17) Let κ be finite, L = 2 and m = 1. If M > 0 (strictly positive entries) then the following statements are equivalent: 1 ( ρ, M ) is invariant by T on Z . 2 G ( M ) is invariant by T on Z / 7 Z . 3 G ( M ) is invariant by T on Z / n Z , for all n ≥ 3. Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 10 / 39
Elements of the proof: algebraization Suppose µ t = ( ρ, M ) . We define ( x ) := ∂ Line M , T ∂ t µ t ( x 1 x 2 . . . x n − 1 x n ) n Mass creation rate of x = − Mass destruction rate of x Definition A ( ρ, M ) MD under its invariant distribution is said to be invariant by T on the line when Line M , T ≡ 0, for all n ∈ N . n Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 11 / 39
Elements of the proof: algebraization Suppose µ t = ( ρ, M ) . We define ( x ) := ∂ Line M , T ∂ t µ t ( x 1 x 2 . . . x n − 1 x n ) n n � � � � � � = T [ u , v | x j , x j + 1 ] ρ x − 1 M x k , x k + 1 M x j − 1 , u M u , v M v , x j + 2 x − 1 , x 0 , ( u , v ) ∈ E 2 j = 0 − 1 ≤ k ≤ n + 1 κ k �∈{ j − 1 , j , j + 1 } xn + 1 , xn + 2 ∈ E κ n + 1 � � � − T [ x j , x j + 1 | u , v ] ρ x − 1 M x k , x k + 1 k = − 1 Definition A ( ρ, M ) MD under its invariant distribution is said to be invariant by T on the line when Line M , T ≡ 0, for all n ∈ N . n Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 12 / 39
Elements of the proof: algebraization Suppose µ t = ( ρ, M ) . We define ( x ) := ∂ Line M , T ∂ t µ t ( x 1 x 2 . . . x n − 1 x n ) n � � n n + 1 � � � = ρ x − 1 M x k , x k + 1 × x − 1 , x 0 , j = 0 k = − 1 xn + 1 , xn + 2 ∈ E κ M x j − 1 , u M u , v M v , x j + 2 � − T out T [ u , v | x j , x j + 1 ] [ x j , x j + 1 ] M x j − 1 , x j M x j , x j + 1 M x j + 1 , x j + 2 ( u , v ) ∈ E 2 κ � �� � Z xj − 1 , xj , xj + 1 , xj + 2 Definition A ( ρ, M ) MD under its invariant distribution is said to be invariant by T on the line when Line M , T ≡ 0, for all n ∈ N . n Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 12 / 39
Elements of the proof: algebraization Suppose µ t = ( ρ, M ) . We define for M > 0 Line M , T ( x ) NLine M , T n ( x ) := n � n − 1 i = 1 M x i , x i + 1 n � � � � = ρ x − 1 M x − 1 , x 0 M x 0 , x 1 × Z x j − 1 , x j , x j + 1 , x j + 2 × M x n , x n + 1 M x n + 1 , x n + 2 x − 1 , x 0 , j = 0 xn + 1 , xn + 2 ∈ E κ Definition A ( ρ, M ) MD under its invariant distribution is said to be invariant by T on the line when Line M , T ≡ 0, for all n ∈ N . n Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 12 / 39
Elements of the proof: algebraization Important message : Line: If M > 0, then ( ρ, M ) is invariant by T on Z ⇐ ⇒ n − 2 � NLine M , T ( x ) = Z L Z x j − 1 , x j , x j + 1 , x j + 2 + Z R for all n ∈ N x 1 , x 2 , x 3 + x n − 2 , x n − 1 , x n = 0 n j = 2 Cycle of length n : If M > 0, then G ( M ) is invariant by T on Z / 7 Z ⇐ ⇒ n − 1 � NCycle M , T ( x ) = Z x j − 1 , x j , x j + 1 , x j + 2 = 0 n j = 0 where i := i mod n Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 13 / 39
Sketch of proof 1) = ⇒ 2) in main theorem: M > 0 1) ( ρ, M ) is invariant by T on Z = ⇒ 2) G ( M ) is invariant by T on Z / 7 Z � NLine M , T ⇒ NCycle M , T ≡ 0 ∀ n ∈ N = ≡ 0 n 7 Consider x ∈ E 7 κ and w = x . . . x � �� � ℓ times NLine M , T ( w ) = Bound. terms + ... = 0 7 ℓ � �� � O ( 1 ) x 1 x 2 x 3 x 4 x 5 x 6 x 2 x 3 x 4 x 5 x 6 x 7 Luis Fredes (Université de Bordeaux) Particle systems and maps 19/09/2019 14 / 39
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