Scaling limits of non-increasing Markov chains and applications to - PowerPoint PPT Presentation

Scaling limits of non-increasing Markov chains and applications to random trees and coalescents Bénédicte HAAS Université Paris-Dauphine based on joint works with Grégory MIERMONT (Orsay) Bénédicte Haas SSP - March 2012 1 / 26

Outline Introduction 1 Scaling limits of non-increasing Markov chains 2 Applications: 3 - scaling limits of Markov branching trees - number of collisions in Λ -coalescents - random walks with barriers Bénédicte Haas SSP - March 2012 2 / 26

Scaling limits: a basic example I.i.d sequence of centered random variables X i ∈ {− 1 , 1 } : − 1 , 1 , 1 , 1 , 1 , − 1 , − 1 , − 1 , 1 , − 1 , − 1 , 1 , 1 , 1 , 1 , ... Centered random walk: S n = X 1 + ... + X n How does S n behave when n is large ? (1) what is the growth rate ? (2) what is the limit after rescaling ? Central limit theorem: S n law → N ( 0 , 1 ) √ n Functional version: D ONSKER ’ S theorem (51): „ S [ nt ] « law √ n , t ∈ [ 0 , 1 ] → ( B ( t ) , t ∈ [ 0 , 1 ]) where B is a standard Brownian motion Bénédicte Haas SSP - March 2012 3 / 26

Another example: Galton-Watson trees Galton-Watson processes are introduced in 1873 to study the extinction of family names generation 3 generation 2 generation 1 ancester=root η : offspring distribution (proba. on Z + = { 0 , 1 , 2 , .. } ), such that η ( 1 ) < 1, with mean m Extinction probability = 1 in subcritical ( m < 1) and critical ( m = 1 ) cases ∈ [ 0 , 1 ) in supercritical cases ( m > 1) Bénédicte Haas SSP - March 2012 4 / 26

Large Galton-Watson trees T GW : critical GW tree conditioned to have n nodes n Offspring distribution η : finite variance σ 2 < ∞ ◮ H ( U ) : height (=generation) of a node chosen uniformly at random n H ( U ) → R n law √ n σ , where R has a Rayleigh distribution: P ( R > x ) = exp ( − x 2 / 2 ) ( M EIR & M OON 78 ) ◮ H n : height of the tree → 2 W H n law √ n σ , where W = maximum of a Brownian excursion with length 1 ( K OLCHIN 86 ) With length 1 on each edge, what does the tree look like when n → ∞ ? Bénédicte Haas SSP - March 2012 5 / 26

Universal limit: the Brownian tree T Br A LDOUS 93: T GW → 2 law n √ n σ T Br (picture by G. Miermont) Compact (random) real tree , i.e. compact metric space with the tree property: ∀ x , y ∈ T Br ∃ ! path from x to y almost surely: binary tree, self-similar, with Hausdorff dimension = 2 Bénédicte Haas SSP - March 2012 6 / 26

Topology on the set of compact rooted real trees Gromov-Hausdorff distance: let ( T , ρ ) , ( T ′ , ρ ′ ) be two compact rooted real trees ( T , ρ ) , ( T ′ , ρ ′ ) := inf d H ( ϕ 1 ( T ) , ϕ 2 ( T ′ )) ∨ d Z ( ϕ 1 ( ρ ) , ( ϕ 2 ( ρ ′ )) ` ´ ` ´ d GH → Z and ϕ 2 : T ′ ֒ the infimum being on all isometric embeddings ϕ 1 : T ֒ → Z into a same metric space ( Z , d Z ) . ( T , ρ ) and ( T ′ , ρ ′ ) are equivalent if ∃ ϕ isometry: T ′ = ϕ ( T ) , ρ ′ = ϕ ( ρ ) d GH : distance on the set of equivalence classes A LDOUS 93 : T GW 2 law n √ n → σ T Br , GH jointly with the convergence of the uniform probability on the nodes of T GW towards a n probability measure on the leaves of T Br Bénédicte Haas SSP - March 2012 7 / 26

Large Galton-Watson trees: when the variance is infinite Assume: η ( k ) = P ( to have k children ) k →∞ Ck − β , 1 < β < 2 ∼ Then ( D UQUESNE 03 ): T GW GH C − 1 /β T β n law → n 1 − 1 /β ◮ “smaller" trees ◮ the limiting tree T β belongs to the family of stable Lévy trees (introduced by Duquesne, Le Gall, Le Jan): - each branching vertex branches in an infinite, countable number of subtrees - it is a self-similar tree, with Hausdorff dimension = β/ ( β − 1 ) Bénédicte Haas SSP - March 2012 8 / 26

Non-increasing Markov chains ( X ( k ) , k ≥ 0 ) : Z + -valued Markov chain, non-increasing ( X n ( k ) , k ≥ 0 ) : chain starting from X n ( 0 ) = n Xn n k 0 An Absorption time: A n = inf { i : X n ( i ) = X n ( j ) , ∀ j ≥ i } , finite X n ( · ) How behave and when n → ∞ ? A n n Bénédicte Haas SSP - March 2012 9 / 26

Scaling limit Assumption: macroscopic jumps are rare starting from n , the probability that the first jump is larger than ≥ n ε behaves like c ε ∼ n γ n →∞ for some γ > 0 ( c ε ր when ε ց ) More precisely, ∃ finite measure µ on [ 0 , 1 ] such that, for ε ∈ ( 0 , 1 ] P ( n − X n ( 1 ) ≥ n ε ) ∼ 1 » n − X n ( 1 ) ∼ µ ([ 0 , 1 ]) Z µ ( d x ) – and E 1 − x , n γ n n γ [ 0 , 1 − ε ] Theorem ( H.-M IERMONT 11 ) Then, ∃ time-continuous Markov process X ∞ such that „ X n ([ n γ t ]) « law , t ≥ 0 → ( X ∞ ( t ) , t ≥ 0 ) , n for the Skorokhod topology on the set D ([ 0 , ∞ ) , [ 0 , ∞ )) . Bénédicte Haas SSP - March 2012 10 / 26

The limit process X ∞ is: ◮ self-similar: starting from X ∞ ( 0 ) = x , the process cX ∞ ( c − γ t ) , t ≥ 0 ` ´ is distributed as X ∞ starting from X ∞ ( 0 ) = cx , ∀ c > 0 ◮ starting from X ∞ ( 0 ) = 1, X ∞ writes X ∞ = exp ( − ξ ρ ) where • ξ is a subordinator • E [ exp ( − λξ t )] = exp ( − t φ ( λ )) , with Z ( 1 − x λ ) µ ( d x ) φ ( λ ) = µ ( { 1 } ) λ + 1 − x + µ ( { 0 } ) , λ ≥ 0 , ( 0 , 1 ) • ρ is an acceleration of time: Z u  ff ρ ( t ) = inf u ≥ 0 : exp ( − γξ r ) d r ≥ t 0 Bénédicte Haas SSP - March 2012 11 / 26

Absorption time Consequently, X ∞ starting from X ∞ ( 0 ) = 1 is absorbed at 0 at time Z ∞ exp ( − γξ r ) d r 0 (almost surely finite) Jointly with the previous convergence, we have Z ∞ A n law exp ( − γξ r ) d r → n γ 0 This is not an immediate consequence of the convergence of the whole process! Remark: extension of these results to regular variation assumptions. Bénédicte Haas SSP - March 2012 12 / 26

Application to Markov branching trees ( T n , n ≥ 1 ) : T n random rooted tree with n nodes Markov branching property: 4 nods 9 nods 4 nods 9 nods 3 nods 3 nods T n : law law law ∼ T 4 ∼ T 3 ∼ T 9 ( n = 17) R R R R Conditional on “the root of T n branches in p sub-trees with n 1 ≥ ... ≥ n p nodes”, these sub-trees are independent, with respective distributions those of T n 1 , ..., T n p Similar definition for sequences of trees indexed by the number of leaves Ex.: Galton-Watson trees conditioned to have n nodes (respectively n leaves) Bénédicte Haas SSP - March 2012 13 / 26

Markov branching trees ( T n , n ≥ 1 ) Markov branching indexed by leaves What does it look like when n is large ? First step: height of a leaf chosen uniformly at random R Bénédicte Haas SSP - March 2012 14 / 26

Markov branching trees ( T n , n ≥ 1 ) Markov branching indexed by leaves What does it look like when n is large ? First step: height of a leaf chosen uniformly at random X n ( 0 ) = 9 R X n ( k ) : size of the sub-tree above generation k containing the marked leaf Bénédicte Haas SSP - March 2012 14 / 26

Markov branching trees ( T n , n ≥ 1 ) Markov branching indexed by leaves What does it look like when n is large ? First step: height of a leaf chosen uniformly at random X n ( 0 ) = 9 , X n ( 1 ) = 5 R X n ( k ) : size of the sub-tree above generation k containing the marked leaf Bénédicte Haas SSP - March 2012 14 / 26

Markov branching trees ( T n , n ≥ 1 ) Markov branching indexed by leaves What does it look like when n is large ? First step: height of a leaf chosen uniformly at random X n ( 0 ) = 9 , X n ( 1 ) = 5 , X n ( 2 ) = 3 R X n ( k ) : size of the sub-tree above generation k containing the marked leaf Bénédicte Haas SSP - March 2012 14 / 26

Markov branching trees ( T n , n ≥ 1 ) Markov branching indexed by leaves What does it look like when n is large ? First step: height of a leaf chosen uniformly at random X n ( 0 ) = 9 , X n ( 1 ) = 5 , X n ( 2 ) = 3 , X n ( 3 ) = 2 R X n ( k ) : size of the sub-tree above generation k containing the marked leaf Bénédicte Haas SSP - March 2012 14 / 26

Markov branching trees ( T n , n ≥ 1 ) Markov branching indexed by leaves What does it look like when n is large ? First step: height of a leaf chosen uniformly at random X n ( 0 ) = 9 , X n ( 1 ) = 5 , X n ( 2 ) = 3 , X n ( 3 ) = 2 , X n ( 4 ) = 1 R X n ( k ) : size of the sub-tree above generation k containing the marked leaf It is a Markov chain! Absorption time at 1= height of the marked leaf Bénédicte Haas SSP - March 2012 14 / 26

Scaling limits of Markov branching trees Let q n ( n 1 , ..., n p ) := P ( the root of T n branches in p sub-trees with n 1 ≥ ... ≥ n p leaves ) and S ↓ = n o X s 1 ≥ s 2 ≥ .. ≥ 0 : s i = 1 i Assumption: ∀ bounded continuous f : S ↓ → R , 1 − n 1 “ n 1 n , ..., n p Z “ ” ” n γ X n , 0 , .. S ↓ ( 1 − s 1 ) f ( s ) ν ( d s ) , q n ( n 1 , ..., n p ) f → n n →∞ ( n 1 ,..., n p ) partition of n with γ > 0 and ν a non-trivial σ − finite measure on S ↓ such that S ↓ ( 1 − s 1 ) ν ( d s ) < ∞ . R Informally: s n 2 ~n s n 1 with proba. ∼ 1 : with proba. ∼ ν ( d s ) : s n 3 n γ size: o(n) R R Bénédicte Haas SSP - March 2012 15 / 26