Some centered random walks on weight lattices conditioned to stay in - PowerPoint PPT Presentation

Context The centered case Some centered random walks on weight lattices conditioned to stay in Weyl chambers Vivien Despax LMPT, Tours Final conference of the Madaca Project Some centered random walks on weight lattices conditioned

Context The multidimensional ballot problem The centered case A generalization W random walk on P = Z d ≥ 0 with set of steps { ε 1 , . . . , ε d } endowed with the probability ν s.t. ν ( { ε i } ) = p i Markov chain with transition matrix : p ( λ, µ ) = p µ − λ 1 { ε 1 ,...,ε d } ( µ − λ ) λ, µ ∈ P � �� � = 1 λ � µ p γ = p γ 1 1 . . . p γ d γ ∈ P d Drift : p = � d i = 1 p i ε i Some centered random walks on weight lattices conditioned

Context The multidimensional ballot problem The centered case A generalization C = { λ 1 ≥ . . . ≥ λ d ≥ 0 } Question � � If P W ∈ C > 0, can we make explicit the law of the RW under the conditioning by this event ? P W∈ C ( W ( i + 1 ) = µ | W ( i ) = λ ) =? λ, µ ∈ P ∩ C Some centered random walks on weight lattices conditioned

Context The multidimensional ballot problem The centered case A generalization Theorem (O’Connell 2003) If the drift is in the open cone , we have � � 1 − p j � � � = p − λ s λ ( p ) P λ W ∈ C λ ∈ P ∩ C p i i < j and the transition matrix of the conditioned random walk is given by p C ( λ, µ ) = p ( λ, µ ) p − µ s µ ( p ) p − λ s λ ( p ) = s µ ( p ) λ, µ ∈ P ∩ C s λ ( p ) 1 λ � µ with s λ ( p ) = det p λ j + d − j i det p d − j i Some centered random walks on weight lattices conditioned

Context The multidimensional ballot problem The centered case A generalization ∅ ↓ ւ ց ւ ց ւ ց ւ ց ւ ց ↓ ց . . . � � � � � � A 1 , ⊂ A 2 , ⊂ A 3 , ⊂ . . . ⊂ Y Some centered random walks on weight lattices conditioned

Context The multidimensional ballot problem The centered case A generalization Settings O’Connell Lecouvey-Lesigne-Peigné gl d ( C ) f.d. simple Lie algebra over C of rank d g → associated root system R W S d Weyl group Z d P weight lattice ≥ 0 C { x 1 > . . . > x d > 0 } open Weyl chamber P + { λ 1 ≥ . . . ≥ λ d ≥ 0 } cone of dominant weights Fix R and a dominant weight δ . Some centered random walks on weight lattices conditioned

Context The multidimensional ballot problem The centered case A generalization P + ← → f.d. irreducible representations of g δ ← → V ( δ ) � � V ( δ ) γ γ ∈ P = weight spaces of δ � V ( δ ) = V ( δ ) γ γ ∈ P � � P ( δ ) = γ ∈ P : K δ γ = dim V ( δ ) γ > 0 � � K δ γ x γ = K δ γ x γ s δ ( x ) = γ ∈ P γ ∈ P ( δ ) δ is minuscule if P ( δ ) ⊂ W δ and then s δ ( x ) = � w ∈ W x w δ Some centered random walks on weight lattices conditioned

Context The multidimensional ballot problem The centered case A generalization Analogue of the partial Young graph for ( R , δ ) Vertices � V ( δ ) ⊗ n = V ( λ ) ⊕ f n λ δ λ ∈ P + � { λ ∈ P + : f n λ δ > 0 } n Arrows � V ( λ ) ⊕ m µ V ( λ ) ⊗ V ( δ ) = λ δ λ ∈ P + ⇒ m µ λ δ > 0 λ � µ ⇐ m µ λ δ − − − → µ λ Some centered random walks on weight lattices conditioned

Context The multidimensional ballot problem The centered case A generalization ∅ ↓ ւ ↓ ց ∅ . . . � � C d , Some centered random walks on weight lattices conditioned

Context The multidimensional ballot problem The centered case A generalization � � C d , Some centered random walks on weight lattices conditioned

Context The multidimensional ballot problem The centered case A generalization    B 3 ,  δ = 1 2 ( ε 1 + ε 2 + ε 3 ) = + + + is minuscule. Geogebra Some centered random walks on weight lattices conditioned

Context The multidimensional ballot problem The centered case A generalization Central probability ν : three parameters θ 1 , θ 2 , θ 3 > 0 − + + 1 3 ր ց × θ 1 × θ 3 3 2 2 P ( δ ) + + + − − − → + + − − − − → + − + − + − − − − → × θ 3 × θ 2 × θ 2 3 1 ν p +++ θ 3 p +++ θ 2 p ++ − ց ր × θ 3 × θ 1 + − − 1 1 x 3 = 1 x 1 = x 2 = θ 1 θ 2 θ 3 θ 2 θ 3 θ 3 x δ p +++ = s δ ( x ) with x = ( x 1 , x 2 , x 3 ) p ++ − p + − + = p + −− p +++ and five other "signs rules" relations Some centered random walks on weight lattices conditioned

Context The multidimensional ballot problem The centered case A generalization d = ( 1 − θ 1 ) θ 2 θ 3 ( 1 + θ 3 ) +( 1 − θ 2 ) θ 3 ( 1 + θ 1 θ 2 θ 3 ) ω 1 ω 2 Σ Σ ���� ���� = ε 1 = ε 1 + ε 2 � � 1 + θ 2 θ 3 + θ 1 θ 2 θ 3 + θ 1 θ 2 2 θ 2 + ( 1 − θ 3 ) 3 ω 3 Σ ���� = 1 2 ( ε 1 + ε 2 + ε 3 ) d = 0 ⇐ ⇒ θ 1 = θ 2 = θ 3 = 1 Some centered random walks on weight lattices conditioned

Context The multidimensional ballot problem The centered case A generalization Important ingredient : a path transformation based on the RS correspondence which generalizes in type A the Pitman transform. Biane, Bougerol and O’Connell defined a generalized Pitman transform P for any type. O’C LLP LLP minuscule p W ( λ, µ ) K δ λ − µ x µ − λ x µ − λ p µ − λ 1 λ � µ s δ ( x ) 1 λ � µ s δ ( x ) p P ( W ) ( λ, µ ) s µ ( p ) s µ ( x ) s µ ( x ) m µ s λ ( p ) 1 λ � µ s λ ( x ) s δ ( x ) 1 λ � µ λ δ s λ ( x ) s δ ( x ) Some centered random walks on weight lattices conditioned

Context The multidimensional ballot problem The centered case A generalization Theorem (LLP 2011) If δ is minuscule and if the drift d = d ( θ ) is in the open Weyl chamber (sup θ < 1), then p P ( W ) is the transition matrix of the random walk W conditioned to never exit the closed Weyl chamber C : p C = p P ( W ) Some centered random walks on weight lattices conditioned

Context The centered case Let d 0 be in C \ C . Fact s µ ( x ( d )) s µ ( x ( d 0 )) s λ ( x ( d )) s δ ( x ( d )) − − − − → s λ ( x ( d 0 )) s δ ( x ( d 0 )) d → d ∈ C d 0 In particular, if d 0 = 0 , s µ ( x ( d )) dim V ( µ ) s λ ( x ( d )) s δ ( x ( d )) 1 λ � µ − − − − → dim V ( λ ) dim V ( δ ) 1 λ � µ d → d ∈ C 0 Question If δ is minuscule and if W is centered, do we have dim V ( µ ) P W ( ≤ n ) ∈ C [ W ( i + 1 ) = µ | W ( i ) = λ ] − − − → dim V ( λ ) dim V ( δ ) 1 λ � µ ? n →∞ Some centered random walks on weight lattices conditioned

Context The centered case Theorem (D 2016) Yes : Law of the random walk with drift in the open Weyl chamber ց conditioned to stay forever d = 0 in the closed Weyl chamber Law of the centered random walk conditioned to stay forever in the closed Weyl chamber Law of the centered random walk ր conditioned to stay until instant n n →∞ in the closed Weyl chamber Some centered random walks on weight lattices conditioned

Context The centered case Sketch of proof : 1 Finite-time conditioning : n ≥ i + 2 P W ( ≤ n ) ∈ C [ W ( i + 1 ) = µ | W ( i ) = λ ] = p W ( λ, µ ) h n − i − 1 ( µ ) h n − i ( λ ) with � � W ( ≤ n ) ∈ C h n : λ ∈ P + �→ P λ n ≥ 1 � � P 0 W ( ≤ n ) ∈ C 2 ( h n ) n converges to a positive harmonic fonction h : delicate, use a probabilistic result du to Denisov-Wachtel 2015 on the exit-time from cones for centered RW. 3 One can compare h to the one given by LLP 2011 for d = 0 : algebraic combinatorics (paths in the cone counted by tensor powers multiplicities). 4 Both coincides everywhere since they coincide at 0. Some centered random walks on weight lattices conditioned

Context The centered case Conjecture Still true for any drift is the frontier of the Weyl chamber. Some centered random walks on weight lattices conditioned