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 centered case The multidimensional ballot problem A generalization

W random walk on P = Zd

≥0 with set of steps {ε1, . . . , εd}

endowed with the probability ν s.t. ν ({εi}) = pi Markov chain with transition matrix : p (λ, µ) = pµ−λ 1{ε1,...,εd} (µ − λ)

• =1λµ

λ, µ ∈ P pγ = pγ1

1 . . . pγd d

γ ∈ P Drift : p = d

i=1 piεi

Some centered random walks on weight lattices conditioned

Context The centered case The multidimensional ballot problem A generalization

C = {λ1 ≥ . . . ≥ λd ≥ 0} Question If P

• W ∈ C
• > 0, can we make explicit the law of the RW under

the conditioning by this event ? PW∈C (W (i + 1) = µ | W (i) = λ) =? λ, µ ∈ P ∩ C

Some centered random walks on weight lattices conditioned

Context The centered case The multidimensional ballot problem A generalization

Theorem (O’Connell 2003) If the drift is in the open cone, we have Pλ

• W ∈ C
• = p−λsλ (p)
• i<j
• 1 − pj

pi

• λ ∈ P ∩ C

and the transition matrix of the conditioned random walk is given by pC (λ, µ) = p (λ, µ) p−µsµ (p) p−λsλ (p) = sµ (p) sλ (p)1λµ λ, µ ∈ P ∩ C with sλ (p) = det pλj+d−j

i

det pd−j

i

Some centered random walks on weight lattices conditioned

Context The centered case The multidimensional ballot problem A generalization ∅ ↓ ւ ց ւ ց ւ ց ւ ց ւ ց ↓ ց

. . .

• A1,
• ⊂
• A2,
• ⊂
• A3,
• ⊂ . . . ⊂ Y

Some centered random walks on weight lattices conditioned

Context The centered case The multidimensional ballot problem A generalization

Settings

O’Connell Lecouvey-Lesigne-Peigné g gld (C) f.d. simple Lie algebra over C of rank d → associated root system R W Sd Weyl group P Zd

≥0

weight lattice C {x1 > . . . > xd > 0}

• pen Weyl chamber

P+ {λ1 ≥ . . . ≥ λd ≥ 0} cone of dominant weights Fix R and a dominant weight δ.

Some centered random walks on weight lattices conditioned

Context The centered case The multidimensional ballot problem A generalization

P+ ← → f.d. irreducible representations of g δ ← → V (δ)

• V (δ)γ
• γ∈P = weight spaces of δ

V (δ) =

• γ∈P

V (δ)γ P (δ) =

• γ ∈ P : Kδ γ = dim V (δ)γ > 0
• sδ (x) =
• γ∈P

Kδ γxγ =

• γ∈P(δ)

Kδ γxγ δ is minuscule if P (δ) ⊂ W δ and then sδ (x) =

w∈W xwδ

Some centered random walks on weight lattices conditioned

Context The centered case The multidimensional ballot problem A generalization

Analogue of the partial Young graph for (R, δ)

Vertices V (δ)⊗n =

• λ∈P+

V (λ)⊕f n

λ δ

• n

{λ ∈ P+ : f n

λ δ > 0}

Arrows V (λ) ⊗ V (δ) =

• λ∈P+

V (λ)⊕mµ

λ δ

λ µ ⇐ ⇒ mµ

λ δ > 0

λ

mµ

λ δ

− − − → µ

Some centered random walks on weight lattices conditioned

Context The centered case The multidimensional ballot problem A generalization ∅ ↓ ւ ↓ ց ∅

. . .

• Cd,
• Some centered random walks on weight lattices conditioned

Context The centered case The multidimensional ballot problem A generalization

• Cd,
• Some centered random walks on weight lattices conditioned

Context The centered case The multidimensional ballot problem A generalization

 B3,  

δ = 1

2 (ε1 + ε2 + ε3) = + + + is minuscule.

Geogebra

Some centered random walks on weight lattices conditioned

Context The centered case The multidimensional ballot problem A generalization

Central probability ν : three parameters θ1, θ2, θ3 > 0

− + +

1

ր

3

ց ×θ1 ×θ3 P (δ) + + +

3

− − − →

×θ3

+ + −

2

− − − →

×θ2

+ − + − + −

2

− − − →

×θ2

ν p+++ θ3p+++ θ2p++−

3

ց

1

ր ×θ3 ×θ1 + − −

x1 = 1 θ1θ2θ3 x2 = 1 θ2θ3 x3 = 1 θ3 p+++ =

xδ sδ(x) with x = (x1, x2, x3)

p++−p+−+ = p+−−p+++ and five other "signs rules" relations

Some centered random walks on weight lattices conditioned

Context The centered case The multidimensional ballot problem A generalization

d = (1 − θ1) θ2θ3 (1 + θ3) Σ ω1

• =ε1

+(1 − θ2) θ3 (1 + θ1θ2θ3) Σ ω2

• =ε1+ε2

+ (1 − θ3)

• 1 + θ2θ3 + θ1θ2θ3 + θ1θ2

2θ2 3

• Σ

ω3

• = 1

2 (ε1+ε2+ε3)

d = 0 ⇐ ⇒ θ1 = θ2 = θ3 = 1

Some centered random walks on weight lattices conditioned

Context The centered case The multidimensional ballot problem 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 pW (λ, µ) pµ−λ1λµ Kδ λ−µ xµ−λ

sδ(x) xµ−λ sδ(x) 1λµ

pP(W) (λ, µ)

sµ(p) sλ(p)1λµ

mµ

λ δ sµ(x) sλ(x)sδ(x) sµ(x) sλ(x)sδ(x)1λµ

Some centered random walks on weight lattices conditioned

Context The centered case The multidimensional ballot problem 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 pP(W) is the transition matrix of the random walk W conditioned to never exit the closed Weyl chamber C : pC = pP(W)

Some centered random walks on weight lattices conditioned

Context The centered case

Let d0 be in C \ C. Fact sµ (x (d)) sλ (x (d)) sδ (x (d)) − − − − →

d →

d∈Cd0

sµ (x (d0)) sλ (x (d0)) sδ (x (d0)) In particular, if d0 = 0, sµ (x (d)) sλ (x (d)) sδ (x (d))1λµ − − − − →

d →

d∈C0

dim V (µ) dim V (λ) dim V (δ)1λµ Question If δ is minuscule and if W is centered, do we have PW(≤n)∈C [W (i + 1) = µ | W (i) = λ] − − − →

n→∞

dim V (µ) dim V (λ) dim V (δ)1λµ?

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 conditioned to stay forever in the closed Weyl chamber ց

d=0

Law of the centered random walk conditioned to stay forever in the closed Weyl chamber Law of the centered random walk conditioned to stay until instant n in the closed Weyl chamber ր

n→∞ Some centered random walks on weight lattices conditioned

Context The centered case

Sketch of proof :

1 Finite-time conditioning : n ≥ i + 2

PW (≤n)∈C [W (i + 1) = µ | W (i) = λ] = pW (λ, µ) hn−i−1 (µ) hn−i (λ) with hn : λ ∈ P+ → Pλ

• W (≤ n) ∈ C
• P0
• W (≤ n) ∈ C
• n ≥ 1

2 (hn)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