Exceptional points of two-dimensional random walks at multiples of - PowerPoint PPT Presentation

Exceptional points of two-dimensional random walks at multiples of the cover time Yoshihiro Abe (Chiba University) The 12th MSJ-SI Stochastic Analysis, Random Fields and Integrable Probability August 1, 2019 Joint work with Marek Biskup (UCLA)

Abstract We have studied the statistics of exceptional points for 2D SRW such as • Avoided points (i.e. points not visited at all, late points) • Thick points (i.e. heavily visited sites) • Thin points (i.e. lightly visited sites) • Light points (i.e. points where the local time is O (1) ) In this talk, we will focus on avoided points. cf. Okada’s talk (tomorrow)

Figure: Avoided points (Simulation by Marek Biskup) 2000 × 2000 square, run-time = 0 . 3 × (cover time) Note: Cover time is the first time at which the SRW visits every vertex.

SRW on D N with wired boundary condition D ⊂ R 2 : “good” bounded open set D N ⊂ Z 2 : “good” lattice approximation of D x x ∈ D N ⇒ N ∈ D ( X t ) t ≥ 0 : Continuous-time SRW on D N with Exp (1) -holding times Technical Assumption: When X exits D N , it re-enters D N through a uniformly-chosen boundary edge. ⇝ Regard ∂D N as a single point ρ We assume this to relate our local times to DGFF with zero boundary conditions via the 2nd Ray-Knight theorem.

Local time L D N t Recall: ( X t ) t ≥ 0 = SRW on D N , ρ = the boundary vertex Local time: ∫ τρ ( t ) 1 ∫ ∫ DN ( x ) := 1 { Xs = x } ds L deg (x) , t 0 where ∫ s 1 { ∫ ∫ } τ ρ ( t ) := inf s ≥ 0 : 1 { Xr = ρ } dr deg ( ρ ) > t . 0 N →∞ tN Let t N be a sequence with − → θ ∈ (0 , 1) . 1 π (log N )2 ⇝ τ ρ ( t N ) ≈ θ × ( cover time of D N ) DN ≈ local time at θ × ( cover time of D N ) ⇝ L tN

Main Result DN Recall: L ≈ local time at θ × ( cover time of D N ) tN 2 tN − 1 π log N = N 2 − 2 θ + o (1) N →∞ tN θ ∈ (0 , 1) , W N := N 2 e − → 1 π (log N )2 ∑ κ D 1 N := 1 N ⊗ δ ( x )=0 } δ x { LDN { LDN WN ( x + z ) : z ∈ Z 2 } tN tN x ∈ DN Main Theorem. (A.-Biskup) law θ ( dx ) ⊗ ν RI κ D N →∞ c θ Z D θ ( dϕ ) , √ − → N (2 λ )2 e 2 λϕD x − 1 2 Var (2 λϕD x ) dx, • Z D λ ( dx )“ = ” r D ( x ) λ ∈ (0 , 1) 2 a Liouville Quantum Gravity measure on D • ν RI θ is the law of occupation time field of the two-dimensional random interlacement at level θ .

Idea of proof. law θ ( dx ) ⊗ ν RI Recall: κD cθ ZD √ θ ( dφ ) . − → N N →∞ Recall: κD 1 ∑ N := 1 ⊗ δ δ x { LDN { LDN WN ( x + z ) : z ∈ Z 2 } ( x )=0 } N x ∈ DN tN tN Thick Avoided 2 nd Ray-Knight Theorem Biskup- points Louidor LQG points for DGFF Pinned Isomorphism Theorem by Local Pinned Rodriguez Occupation time field structure DGFF of RI

Heuristics of Avoided points ↔ Thick points for DGFF

Discrete Gaussian Free Field (DGFF) and the maximum DN Definition. h DN = ( h ) x ∈ DN is DGFF on D N x def ⇔ h DN is centered Gaussian with ∫ H∂DN [∫ ∫ ] 1 [ ] h DN h DN = G DN ( x, y ) := E x 1 { Xs = y } ds deg ( y ) . E x y 0 Theorem. (Bolthausen-Deuschel-Giacomin (2001)) DN √ max x ∈ DN h 2 x N →∞ − → in probab . log N π DN 1 Note: Var ( h ) = 2 π log N + O (1) x Remark. 2 nd order: Bramson-Zeitouni (2011) Convergence in law: Bramson-Ding-Zeitouni (2016)

Convergence of λ -thick points λ ∈ (0 , 1) max x ∈ DN hDN √ N →∞ x 2 Recall: − → in probab . log N π η D 1 ∑ ∑ ∑ N := N ⊗ δ ⊗ δ x ∈ DN δ x hDN { hDN − hDN KN x + z : z ∈ Z 2 } − aN x x a 2 N − 1 π log N = N 2 − 2 λ 2+ o (1) . √ N 2 N →∞ aN 2 → λ π , K N := √ log N e log N Theorem. (Biskup-Louidor (2016)) √ law λ ( dx ) ⊗ e − 2 η D N →∞ c ( λ ) Z D 2 πλh dh ⊗ ν λ , − → N (2 λ )2 e 2 λϕD x − 1 2 Var (2 λϕD x ) dx LQG on D , where Z D λ ( dx )“ = ” r D ( x ) 2 √ 2 π λ a , ϕ is DGFF on Z 2 pinned to zero at the origin. ν λ is the law of ϕ + 2

Heuristics of avoided points ↔ thick points √ DN π (log N ) 2 , x is a λ -thick point ⇔ h Recall: t N ≈ θ 1 2 ≈ λ π log N x Key: 2nd Ray-Knight Theorem (Eisenbaum-Kaspi-Marcus-Rosen-Shi (2000)) tN ( x ) + 1 { } ) 2 : x ∈ D N under P ρ ⊗ P DN 2 ( h DN L x { 1 √ ) 2 : x ∈ D N } law ( h DN = + 2 t N . x 2 Thus, √ Z D θ ↔ x is θ -thick point √ + √ 2 t N ≈ 0 ↔ h D N x ↔ L D N t N ( x ) = 0 . i.e. x is an avoided point.

Heuristics of Local structure of avoided points ↔ 2D random interlacements

Two-dimensional random interlacements Poissonian soup of trajectories of SRWs conditioned on never hitting the origin. 0 2 Z

Two-dimensional random interlacements Poissonian soup of trajectories of SRWs conditioned on never hitting the origin. A 0 2 Z

0 ∈ A ⊂ Z 2 finite What 2D RI at level θ looks like Take i.i.d. Poi ( πθ cap ( A )) samples from the law e A ( · ) cap ( A ) , where e A is the equilibrium measure and cap is the capacity: | y |→∞ P y [ X HA = x ] , x ∈ A , e A ( x ) := 4 a ( x ) hm A ( x ) := 4 a ( x ) lim ∑ cap ( A ) := ∑ ∑ x ∈ A e A ( x ) . A

0 ∈ A ⊂ Z 2 finite What 2D RI at level θ looks like From each point, start indep two walks (blue and green); Blue paths avoid 0 and green paths never return to A . 0 A

Construction of 2D RI Two-dimensional random interlacements was constructed by Comets-Popov-Vachkovskaia (2016) and Rodriguez (2019). One of the main motivations is to study the local structure of the uncovered set by SRW on 2D torus ( Z /N Z ) 2 . cf. Sznitman (’10) : Z d , d ≥ 3 , Teixeira (’09) : general transient weighted graphs

Occupation time field for 2D RI at level θ Let ( w i ) i ∈ N be the doubly-infinite trajectories in the two-dimensional random interlacements at level θ . The occupation time field is defined by ∫ ∞ 1 ∫ ∫ ∑ x ∈ Z 2 . ℓ RI ∑ ∑ θ ( x ) := 1 { w i ( t )= x } dt, 4 −∞ i ∈ N law θ ( dx ) ⊗ ν RI Recall the main theorem: κ D N →∞ c θ Z D θ ( dϕ ) √ − → N 1 with κD ∑ N = 1 ⊗ δ , δ x { LDN { LDN WN ( x + z ) : z ∈ Z 2 } ( x )=0 } N x ∈ DN tN tN π (log N )2 , LDN tN ≈ θ 1 ≈ local time at θ × ( cover time of DN ) tN ν RI θ = the law of ( ℓ RI θ ( x )) x ∈ Z 2 .

Heuristics of local picture ↔ RI Recall: t N ≈ θ 1 π (log N ) 2 , ( ℓ RI θ ( z )) z ∈ Z 2 : Occupation time field of RI Key: Pinned Isomorphism Theorem (Rodriguez (2019)) { 1 √ ) 2 : z ∈ Z 2 θ ( z ) + 1 { } } 2 ( ϕ z ) 2 : z ∈ Z 2 law ( ℓ RI = ϕ z + 2 2 πθ a , 2 where ( ϕ z ) z ∈ Z 2 be DGFF on Z 2 pinned to zero at the origin. Recall: LDN ≈ local time at θ × ( cover time of DN ) tN √ ( ) ( ) ( LDN ( x + z )) z | LDN ( hDN x + z − hDN ) z | hDN ( x ) = 0 ↔ ≈ θ × max x x tN tN √ ↔ ( φz + 2 2 πθ a ) z. ) law ( DN DN ≈ ( ℓ RI ( L tN ( x + z )) z ∈ Z 2 | L tN ( x ) = 0 θ ( z )) z ∈ Z 2 . ∴ ∴ ∴

Thank you. Thick Avoided 2 nd Ray-Knight Theorem Biskup- points Louidor LQG points for DGFF Pinned Isomorphism Theorem by Local Pinned Rodriguez Occupation time field structure DGFF of RI