Exceptional points of two-dimensionalrandom walks at multiples of thecover 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 DN with wired boundary condition
D ⊂ R2 : “good” bounded open set
DN ⊂ Z2 : “good” lattice approximation of Dxx ∈ DN ⇒ N∈D(Xt )t≥0 : Continuous-time SRW on DN withExp(1)-holding times

Technical Assumption: When X exits DN , it re-enters DN through a
uniformly-chosen boundary edge.
⇝ Regard ∂DN as a single point ρWe assume this to relate our local times to DGFF with zero boundary
conditions via the 2nd Ray-Knight theorem.

DNLocal time Lt
Recall: (Xt )t≥0 = SRW on DN ,Local time:
DLt N (x) :=ρ = the boundary vertex∫ τρ (t)01
1{Xs =x} ds,
deg (x)where{∫ sτρ (t) := inf s ≥ 0 :1{Xr =ρ} dr0Let tN be a sequence withtN

1 (log N )2π

N →∞−→ θ ∈ (0, 1).

⇝ τρ (tN ) ≈ θ × (cover time of DN )D⇝ Lt N ≈ local time at θ × (cover time of DN )N

1>tdeg(ρ)}.

Main ResulttN1 (log N )2π

DRecall: Lt N ≈ local time at θ × (cover time of DN )N2tN− 1
N →∞−→ θ ∈ (0, 1), WN := N 2 e π log N1
κD:=NWN∑x∈DN1δxD{Lt N (x)=0} N
N= N 2−2θ+o(1)⊗δD{Lt N (x+z) : z∈Z2 }N

Main Theorem. (A.-Biskup)
κDN•lawRID(dx)⊗ν
−→ cθ Z√θ (dϕ),
θN →∞(2λ)22λφD− 1 Var(2λφDDDxx ) dx,2Zλ (dx)“ = ”r (x) 2 ea Liouville Quantum Gravity measure on D

• νθRI is the law of occupation time field of
the two-dimensional random interlacement at level θ.

λ ∈ (0, 1)

Idea of proof.
1
Recall: κD
N := WN

Recall: κDNlaw
D (dx) ⊗ ν RI (dϕ).
−→ cθ Z√θθN →∞∑1δ x ⊗δ
DNDN2N{L(x)=0}{Lx∈DNtNtN (x+z) : z∈Z }Avoidedpoints

Localstructure

2nd Ray-Knight Theorem

Thickpoints
for DGFF

PinnedDGFF

BiskupLouidor

LQG

PinnedIsomorphism
Theorem byRodriguez Occupation

time fieldof RI

Heuristics of
Avoided points↔ Thick points for DGFF

Discrete Gaussian Free Field (DGFF) and the maximum
DDefinition. hDN = (hx N )x∈DN is DGFF on DN
def⇔ hDN is centered Gaussian withE[DDhx N hy N]=GDN(x, y)

:= Ex

[∫ H

∂DN01
1{Xs =y} ds.
deg(y)

Theorem. (Bolthausen-Deuschel-Giacomin (2001))
Dmaxx∈DN hx Nlog NN →∞√−→2πD

1Note: Var(hx N ) = 2πlog N + O(1)Remark. 2nd order: Bramson-Zeitouni (2011)
Convergence in law: Bramson-Ding-Zeitouni (2016)]

in probab.

Convergence of λ-thick points λ ∈ (0, 1)
Recall:maxx∈DD
ηN:= K1
NNlog NDhx N∑

N →∞xx∈DN δ N

N →∞aN→ λlog N√−→√2π⊗ δ DNhx

in probab.
−aN⊗δDD{hx N −h N : z∈Z2 }
x+za2
NN22√
π , KN :=log N− 1e π log N=N2−2λ2 +o(1).

Theorem. (Biskup-Louidor (2016))
DηNlaw−→N →∞√D−2 2πλhc(λ)Zλ (dx) ⊗ edh ⊗ νλ ,(2λ)22λφD− 1 Var(2λφDDDxx ) dx
22ewhere Zλ (dx)“ = ”r (x)νλ is the law of ϕ + 2√LQG on D,

2π λa, ϕ is DGFF on Z2 pinned to zero at the origin.

Heuristics of avoided points ↔ thick points
Dx is a λ-thick point ⇔ hx N√12Recall: tN ≈ θ π(log N )2 ,≈λ πlog N
Key: 2nd Ray-Knight Theorem (Eisenbaum-Kaspi-Marcus-Rosen-Shi (2000))

{}
1DD2ρ
Lt N (x) + (hx N ) : x ∈ DNunder P ⊗ PN
2law=Thus,{)2√
1 ( DNhx+ 2tN: x ∈ DN2√}.

D↔ x is θ-thick pointZ√θ√DN↔ hx + 2tN ≈ 0N
↔ LDtN (x) = 0. i.e. x is an avoided point.

Heuristics ofLocal structure of avoided points↔ 2D random interlacements

Two-dimensional random interlacements
Poissonian soup of trajectories ofSRWs conditioned on never hitting the origin.

0Z

2

Two-dimensional random interlacements
Poissonian soup of trajectories ofSRWs conditioned on never hitting the origin.

0AZ

2What 2D RI at level θ looks like 0 ∈ A ⊂ Z2 finiteeA ( · )Take i.i.d. Poi (πθcap(A)) samples from the law cap(A) ,where eA is the equilibrium measure and cap is the capacity:eA (x) := 4a(x)hmA (x) := 4a(x)cap(A) :=∑lim|y|→∞yP [XHA = x], x ∈ A,x∈A eA (x).

AWhat 2D RI at level θ looks like 0 ∈ A ⊂ Z2 finite
From each point, start indep two walks (blue and green);Blue paths avoid 0 and green paths never return to A.

0A

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) : Zd , d ≥ 3,Teixeira (’09) : general transient weighted graphs

Occupation time field for 2D RI at level θLet (wi )i∈N be the doubly-infinite trajectories in thetwo-dimensional random interlacements at level θ.
The occupation time field is defined by∑1∫ ∞1{wi (t)=x} dt,ℓRI
θ (x) :=
4−∞i∈Nx∈Z .

lawDRI√⊗−→cZ
Recall the main theorem: κD
ν
(dx)
θN
θ (dϕ)
θwith κD
N =1WN∑N →∞1
δ x ⊗δ,DNDN
2{Lt(x+z) : z∈Z }x∈DN {LtN (x)=0} NNDtN ≈ θ 1 (log N )2 , Lt N ≈ local time at θ × (cover time of DN )πNνθRI = the law of (ℓRIθ (x))x∈Z2 .

2

Heuristics of local picture ↔ RI
1Recall: tN ≈ θ π(log N )2 , (ℓRIθ (z))z∈Z2 : Occupation time field of RI

Key: Pinned Isomorphism Theorem (Rodriguez (2019)){ ({}})
√2
1
1law
22RI
2=ϕz + 2 2πθ aℓθ (z) + (ϕz ) : z ∈ Z:z∈Z,22where (ϕz )z∈Z2 be DGFF on Z2 pinned to zero at the origin.
DRecall: Lt N ≈ local time at θ × (cover time of DN )N)()(√DN
DN
DN
DNDNθ × max(Lt(x + z))z | Lt(x) = 0 ↔ (h− hx) z | hx
≈x+zNN↔ (ϕz + 2√2πθ a)z .

) law(DNDN
RI(x + z))z∈Z2 | Lt(x) = 0 ≈ (ℓθ (z))z∈Z2 .
∴ (LtNNThank you.

Avoided 2 Ray-Knight Theorem ThickpointspointsBiskupLouidor

nd

LQG

for DGFF

Local
structure

PinnedDGFF

PinnedIsomorphism
Theorem byRodriguez Occupation

time fieldof RI