On the Liouville heat kernel for k-coarse MBRW and nonuniversality

0.27 MB·

by  Jian Ding

k
On the Liouville heat kernel for -coarse MBRW and
nonuniversality
Jian Ding Ofer Zeitouni Fuxi Zhang
∗ † ‡
University of Chicago Weizmann Institute Peking University
7
1 Courant Institute
0
2

n
a
J Abstract

5 WestudytheLiouvilleheatkernel(intheL2phase)associatedwithaclassoflogarithmically

correlated Gaussian fields on the two dimensional torus. We show that for each ε > 0 there
]
R exists such a field, whose covariance is a bounded perturbation of that of the two dimensional
P Gaussian free field, and such that the associated Liouville heat kernel satisfies the short time
. estimates,
h
at exp −t−1+112γ2−ε ≤pγt(x,y)≤exp −t−1+112γ2+ε ,
m (cid:18) (cid:19) (cid:18) (cid:19)
for γ <1/2. Inparticular,these are differentfrompredictions,due to Watabiki,concerningthe
[
Liouville heat kernel for the two dimensional Gaussian free field.
1
v
1
1 Introduction
0
2
1 In recent years, there has been much interest and progress in the understandingof two dimensional
0
Liouville quantum gravity, and associated processes. We do not provide an extensive bibliography
.
1 and refer instead to the original articles and surveys [9, 10, 5] for background. The starting point
0
for this study is the construction of Liouville measure, which is the exponential of the Gaussian
7
1 free field and is constructed rigorously using Kahane’s theory of Gaussian multiplicative chaos [17].
v: One aspect that has received attention is the construction of Liouville Brownian motion using
i the Liouville measure and the theory of Dirichlet forms. Mathematically, this has been achieved
X
in [11] (see also [4]), and properties of the associated Liouville heat kernel have been discussed in
r
a [12, 15, 2]. One important motivation behind the study of the Liouville heat kernel is that it can
be used to study the geometry (and critical exponents) of Liouville quantum gravity. Indeed, a
particularly nice application of the construction of the Liouville heat kernel is that it allows for
a clean derivation of the so-called KPZ relations [3]. Another important motivation, discussed in
[15], are the predictions of Watabiki [18] concerning the short time behavior of the Liouville heat
kernel. See the discussion in [15, 2] for existing (weak) estimates on the diffusivity exponents of
the Liouville heat kernel.
∗Partially supported by an NSFgrant DMS-1455049, an Alfred Sloan fellowship, and NSFof China 11628101.
Partially supported by the ERC advanced grant LogCorrelatedFields and by the Herman P. Taubman chair at
†
theWeizmann Institute.
‡Supported byNSFof China 11371040.
1
An important aspect of the class of logarithmically correlated Gaussian fields (of which the
2D Gaussian free field is arguably the prominent example) is the universality of many quantitites,
e.g. Hausdorff dimensions, statistics of the maximum, etc., see [17, 7]. One could naively expect
that for Gaussian fields in this class, the predicted exponents of the Liouville heat kernel would be
universal.
Ourgoalinthispaperistoshowthatthisisnotthecase,inthesensethattheexplicitpredictions
on Liouville heat-kernel exponents (appearing in [18] and discussed in [15, 2]) do not hold for some
two dimensional logarithmically correlated Gaussian fields which are bounded perturbations of the
Gaussian free field. Namely, we study in this paper the heat kernel for Liouville Brownian motion
constructed with respect to a particular logarithmically correlated field, introduced in [6] under the
name k-coarse modified branching random walk (MBRW for short). Given k > 0 integer, this is
the centered Gaussian field on the torus T = R2/(4Z)2, denoted h = h(x) , with covariance
x T
{ } ∈
∞
G(x,y) = klog2 A(x,y;2 kj),
−
j=0
X
where A(x,y;R) = B(x,R) B(y,R)/B(x,R), B(z,R) is the (open) ball centered at z with
| ∩ | | |
radius R with respect to the natural metric on the torus, and B is the Lebesgue measure of a set
| |
B. Theparticular choice of thescaling of the torus is not importantand only done for convenience.
We will show in Section 2.1 that for all k,
1
G(x,y) = log +λ(x y ), (1)
x y | − |
| − |
where λ is continuous in (0,2] and λ 6k. Fixing γ (0,2), we introduce in Section 2.3,
| | ≤ ∈
following [11], the Liouville measure µγ, Liouville Brownian motion (LBM) Y , and Liouville
t
γ { }
heat kernel (LHK) p (x,y), associated with (γ,h). Formally, the Liouville measure on T is defined
t
as µγ(dx) := eγh(x)−12γ2Eh2(x)dx; one then introduces the positive continuous additive functional
(PCAF) with respect to µγ as
F(v) := veγh(Xu)−γ22Eh(Xu)2du,
Z0
where X denotes a standard Brownian motion (SBM) on T. The LBM is then defined formally
t
{ } γ
as Y := X , and the LHK p (x,y) is then the density of the Liouville semigroup with respect
t F−1(t) t
to µγ, i.e.
Exf(Y )= pγ(x,y)f(y)µγ(dy),
t t
Z
where the superscript x is to recall that Y = X = x.
0 0
Let P denote the Gaussian law of h. The main result of this paper is as follows.
Theorem 1.1. Suppose 0 γ < 1, and x,y T with x = y. For any ε > 0, there exist k(ε,x,y)
≤ 2 ∈ 6
and a random variable T depending on (x,y,γ,k,ε,h) only so that for any k k(ε,x,y) and
0
≥
t < T ,
0
1 ε 1 +ε
exp −t−1+21γ2− ≤ pγt(x,y) ≤ exp −t−1+12γ2 , P-a.s.. (2)
(cid:18) (cid:19) (cid:18) (cid:19)
2
Remark 1.2. Ourresult shows that the exponent of the LHK with respect to the k-coarse MBRW
is for large k and small γ, roughly (1+o (1))/(1+γ2/2). In particular, it does not match values
k
one could guess from Watabiki’s formula, see [18, 15], based on which one would predict that for γ
small, theexponentis(1+o(γ))/(1+7γ2/4). Thisisyetanothermanifestation oftheexpectednon-
universality of exponents related to Liouville quantum gravity, across the class of logarithmically
correlated Gaussian fields. See [6, 8] for other examples.
Heuristic. We describe the strategy behind the proof of the lower bound, and the upper bound
is similar. First, represent hierarchically the MBRW as follows. Let h be independent centered
j
Gaussian fields on T with covariance
Ehj(x)hj(y) =klog2 A(x,y;2−kj) =:gj(x,y). (3)
×
Formally, h = ∞j=0hj. For given t, choose r such that t = 2−kr(1+12γ2−o(1)), and decompose the
field h into a coarse field ϕ and a fine field ψ , with
r r
P
r 1
− ∞
ϕ := h , ψ := h , (4)
r j r j
j=0 j=r
X X
with respective covariances
r 1
− ∞
G(1)(x,y) = klog2 A(x,y;2 kj), G(2)(x,y) = klog2 A(x,y;2 kj). (5)
r − r −
j=0 j=r
X X
Note that much like the MBRW, the fine field is not defined pointwise but only in the sense of
distributions.
With k,r fixed, we partition T into 22(kr+2) boxes of side length s = 2 kr, elements of
−
= [a2 kr,(a+1)2 kr) [b2 kr,(b+1)2 kr) .
BDr { − − × − − }a,b∈[0,2kr+2)∩Z
We call theelements of s-boxes. Similarly to [6], we willfindasequence of neighborings-boxes
r
BD
B , 1 i I (with I 2kr(1+δ), δ chosen below) connecting x to y, sothat the following properties
i
≤ ≤ ≤
(of the B ’s) hold. The coarse field ϕ throughout each B is bounded above by δkrlog2, where
i r i
δ > 0 is small and will be chosen according to ε in Theorem 1.1. With probability at least sδ,
the LBM associated with the fine field ψ crosses each B within time s2 δ. Forcing the original
r i −
LBM to pass through this sequence of boxes, we will then conclude that it spends time at most
2kr(1+δ) 2δγkr−12γ2krs2−δ = 2−kr(1+12γ2−(2+γ)δ) = t1+O(ε) crossingfrom x tothe s-boxcontaining
≤ × 1 +ε
y. This happens with probability at least (sδ)−2kr(1+δ) exp( t−1+21γ2 ), and, modulu a
≥ ≥ −
localization argument, completes the proof of the lower bound.
Structure of the paper. The preliminaries Section 2 is devoted to the study of the covariance of
the k-coarse MBRW h, and in particular to verifying that its covariance is a bounded perturbation
of that of theGaussian freefield. We also discussthepower law spectrumof handtheconstruction
of the LBM with its corresponding PCAF. In addition, Section 2.2 is devoted to a study of the
coarse field ϕ , and results in estimates on its fluctuations and maximum in a box. Section 3 is
r
devoted to a study of the fine field; we introduce the notions of slow and fast points/boxes and
estimate related probabilities. (The property of being fast is used in the proof of the lower bound,
3
and that of being slow is used in the upper bound.) Finally, the proof of lower bound is contained
in Section 4, and that of upper bound is contained in Section 5. Both these sections borrow crucial
arguments from [6].
Notation convention. Throughoutthepaper,werestrictattention to0 γ < 1/2. Tisequipped
≤
with the natural metric inherited from the Euclidean distance. We choose δ > 0 small and k large
integer (as functions of ε) and keep them fixed throughout. We let C , i = 0,1,... be universal
i
positive constants, independent of all other parameters. With r as described above, we let BD (x)
r
denote the unique element of containing x. For ℓ > 0, an ℓ-box means a box of side length
r
BD
ℓ. Let B (x) denote the ℓ-box centered at x, and let B(x,ℓ) denote the ball centered at x with
ℓ
radius ℓ. For any box B, let c denote the center of B. If B is an ℓ-box, denote by B the (5ℓ)-box
B ∗
centered at c . We use P and E to denote the probability and expectation related to the Gaussian
B
field h. Let Px and Ex bethe probability and expectation related to theSBM starting at x. We let
Fx and Fx be the PCAFs for the LBM and ψ -LBM started at x, respectively. When the starting
r r
point x needs not be emphasized, we drop the superscript x.
2 Preliminaries
Subsection 2.1 is devoted to the proof of (1). In Subsection 2.2, we study the coarse field ϕ and
r
bound its maximum on small boxes as well as the fluctuation across such boxes. Subsection 2.3 is
devoted to a quick review of the construction and existence of the LBM and the LHK.
2.1 Proof of (1)
Let d denote the T distance between x,y, and fix r := r (d) 0 integer so that
0 0
≥
d
2 k(r0+1) < 2 kr0.
− −
2 ≤
Denote
θ := arcsin(2kjd/2), j = 0,1,...,r .
j,d 0
We compute the covariance g (x,y), c.f. (3). For j r , note that R := 2 kj d; set θ = θ .
j ≤ 0 − ≥ 2 j,d
Then B(x,R) B(y,R) = (π 2θ)R2 2R2sin(θ)cos(θ) = πR2 R2(2θ+sin(2θ)), which implies
| ∩ | − − −
that A(x,y;R) = 1 1(2θ+sin(2θ)). It follows that with j Z ,
− π ∈ +
klog2 klog2 2θ +sin(2θ ) , if j r ,
gj(x,y) = − π j,d j,d ≤ 0 (6)
0, otherwise.
(cid:26) (cid:0) (cid:1)
We now write
∞ r0 1 r0
G(x,y) = g (x,y) = g (x,y) = klog2 (r +1) 2θ +sin(2θ ) . (7)
j j 0 j,d j,d
 − π 
j=0 j=0 j=0
X X X(cid:0) (cid:1)
 
Since r = r (d), we obtain that G(x,y) = g(d) for some function g : (0,2] R . We now show
0 0 +
→
that g is continuous. Indeed, note that for any fixed j, d θ is continuous (in d [0,21 kj]).
j,d −
7→ ∈
Thus the only possible discontinuities of g on (0,2] are whenever log (d/2)/k is an integer (i.e.
− 2
4
equals r (d)); however, for such d we obtain that θ = π/2, which together with the continuity
0 r0(d),d
of d θ , yields the continuity of g.
j,d
7→
To estimate g(d), note that for all θ [0, π], 0 sin(2θ) 2sin(θ) and θ 2sin(θ), and
∈ 2 ≤ ≤ ≤
therefore
0 2θ+sin(2θ) 6sin(θ). (8)
≤ ≤
In particular,
1 r0 6 r0 6 ∞ 12
(2θ +sin(2θ )) 2 k(r0 j) 2 ki 4.
j,d j,d − − −
π| | ≤ π ≤ π ≤ π ≤
j=0 j=0 i=0
X X X
On the other hand, k(r +1)log2+logd (k +1)log2 2k. Combining the last two displays
0
| | ≤ ≤
with (7) shows that
g(d)+logd 6k,
| | ≤
yielding (1).
2.2 The coarse field
Note that g (x,y) is a positive definite kernel on L2(T), since, with R = R = 2 kj,
j j −
gˆ (x,y) = B(0,R)g (x,y) = dz 1 1
j j z x R z y R
| | ZT | − |≤ | − |≤
and therefore, for any f L2(T),
∈
2
f(x)f(y)gˆ (x,y)dxdy = dz dx f(x)1 0.
j x z R
Z(T)2 ZT (cid:18)ZT | − |≤ (cid:19) ≥
Since g (x,y) is Lipshitz continuous, Kolmogorov’s criterion implies that the associated Gaussian
j
field x h (x) is continuous almost surely (more precisely, there exists a version of the field which
j
7→
is continuous almost surely). Consequently, the coarse field ϕ is also smooth. In this subsection,
r
we estimate the maximum value as well as the fluctuations of ϕ in a box.
r
We begin by recalling an easy consequence of Dudley’s criterion.
Lemma 2.1. ([1, Theorem 4.1]) Let B Z2 be a box of side length ℓ and η :w B be a mean
w
⊂ { ∈ }
zero Gaussian field satisfying
E(η η )2 z w /ℓ for all z,w B.
z w
− ≤ | − |∞ ∈
Then Emax η C , where C is a universal constant.
w B w 0 0
∈ ≤
Thenextlemmais usuallyreferredtoas theBorell, orIbragimov-Sudakov-Tsirelson, inequality.
See, e.g., [14, (7.4), (2.26)] as well as discussions in [14, Page 61].
Lemma2.2. Let η : z B beaGaussianfieldonafiniteindexsetB. Setσ2 = max Var(η ).
z z B z
{ ∈ } ∈
Then for all λ,a > 0,
λ2σ2 a2
E[exp λ(maxηz Emaxηz) ] e 2 , and P( maxηz Emaxηz a) 2e−2σ2.
{ z B − z B } ≤ | z B − z B | ≥ ≤
∈ ∈ ∈ ∈
5
Proposition 2.3. Suppose k is large. For all r 1,
≥
E(ϕ (x) ϕ (y))2 2kr x y , x,y T.
r r
− ≤ | − | ∀ ∈
Proof. Use the notation in Subsection 2.1. Let d= x y , r = r (d). By (6) and (8),
0 0
| − |
2klog2 2kd2kj, j r ,
E(hj(x)−hj(y))2 = π 2θj,d+sin(2θj,d)) ≤ 2k, ∀j >≤ r00,
(cid:26) ∀
(cid:0)
where we use sin(θ ) = 2kjd/2 in the case j r .
j,d 0
≤
If r r 1,
0
≥ −
r 1 r 1
− −
E(ϕ (x) ϕ (y))2 = E(h (x) h (y))2 2kd 2kj 2krd.
r r j j
− − ≤ ≤
j=0 j=0
X X
Otherwise, r r 2.
0
≤ −
r0
E(ϕ (x) ϕ (y))2 =2k(r r 1)+ 2kd2kj 2k(r r 1)+4kd2kr0.
r r 0 0
− − − ≤ − −
j=0
X
Note 2krd 2k(r r0 1)+1 and r r 1 1. It follows that
− − 0
≥ − − ≥
k(r r 1) 4k
E(ϕ (x) ϕ (y))2 − 0− 2krd+ 2krd 2krd,
r − r ≤ 2k(r r0 1) 2k(r r0) ≤
− − −
since k is large enough.
Corollary 2.4. Suppose k islarge. LetB denote aboxof sidelengthℓ, andsetM := max ϕ (z).
z B r
Then, EM √2C √2krℓ. ∈
0
≤
Proof. We discretize B by dividing B into 22n identical boxes B˜’s and identifying the lower left
corner c˜ of each B˜ as a point in Z2. Denote by M the maximum value of ϕ over these c˜’s.
n r
By the continuity of the coarse field, M increases to M as n . By Proposition 2.3, we can
n
→ ∞
apply Lemma2.1 toϕ /√2kr2ℓ andconclude thatEM √2C √2krℓ. Themonotone convergence
r n 0
≤
theorem yields the result.
Corollary 2.5. There exist r = r (k,δ) such that the following holds for k large and r r .
0 0 0
≥
Enumerate the boxes in arbitrarily as B , i = 1,...,22(kr+2). Denote M = max ϕ (x),
BDr i i x∈Bi∗ r
Mf = sup ϕ (x) ϕ (c ), and Mf = max Mf. Then
i x∈Bi∗| r − r Bi | 1≤i≤22(kr+2) i
P(Mi δkrlog2) 2e−18δ2krlog2, P(Mf δkrlog2) e−r.
≥ ≤ ≥ ≤
Proof. Note that, for all x, Eϕ (x)2 = krlog2. By Corollary 2.4, EM √2C √5 1δkrlog2 for
r i ≤ 0 ≤ 2
r r (k,δ). By Lemma 2.2,
0
≥
1
P(Mi δkrlog2) P(Mi EMi δkrlog2) 2e−(12δkrlog2)2/(2krlog2) = 2e−81δ2krlog2.
≥ ≤ − ≥ 2 ≤
6
DenoteMˆif := supx∈Bi∗(ϕr(x)−ϕr(cBi)). Similarly,wehaveP(Mˆif ≥ δkrlog2) ≤2e−312(δkrlog2)2,
noting EMˆf = EM and by Proposition 2.3, E(ϕ (x) ϕ (c ))2 2kr x c 4 for all x B .
i i r − r Bi ≤ | − Bi|≤ ∈ i∗
Furthermore, by a union bound and symmetry,
22(kr+2)
P(Mf ≥ δkrlog2) ≤ 2P(Mˆif ≥ δkrlog2) ≤ 64×22kre−(δkl3o2g2)2r2 ≤ e−r,
i=1
X
where in the last inequality we use r r (k,δ).
0
≥
2.3 Construction of the LBM and LHK
Thereare several ways to construct the Liouville measure µγ with respect to h, say, via the method
of Gaussian multiplicative chaos [13]. In our case, since we deal with γ < 1/2, it is particulaly
simple since L2 methods apply. So, in the rest of this section we concentrate on the construction
of the LBM and LHK.
Suppose ε= 2 kr. Then,
−
G(x,y) = G(2)(εx,εy), i.e. G(εx,εy) = G(x,y)+G(1)(εx,εy) (9)
r r
since A(εx,εy;2 k(r+j))= A(x,y;2 kj). By (6),
− −
1
G(1)(εx,εy) G(1)(εx,εx) = krlog2 = log .
r ≤ r ε
It follows that
1
G(εx,εy) G(x,y)+log . (10)
≤ ε
(1)
Let Ω be a Gaussian field independent of h, with EΩ = 0 and EΩ (x)Ω (y) = G (εx,εy).
ε ε ε ε r
Actually, Ω is a copy of the coarse field ϕ if we regard x as εx. Then
ε r
d d
h(εx) = h(x)+Ω (x) , Ω (x) = ϕ (εx) .
x ε x ε x r x
{ } { } { } { }
Let M = max Ω (x). It follows that for q [0,4/γ2],
x [ 1,1]2 ε
∈− ∈
Eµγ(B(0,ε))q ε(2+12γ2)qEeγqMEµγ(B(0,1))q.
≤
Note M =d maxx [ ε,ε]2ϕr(x). By Lemma 2.2 and Corollary 2.4, EeγqM C˜(q)ε−12γ2q2 , where
C˜(q) is a constan∈t −depending on q (as well as γ). Thus ≤
Eµγ(B(0,ε))q Cˆ(q)εξ(q),
≤
where Cˆ(q) = C˜(q)Eµγ(B(0,1))q, and
γ2 γ2
ξ(q) = (2+ )q q2.
2 − 2
For any 2 k(r+1) < ε 2 kr, we take C(q)= Cˆ(q)2 kξ(q) and conclude that
− − −
≤
Eµγ(B(0,ε))q Eµγ(B(0,2−kr))q Cˆ(q)2−krξ(q) C(q)εξ(q). (11)
≤ ≤ ≤
7
Recall that the coarse field ϕ is smooth, so
r
u
Hr(u) := eγϕr(Xv)−21γ2Eϕr(Xv)2dv
Z0
is well-defined.
With (10) and (11), one can follow the arguments in [11, Section 2] and obtain the following
conclusions. Let F denote the PCAF associated with µγ. Then, P-a.s., the limit of H in Px-
r
probability exists and it is the PCAF F; that is, Px(sup F(u) H (u) > a) 0, for all
0≤t≤T | − r | →r→∞
a > 0 and T > 0. Further, the process Y := X is a strong Markov process, which is called the
t F 1(t)
LBM with respect to µγ. The LHK pγ(x,y) ex−ists and satisfies Exf(Y ) = f(y)p (x,y)µγ(dy).
t t t
γ
Furthermore, by [12, Theorem 0.1] and parallel arguments in [15], p (x,y) is continuous in (t,x,y).
t R
3 Fast/slow points/boxes of the fine field
This section is devoted to the study of properties of the fine field. For the lower bound on the
LHK, we need to construct regions which are fast to cross for the LBM, while for the upper bound
we will need to create obstacles, i.e. regions which force the LBM to be slow. Toward this end,
we introduce in Definitions 3.1 and 3.2 the notions of fast/slow points and boxes, and estimate, in
Lemma 3.3 and 3.4, the probability that a point/box is fast/slow.
Throughout, we fix s = 2 kr for an appropriate integer r 1 (as explained in the introduction,
−
≥
r, and hence s, are chosen so that t = s1+12γ2+o(1)). This choice determines the fine field ψr,
see (4). With this choice, one can construct the PCAF F based on ψ in the same way as F
r r
was constructed, by replacing the measure µγ with the truncated measure µγ written formally
r
as µγr(dx) = eγψr(x)−γ22Eψr2(x)dx (as before, the actual construction involves the smooth cutoff
ψ := w h and taking the limit as w ). Formally, we write
r,w j=r j → ∞
P v
Fr(v) = eγψr(Xu)−12γ2Eψr(Xu)2du. (12)
Z0
We note also that the sequence of approximating PCAF
v
Fr,w(v) := eγψr,w(Xu)−12γ2Eψr,w(Xu)2du
Z0
converges as w , in the sense described at the end of Section 2, to F .
r
→ ∞
Fix δ ,δ ,δ ,ε ,ε ,ε > 0 small, possibly depending on k,γ and s. Fix z T and recall that
1 2 3 1 2 3
∈
B (z) denotes the ℓ-box centered at z. Let σ denote the time that the SBM (starting from z)
ℓ z,ℓ
hits ∂B (z).
ℓ
Definition 3.1 (Fast points and boxes). A point z is said to be fast if
Pz(F (s2 σ ) s2/δ ) 1 δ . (13)
r z,6s 1 2
∧ ≤ ≥ −
The set of fast points is denoted by . An s-box B is said to be fast if B δ s2.
3
F | ∩F| ≥
Definition 3.2 (Slow points and boxes). A point z is said to be slow if
Pz(F (σ ) ε s2) ε . (14)
r z,s 1 2
≥ ≥
The set of slow points is denoted by . An s-box B is said to be slow if B ε s2.
3
S | ∩S|≥
8
We emphasize that the notions of fast/slow points and boxes depend on the fine field ψ only.
r
Further, a point (or box) may be fast and slow simultaneously.
Our fundamental estimate concerning fast/slow points is contained in the next lemma.
Lemma 3.3. There exist universal positive constants C ,C ,C such that the following hold.
1 2 3
(i) P(z ) 1 C δ1.
∈ F ≥ − 1δ2
(ii) For ε C and ε C e 6kγ2, we have P(z ) 120C e 6kγ2.
1 2 2 3 − 3 −
≤ ≤ ∈ S ≥
Proof. (i) Set ξ = Fz(s2 σ ) and η = Pz(ξ > s2/δ ). By definition,
r ∧ z,6s 1
P(z / ) = P(η > δ ) Eη/δ . (15)
2 2
∈F ≤
Note that
δ δ
Eη = EzP(ξ > s2/δ ) 1EzEξ = 1Ez(s2 σ ).
1 ≤ s2 s2 ∧ z,6s
Define C := E0(1 σ ), where σ is the time that the SBM in R2 hits the boundary of [ 3,3]2.
1 6 6
∧ −
Then, by scale invariance of Brownian motion, Ez(s2 σ ) = C s2. Combining the last two
z,6s 1
∧
displays with (15), one obtains P(z / ) C δ /δ , completing the proof.
1 1 2
∈ F ≤
(ii)Weusetheabbreviationσ =σ andsetnowξ = Fz(σ)andη = Pz(ξ ε s2). Withoutloss
z,s r ≥ 1
of generality, we suppose z = (0,0) and consistently drop z from the notation, writing B = B (z).
s s
Since η 1, we have Eη = Eη1 +Eη1 P(η ε )+ε . By definition,
≤ η≥ε2 η<ε2 ≤ ≥ 2 2
P((0,0) )= P(η ε ) Eη ε = EP(ξ ε s2) ε . (16)
2 2 1 2
∈ S ≥ ≥ − ≥ −
We are going to estimate P(ξ ε s2) via the second moment method. Recall that Eξ = σ, which
1
≥
has order s2. To compute the second moment, note that since γ < 1/2, the sequence of squares
of approximating PCAFs (F )2 are uniformly (in w) integrable (see the argument just after (17)
r,w
below) and therefore
σ σ
Eξ2 = EFr(σ)2 = Eeγψr(Xu)−12γ2Eψr(Xu)2+γψr(Xv)−21Eψr(Xv)2dudv
Z0 Z0
σ σ
= eγ2G(r2)(Xu,Xv)dudv = eγ2G(r2)(w,w′)ν(dw)ν(dw′) =:Iγ2,
Z0 Z0 Zw,w′∈Bs
(2)
where X is the SBM starting from (0,0), G is defined in (5), and ν denotes the occupation
u r
{ }
measure of X before exiting B , i.e.
u s
{ }
σ
f(w)ν(dw) = f(X )du.
u
Zw∈Bs Z0
Let wˆ = 2krw and wˆ = 2krw , with wˆ,wˆ T. By (1) and (9),
′ ′ ′
∈
1 s
G(2)(w,w ) = G(wˆ,wˆ ) log +6k = log +6k.
r ′ ′ ≤ wˆ wˆ w w
′ ′
| − | | − |
Consequently,
1 σ σ 1
I e6kγ2sγ2 ν(dw)ν(dw ) = e6kγ2sγ2 dudv.
γ2 ≤ w w γ2 ′ X X γ2
Zw,w′∈Bs | − ′| Z0 Z0 | u − v|
9
LetXˆ = 1X ,andletσˆ = σ/s2 bethetimethattheSBM Xˆ startedat(0,0)exits[ 1/2,1/2]2.
u s s2u { } −
Then
σˆ σˆ 1
I e6kγ2s4 dudv.
γ2 ≤ Xˆ Xˆ γ2
Z0 Z0 | u − v|
Note Xˆu Xˆv γ2 Xˆu Xˆv 1/4, since Xˆ Xˆ √2 and γ2 1/4. Thus,
| √−2 | ≥ | √−2 | | u − v|≤ ≤
1
Xˆ Xˆ γ2 Xˆ Xˆ 1/4.
u v u v
| − | ≥ 2| − |
It follows that
σˆ σˆ 1
I 2e6kγ2s4Iˆ, where Iˆ= dudv. (17)
γ2 ≤ Z0 Z0 |Xˆu −Xˆv|1/4
NotethatIˆisarandomvariabledependingonlyontheSBM Xˆ . By[16,Theorem4.33], EIˆ< .
{ } ∞
Consequently, there exists a universal constant C˜ such that P(Iˆ 1C˜ ) 3/4. Hence, the event
1 ≤ 2 1 ≥
E := Eξ2 C˜ e6kγ2s4 has probability P(E ) 3/4. By the scaling invariance of the SBM,
1 1 1
{ ≤ } ≥
there exists a universal positive constant C such that the event E = σ 2C s2 has probability
2 2 2
{ ≥ }
3/4. Thus, P(E E ) 1/4.
1 2
≥ ∩ ≥
Assume E E happens. On the one hand, on E ,
1 2 1
∩
2
P(ξ ≥ ε1s2)≥ (cid:0)Eξ1Eξ≥ξε21s2(cid:1) ≥ C˜1e61kγ2s4 Eξ1ξ≥ε1s2 2.
(cid:0) (cid:1)
On the other hand, on E , ξ = F (σ) F (2C s2)=:ζ. Note that 2C s2 = Eζ Eζ1 +ε s2.
We have Eξ1 Eζ21 r (2≥C r ε )2s2 C s2, where we u2se the as≤sumptζio≥nε1sε2 1C .
ξ≥ε1s2 ≥ ζ≥ε1s2 ≥ 2 − 1 ≥ 2 1 ≤ 2
Thus,
C s2 2 C2
P(ξ ≥ ε1s2) ≥ C˜ e26kγ2s4 = C˜2e−6kγ2, on E1∩E2.
(cid:0)1 (cid:1) 1
Consequently,
C2 C2
EP(ξ ≥ ε1s2) ≥ E P(ξ ≥ ε1s2)1E1∩E2 ≥ C˜2e−6kγ2 ×P(E1∩E2)≥ 4C˜2 e−6kγ2.
1 1
(cid:0) (cid:1)
Take C := C2/(484C˜ ). Then EP(ξ ε s2) 121C e 6kγ2. This, together with (16) and the
assump3tion ε 2 C e 61kγ2, implies the≥resu1lt. ≥ 3 −
2 3 −
≤
The next lemma estimates the probability that an s-box B is fast/slow.
Lemma 3.4. (i) P(B is fast) 1 C δ1 δ .
≥ − 1δ2 − 3
(ii) Suppose ε C e 6kγ2 and ε C2e 12kγ2. Then, P(B is slow) 1 εC3e−6kγ22−2k if ε is
2 ≤ 3 − 3 ≤ 3 − ≥ − 1 1
less than some constant ε (γ,k).
1
Proof. (i) By Lemma 3.3(i) and the translation invariance of the fine field ψ , E B (1
r
| ∩F| ≥ −
C δ1)s2. Since B B s2, B B 1 + B 1
1δ2 | ∩ F| ≤ | | ≤ | ∩ F| ≤ | ∩ F| |B∩F|<δ3s2 | ∩ F| |B∩F|≥δ3s2 ≤
δ s2 + s21 . Hence, E B δ s2 s2P(B δ s2) = s2P(B is fast). There-
fo3re, P(B i|sBf∩aFs|t≥)δ3s21 E B | ∩δ Fs2| − 13 C≤ δ1 |δ .∩ F| ≥ 3
≥ s2 | ∩F|− 3 ≥ − 1δ2 − 3
(cid:0) (cid:1)
10