... E
2
)=E
1
(assuming this
Annals of Mathematics
Geometry of the uniform
spanning forest: Transitions
in dimensions 4, 8, 12, . . .
By Itai Benjamini, Harry Kesten, Yuval Peres, and
Oded Schramm
GEOMETRY ... uniform spanning forest:
Transitions in dimensions 4, 8, 12,
By Itai Benjamini, Harry Kesten, Yuval Peres, and Oded Schramm*
Abstract...
... Physicians. And in the Plague
time they took upon them the whole
Practice of Physic, which ever since
they have continued, being much helped
also therein by the dispersing of
Physicians into places ... return, some
of them have kept up the honour of the
Faculty; which manifestly appears by the
great esteem Foreigners have of their
Books, by often printing them, and
tra...
... obtained by reflecting Q
λ
in a line of slope −1.)
Theorem 2. Let K(λ) be the set of nonergodic directions that can be
obtain using Lemma 1.1. If λ is Diophantine, then H.dim K(λ)=1/2.
Proof of Theorem ... attached to the lower edge of the slit in the other,
and similarly for the remaining edges. The induced map π : X
λ
→ T is the
branched double cover obtained by mak...
... and
generalization of the proof of Proposition 3 in [3]. We outline the main steps,
while skipping the details.
Step 1. One of the basic ingredients in the proof in [3] is to approximate the
volume of the Wiener ... second
term on the right-hand side of the analogue of (2.93).
4. Proof of Theorem 3
In Sections 4–6 we prove Theorems 3–5. The proof foll...
... set, then
|Λ
E,T
(f
1
,f
2
,f
3
)|≤C
q
1
2
−(
1
p
1
+
1
p
2
2
q
2
+
1
p
3
2
q
3
)µ
|I
T
| if j =2, 3,(4.20)
where C, C
q
1
are independent of m.
The core of the proof consists of the proofs of these ... bounds on Λ
E,S
for all finite convex sets S of triples
of integers, provided the bound is independent of S and of course m.
4. The selection of the trees
Definiti...
... of normal tubular neighborhoods of curves lying in the intersection of the
surface and the boundary of an extrinsic ball. These domains arise naturally
in our main result and are unfortunately ... 8: The separation w for a multi-valued graph in (II.1.1).
PLANAR DOMAINS
535
The main result of the next two sections is the following theorem (Γ
1
(∂)
is the compone...
... describe the general
strategy behind the proofs of Theorems A and B.
(1) We consider a certain set of nonregular parameters of full measure
and describe (in a probabilistic way) the dynamics of the ... continuous way. Thus, loosely speaking, the domain of L
i
induces a persistent
partition of the interval I
i
.
STATISTICAL PROPERTIES IN THE QUADRATIC FAMILY
841
Al...
... is the
diameter of ϕ(V )inM; on contracted components one may take the smallest
ε with V contained in the ε-thin part. The latter consists of endpoints of loops
around the singular points of ... It is instructive to compare the linearizations of the equa-
tions characterizing J-holomorphic cycles of the coordinate dependent descrip-
tion in this proposition and...
... desired in the proof of (93). The remaining term on the left-hand side of
(93) is easier to treat.
7. Proof of the L
2
− L
∞
decay estimate; Theorem 2.1
In this section we rely on the Boundedness Theorem ... u.
With the help of these constructions the proof of the L
2
− L
∞
decay
estimate stated in Theorem 2.1 can be reduced to the following:
Theorem 3.4 (Bou...
... step in the derivation of the decay estimates needed in the
first paper.
1. Introduction
This is the second in a series of three papers in which we initiate the study
of very rough solutions of the ... require the development of bilinear techniques for the Ein-
stein equations; see the discussion in the introduction to [Kl-Ro1].
2
We denote by R
αβ
the...