6–10 May 2013, Venice, Italy
Random Combinatorial Structures and Statistical Mechanics,
Joint work with Jesse Goodman
Frank den Hollander
Leiden University
The Netherlands
EXTREMAL GEOMETRY OF A
BROWNIAN POROUS MEDIUM
Describe the geometry of the complement for d ≥ 3 and
reflect on what happens for d = 2.
GOAL OF THIS TALK
Run a Brownian motion on a d-dimensional torus for a long
time. What does its complement look like?
QUESTION
Simulation of W [0, t] for t = 15 in d = 2.
Let W = (W (t))t≥0 denote standard Brownian motion
wrapped around the unit torus Td = Rd (mod Zd). Write
W [0, t] for the path of W up to time t.
A deranged builder runs a heating coil through your house
in an erratic manner and switches it on. How fast does
the temperature equilibrate?
• The Brownian heating coil
You and a mosquito are trapped in a room for one hour.
How should you position yourself to minimize the chance
of being bitten?
• The Brownian mosquito
§ TWO FUN APPLICATIONS
x ∈ Td fixed.
n
o
d
d
∃ x ∈ T : x + φE ⊂ T \ W [0, t] .
Fix E ⊂ Rd compact. For suitably chosen φ = φglobal(t),
consider the event
Global properties:
n
o
d
x + φE ⊂ T \ W [0, t] ,
Fix E ⊂ Rd compact. For suitably chosen φ = φlocal(t),
consider the event
Local properties:
§ PROPERTIES OF INTEREST
,
φlocal(t) φglobal(t),
i.e., the largest holes are much larger than the typical holes.
Note that
t
log t 1/(d−2)
φglobal(t) =
.
t
φlocal(t) =
1/(d−2)
1
We will see that the proper scales for these events to be
typical are
φd(t) =
d
log t
(d − 2)κd t
Rin(t)
=1
t→∞ φd (t)
lim
!1/(d−2)
a.s.
,
where κd = 2π d/2/Γ(d/2 − 1) is the Newtonian capacity of
the unit ball.
with
For d ≥ 3,
THEOREM 1 Dembo, Peres, Rosen 2003
x∈Td
Rin(t) = sup d(x, W [0, t]).
The largest inradius at time t is defined as
§ LITERATURE
with
namely,
o
ψd() =
a.s.
1 log(1/)
.
d−2
κd C()
=d
↓0 ψd ()
lim
C() = inf t ≥ 0 : ρin(t) ≤ ,
n
Theorem 1 can be reformulated in terms of the -cover
time
o
T (x, ) = inf t ≥ 0 : d(x, W [0, t]) ≤ .
n
T (x, )
=α
ψd()
a.s.
Theorem 2 identifies the fractal dimension of late points.
n
o
d
dim x ∈ T : x is α-late = d − α
For d ≥ 3 and 0 ≤ α ≤ d,
THEOREM 2 Dembo, Peres, Rosen 2003
Since C() = supx∈Td T (x, ), Theorem 1 states that no
point is α-late for any α > d.
with
↓0
lim sup
A point x ∈ Td is called α-late when
(3) Does Td \ W [0, t] have some kind of component structure?
(2) Is W [0, t] sparse or dense in the neighborhood of this
copy?
(1) Given E ⊂ Rd compact, does Td \ W [0, t] contain a
shifted scaled copy x + φd(t)E for some x ∈ Td?
In the remainder of the talk we will consider the following
three quantitative questions:
Goodman & dH 2013
van den Berg, Bolthausen & dH 2013
§ EXTREMAL GEOMETRY
dense picture
sparse picture
• Spatial avoidance: W [0, t] stays a typical amount of
time near x + φd(t)E but moves in an atypical manner.
• Temporal avoidance: W [0, t] stays an atypically short
amount of time near x + φd(t)E but moves in a typical
manner.
Two possible strategies to avoid sets are:
x + ϕd (t)E
x + ϕd (t)E
where Cap(E) is the Newtonian capacity of E.
lim P ∃ x ∈ Td : x + φd(t)E ⊂ Td \ W [0, t]
t→∞

1 if Cap(E) < κ ,
d
=
0 if Cap(E) > κ ,
d
For d ≥ 3, let E ⊂ Rd be a compact set satisfying a certain
regularity condition. Then
THEOREM 3
(1) SHAPE OF LARGE HOLES?
d
Cap(E)
log χ(t, E)
=
1−
t→∞
log t
d−2
κd
a.s.
Then subsets of the form x + φd(t)E 0 are much less numerous than subsets of the form x + φd(t)E, and so dense
picture with spatial avoidance applies.
Cap(E 0) ≥ Cap(E) + δ.
Pick any δ > 0, and any E ⊂ E 0 with
lim
!
For d ≥ 3, if Cap(E) < κd, then the maximal number χ(t, E)
of disjoint shifts x + φd(t)E inside Td \ W [0, t] satisfies
THEOREM 4
(2) SPARSE OR DENSE?
Wρ(t)[0, t] =
x ∈ Td : d(x, W [0, t]) ≤ ρ(t)
n
ρ(t) φd(t).
be the Wiener sausage of radius ρ(t).
Let
(log t)1/d
φd(t)
Let ρ(t) be such that
o
The set Td \ W [0, t] is connected a.s. However, large holes
are rare and tend to be surrounded by W [0, t]. They can
therefore be viewed as lakes connected by narrow channels,
which can be sealed off by thickening up W [0, t].
(3) COMPONENT STRUCTURE?
maxi∈I Cap(Ci)
lim
= κd.
t→∞
φd(t)d−2
Note that the almost-component structure of Td \ W [0, t]
is well-defined because the scaling does not depend on the
fine details of the thickening radius ρ(t).
For d ≥ 3,
THEOREM 5
Write (Ci)i∈I to denote the components of Td \ Wρ(t)[0, t].
lim φd(t)2λmin = λ(unit ball).
t→∞
Let λmin = mini∈I λ(Ci) denote the minimal principal Dirichlet eigenvalue of the components of Td \ Wρ(t)[0, t]. Then
COROLLARY 7
lim
Vmax
= Vol(unit ball).
d
t→∞ φd (t)
Let Vmax = maxi∈I Vol(Ci) denote the maximal volume of
the components of Td \ Wρ(t)[0, t]. Then
COROLLARY 6
The proofs of Corollaries 6–7 rely on the fact that, among
all regions of a given volume, respectively, principal Dirichlet eigenvalue, the ball has the smallest capacity.
E
The proofs of Theorems 3–5 rely on a careful analysis
of excursions between boundaries of small concentric balls
that fill the torus, and on lattice approximations of compact sets that allow for a proper counting of shifts of scaled
copies of sets.
§ KEY INGREDIENTS OF PROOFS
Id (κ)
∞
I(r) = Id(κdrd−2),
κd
κ
r ∈ (0, ∞).
The proofs of Theorems 3–5 also imply large deviation
principles. For instance, Rin(t)/φd(t) satisfies the LDP on
(0, ∞) with rate log t and ]rate function I given by
C(E) = 1 − e−Cap(E)
for all finite sets E ⊂ Zd, where CapZd (E) is the discrete
capacity of E and u ∈ (0, ∞) is a parameter.
Pu(Xu ∩ E = ∅) = e
−uCapZd (E)
The discrete analogue of X is a random finite set Xu ⊂ Zd
such that
for all closed sets E ⊂ Rd.
P(X ∩ E = ∅) = e−Cap(E)
is a Choquet capacity with 0 ≤ C(E) ≤ 1 and C(∅) = 0,
there is a random closed set X ⊂ Rd such that
Since
§ RANDOM INTERLACEMENTS
N ≈ t1/(d−2).
The link between the discrete and the continuum model is
obtained by picking
Interestingly, Xu undergoes a percolation transition at a
critical threshold u∗ ∈ (0, ∞).
TdN \ S[0, uN d] seen locally from a uniformly chosen
point of TdN converges in law to Xu as N → ∞.
Random interlacements arise from running a random walk
S = (S(n))n∈N on the N -torus TdN = Zd(modN ) up to time
uN d, namely,
Sznitman & co-workers
Xu is called the random interlacement with parameter u.
ρ(t) (log t)1/d
φd(t)
• Is there a percolation transition in the continuum model
for some critical choice ρ∗(t) of the thickening radius?
• Does the link N ≈ t1/(d−2) for random interlacements
carry over to large holes, i.e., is this the link for local
and global properties?
of the Wiener sausage optimal or can it be pushed
down to ρ(t) t−1/(d−2)?
• Is the scale
§ OPEN QUESTIONS
Goodman & dH work in progress
Expect sparse picture and temporal avoidance, together
with strong correlations between times spent near different
points.
§ WHAT HAPPENS IN d = 2?