David Croydon (University of Warwick)
THE GEOMETRY OF DISCRETE RANDOM STRUCTURES
EPSRC SYMPOSIUM WORKSHOP
UNIVERSITY OF WARWICK, 18-22 JUNE 2012
critical random trees
on random paths and
Biased random walks
1
β
(β − 1)ne1
Xn =
+ cβ,dN (0, I)n1/2 + o(n1/2).
β + 2d − 1
That is, the random walk that is β times more likely to jump in
first-coordinate direction than in any other direction. We have
1
1
Consider the β-biased random walk (Xn)n≥0 on the integer lattice Zd (we will always assume β ≥ 1):
BIASED RANDOM WALKS
If β > 1, then the walk is directionally transient for P-a.e. environment. Moreover, there exists a βc ∈ (1, ∞) such that:
if β < βc, then the biased random walk has positive speed,
if β > βc, then the biased random walk has zero speed,
[Berger/Gantert/Peres, Sznitman, Fribergh/Hammond].
If β = 1, then the random walk is diffusive for P-a.e. environment [Barlow, Sidoravicius/Sznitman, Biskup/Berger, Mathieu/Piatnitski].
Bond percolation on integer lattice Zd (d ≥ 2), parameter p > pc:
e.g. p = 0.53,
SUPERCRITICAL PERCOLATION
Motivated by this, we consider biased random walk on:
• a critical Galton-Watson tree conditioned to survive,
• the range of a random walk.
‘Traps along the backbone’
‘Trapping in branches’
Close to p = pc, physicists [Barma/Dhar] have identified two
types of trapping:
CRITICAL PERCOLATION
How does the biased random walk behave?
(Joint with Alexander Fribergh and Takashi Kumagai.)
αn ∼ n(α−1)/αℓ(n).
converges in distribution to a non-trivial diffusion, where
)
−1
αn X⌊tnαn⌋
,
t≥0
(
[cf. C.] The unbiased walk (β = 1), when rescaled as
T ∗ - family tree of critical Galton-Watson process
conditioned to survive. Offspring distribution in
domain of attraction of α-stable law, α ∈ (1, 2].
Bias away from root.
TRAPPING IN CRITICAL BRANCHES
x∼y∈T
∑
c(x, y) ≈ β h(T ),
h(T ) has exponential tails ⇒ time in traps has polynomial ⇒
polynomial rate of escape. [cf. Zindy]
where c(x, y) := β min{gen(x),gen(y)}.
1 + β −i
Expected time to leave trap
with base at level i:
Linear progress on backbone
⇒ regeneration arguments.
Decompose into backbone
(solid lines) and traps (dashed
lines).
SUPERCRITICAL ARGUMENT
[BEN AROUS/FRIBERGH/GANTERT/HAMMOND]
1
β > βc := ′
,
f (q)
log |Xn|
log βc
→ γ :=
, Pρ-a.s.
log n
log β
Finer subsequential distributional limits established. Also distributional limits in randomly biased case [Ben Arous/Hammond,
Hammond].
where q is the extinction probability and f (x) := ExZ , then
and the drift satisfies:
EZ > 1, EZ 2 < ∞, P(Z = 0) > 0,
If offspring distribution Z satisfies:
SUPERCRITICAL RESULT
[BEN AROUS/FRIBERGH/GANTERT/HAMMOND]
ρ1
)
...
ρi1
ρij
ρi
Tij are unconditioned Galton-Watson trees, offspring dist. Z.
k ≥ 1.
...
i
ρi(Z̃ −1)
Ti1 . . . Tij . . . Ti(Z̃i−1)
P Z̃ = k = kP(Z = k),
(
Backbone
ρ = ρ0
Buds
Leaves
Let T ∗ be a critical Galton-Watson tree (i.e. EZ = 1), with
offspring distribution in domain of attraction of α-stable law,
α ∈ (1, 2], conditioned to survive:
CRITICAL STRUCTURE
}
P (Nn(i) = 0) ∼ 1 −
ρi
α
,
(α − 1)hn
i
ρi(Z̃ −1)
the number of backbone vertices from which big traps emanate
up to level n is distributed as Binomial(n, cαn−1 log n). In particular, it grows like cα log n.
Then, since
ρi1
Nn(i) := # 1 ≤ j ≤ Z̃i − 1 : h(Tij ) ≥ hn .
{
Define a level n critical height hn := n(log n)−1, and set
BIG TRAP LOCATIONS
i=0
ti1{Nn(i)≥1}.
Since jump process on backbone does not backtrack more than
(log n)2, and big traps are separated by at least nε, summands
are asymptotically independent.
∆n ≈
n−1
∑
Let ti be total time spent by random walk in traps emanating
from backbone vertex ρi. Can show time spent in small traps
(height less than hn) is negligible. Therefore
∆n := inf{m ≥ 0 : Xm = ρn}.
Define a hitting time ∆n by setting
APPROXIMATION BY I.I.D. SUM
ρi
ρij
xij
yij
Tij
zi
}
Vi := j ∈ Bi : τxij < τzi ,
{
Bi := j = 1, . . . , Z̃i − 1 : h(Tij ) ≥ hn
{
}
∗
PρTi (Vi = A) =
(#B )−1
1
i
.
1 + #Bi #A
where xij is ‘trap entrance’ and zi := ρi+1+hδ , then
n
and
Let
# BIG TRAPS VISITED
j=1,...,Z̃i −1
max
(
)
α
.
(α − 1)x
log β
.
≥x ≈
(α − 1) log x
)
∼ αqxα−1L(qx) ∼
maxj∈Vi h(Tij )
h(Tij ) ≥ x
Pρ (ti ≥ x) ≈ Pρ β
It follows that
Pρ
(
NB. This is different to
By first conditioning on Bi and then Vi, it is possible to check
that
(
)
1
Pρ max h(Tij ) ≥ x = qxα−1L(qx) ∼
.
j∈Vi
(α − 1)x
TIME SPENT IN BIG TRAP CLUSTER
1

nt
∑

where (m(t))t≥0 is an extremal process. In particular, m can be
defined as the maximum process of the Poisson point process
with intensity measure x−2dxdt. See [Darling, Kasahara].
 L
Xi
→ (m(t))t≥0 ,
n
i=1
t≥0

for any v > 0, and F̄ (u) → 0 as u → ∞. If L(x) := 1/F̄ (x), then
F̄ (uv)
lim
= 1,
u→∞ F̄ (u)
Let (Xi)∞
i=1 be independent random variables, with distributional
tail F̄ (u) = 1 − F (u) = P(Xi > u) satisfying: F̄ (0) = 1, F̄ (u) > 0
for all u > 0,
SUMS OF SLOWLY-VARYING VARIABLES
(α − 1) log+ ∆nt
→ (m(t))t≥0 .
n log β
t≥0
)
(α − 1)n
π(X⌊ent⌋) log β
)
t≥0
)
−1
→ m (t)
.
t≥0
(
Arguments also demonstrate localisation and extremal aging.
(
Moreover, if (π(Xn))n≥0 is the projection of the biased random
walk onto the backbone, then
(
If β > 1, then (∆n)n≥0 satisfies
EXTREMAL CONVERGENCE
[C./FRIBERGH/KUMAGAI]
(
)
(
)
1
−1
XF̄ −1(1/nt)
→ m (t)
.
t≥0
n
t≥0
and c(x, y) = 0 otherwise, then
 (
)
β
−1 , if y = x + 1,

τ
x
)
c(x, y) := ( β+1
1
−1 , if y = x − 1,

τ
x
β+1
Conditional on τ , let X be a Markov chain on Z with jump rates
Let τ = (τx)x∈Z be a family of independent and identically distributed strictly positive (and finite) random variables whose distribution has a slowly-varying tail F̄ (u) = P(τ0 > u).
RELATED ONE-DIMENSIONAL TRAP MODEL
‘Traps along the backbone’
‘Trapping in branches’
RECALL CRITICAL TRAPPING MECHANISMS
{
}
For P-a.e. random walk path, the graph G is infinite, connected
and clearly has bounded degree.
E(G) := {Sn, Sn+1} : n ∈ Z .
and edge set
V (G) := {Sn : n ∈ Z} ,
Let S = (Sn)n∈Z be the two-sided simple random walk on Zd
starting from 0, built on an underlying probability space with
probability measure P. Define the range of the random walk S
to be the graph G = (V (G), E(G)) with vertex set
RANGE OF A RANDOM WALK
NB. Result does not hold in d = 3, 4 [C., Shiraishi].
tκ2 (d)
(d)
under P, converges as n → ∞ to the law of (WB
)
−1/4
,
n
X⌊tn⌋
t≥0
(
)t≥0.
under PG
0 , converges as n → ∞ to the law of (Btκ1(d))t≥0.
Furthermore, the law of
)
−1/2
n
sgn(X⌊tn⌋)(dG (0, X⌊tn⌋)
,
t≥0
(
Let d ≥ 5. For P-a.e. realisation of G, the law of
UNBIASED RANDOM WALK ON G [C.]
Gn−1
Gn
Cn := STn
[Bolthausen/Sznitman/Zeitouni] applied this kind of decomposition of the path of S to deduce the diffusivity of a random walk
in a particular high-dimensional random environment.
Gn−2
Under P̂ := P(·|0 ∈ T ), G is made up of a string of stationary
ergodic finite graphs.
which will be denoted . . . T−2 < T−1 < T0 ≤ 0 < T1 < T2 < . . . .
For d ≥ 5, P-a.s., the two-sided process S admits an infinite set
of cut-times
T := {n : S(−∞,n] ∩ S[n+1,∞) = ∅},
GEOMETRY OF TWO-SIDED RANGE
(1)
is a stationary, ergodic sequence.
−, ω 0, ω +)
(ωn
n n n∈Z
From this, one can check that
± := PG (J
ωn
0 m+1 = n ± 1|Jm = n) =
,
1
.
µ({Cn})RG (Cn, Cn±1)
is the first coordinate of e±.
For jump chain, we have
where e±
µe := β
max{e− ,e+ }
(1) (1)
Given β ≥ 1, assign to each edge e = {e+, e−} ∈ E(G) a conductance
BIASED RANDOM WALK ON G
i=1
log ρi,
(1)
(1)
log β ∼ −Snτ log β,
a(n)
b(n)
c(n)
log n
which has Brownian scaling ⇒ Sinai’s regime, in which the walker
is trapped for long periods in valleys of the potential [Sinai].
Rn = log RG (Cn, Cn+1)−log RG (C0, C1) ∼ −Cn
where ρi := ωi−/ωi+. This satisfies
Rn :=
n
∑
A key quantity in understanding the behaviour of a RWRE is
the potential function:
POTENTIAL OF RWRE
L
,
log β
in distribution under P, where L is a random variable taking
values in Rd whose distribution can be characterized explicitly.
Ln → Lβ :=
log n
for any ε > 0. Moreover,
(
)
Xn
P − Ln > ε → 0,
Fix a bias parameter β > 1 and d ≥ 5. If X = (Xn)n≥0 is
the biased random walk on the range G of the two-sided simple
random walk S in Zd, then there exists an S-measurable random
variable Ln taking values in Rd such that
LOCALIZATION RESULT [C.]