Some Results and Conjectures for Tree
Polymers under Weak/Strong Disorder
Ed Waymire
Department of Mathematics
Oregon State University
3RD FRONTIER PROBABILITY DAYS
SALT LAKE CITY, UT MARCH 7-22, 2011
Based on joint work with Stanley C. Williams, Torrey Johnson
[W&W,2010] Tree polymers and multiplicative cascades,
in: Fractals and Related Fields, ed Julien Barrel, Birkhauser
[J&W,2011] Tree polymers in the infinite volume limit at critical
strong disorder, (PREPRINT)
[http://www.math.oregonstate.edu/people/view/waymire]
3 Types POLYMER
TREE PATHS
(−1)
(−1, −1)
v|2 = (−1, +1)
(+1)
(+1, +1)
(+1, −1)
n
v
=
(S|n)
=
(s
,
s
,
.
.
.
,
s
)
∈
{−1,
+1}
1 2
n
(FINITE TREE PATH):
∞
S
=
(s
,
s
,
.
.
.)
∈
∂T
=
{−1,
+1}
(INFINITE TREE PATH):
1 2
(POLYGONAL TREE PATH):
n → (S)n =
n
!
j=1
sj
Path Probability
Weights
IID POSITIVE
X(−1)
X(−1, −1)
(Ω, F, P )
X(−1, +1)
X(v)
EX = 1
X(+1)
X(+1, −1)
X(+1, +1)
v = (S|n) = (s1 , s2 , . . . , sn ) ∈ {−1, +1}n
S = (s1 , s2 , . . .) ∈ ∂T = {−1, +1}∞
n
!
sj
n → (S)n =
j=1
Path Probability
Weights
IID POSITIVE
X(−1)
X(−1, −1)
(Ω, F, P )
X(−1, +1)
X(v)
EX = 1
X(+1)
X(+1, −1)
X(+1, +1)
v = (S|n) = (s1 , s2 , . . . , sn ) ∈ {−1, +1}n
∞
S
=
(s
,
s
,
.
.
.)
∈
∂T
=
{−1,
+1}
1 2
(PATH PROBABILITIES):
probn (∆n (S)) = Zn−1
∆n (S) = {s ∈ ∂T : s|n = S|n}
n
!
j=1
X(S|j)2−n
Normalized Haar measure: λ(ds) on {−1, 1}∞
n
!
dprobn
(S) = Zn−1
X(S|j)
dλ
j=1
X(v)
S = (s1 , s2 , . . .) ∈ ∂T = {−1, +1}∞
(S)n =
n
!
sj
j=1
SIMPLE
SYMMETRIC X ≡ 1 ⇒ probn (ds) = λ(ds)
RW
PROBABILITY LAWS OF n → (S)nWELL UNDERSTOOD !
Bolthausen’s Disorder Parameterization -Weak Disorder:
Z∞ = lim Zn > 0 a.s.
n→∞
Strong Disorder:
Z∞ = lim Zn = 0 a.s.
n→∞
Bolthausen’s Disorder Parameterization -Weak Disorder:
Z∞ = lim Zn > 0 a.s.
n→∞
Strong Disorder:
Z∞ = lim Zn = 0 a.s.
n→∞
Theorem (Kahane-Peyriere `76, Biggins `76, W., Williams, `94)
Weak Disorder if and only if EX ln X < ln 2
Strong Disorder if and only if EX ln X ≥ ln 2
Bolthausen’s Disorder Parameterization -Weak Disorder:
Z∞ = lim Zn > 0 a.s.
n→∞
Strong Disorder: Z∞ = lim Zn = 0 a.s.
n→∞
THEOREM (Kahane-Peyriere 1976 ``Live/Die’’ Criteria)
Weak Disorder if and only if EX ln X < ln 2
Strong Disorder if and only if EX ln X ≥ ln 2
(Disorder Strength vs Branching Rate)
TYPICAL POLYMER PROBLEMS
FOR:
n
(S)n =
!
sj
j=1
A. a.s.Non-ballistic ? (a.s. LLN---Weak/Strong ?)
B. Infinite Volume Limit ? prob∞ (ds) = lim probn (ds) a.s.?
n→∞
C. a.s. Diffusive (sub/super) ? (a.s.CLT---Universal/Variance ?)
X(v) ≡ 1,
v∈T =
n
∞
∪n=1 {−1, +1}
TYPICAL POLYMER PROBLEMS
FOR:
n
(S)n =
!
sj
j=1
A. a.s.Non-ballistic ? (a.s. LLN---Weak/Strong ?)
B. Infinite Volume Limit ? prob∞ (ds) = lim probn (ds) a.s.?
n→∞
C. a.s. Diffusive (sub/super) ? (a.s.CLT---Universal/Variance ?)
! x
(s)n
1
− 12 z 2
dz a.s. ?
probn ({s ∈ ∂T : √ ≤ x}) → √
e
n
2π −∞
TYPICAL POLYMER PROBLEMS
FOR:
n
(S)n =
!
sj
j=1
A. a.s.Non-ballistic ? (a.s. LLN---Weak/Strong ?)
B. Infinite Volume Limit ? prob∞ (ds) = lim probn (ds) a.s.?
n→∞
C. a.s. Diffusive (sub/super) ? (a.s.CLT---Universal/Variance ?)
! x
(s)n
1
− 12 z 2
dz a.s. ?
probn ({s ∈ ∂T : √ ≤ x}) → √
e
n
2π −∞
SIMPLE SYMMETRIC RW BENCHMARK:
probn (ds) = λ(ds)
ANSWER A: Always non-ballistic (in a weak sense)
regardless of disorder type.
(S)n
| → 0 a.s. as n → ∞
Eprobn |
n
ANSWER A: Always non-ballistic (in weak sense)
regardless of disorder type.
(S)n
| → 0 a.s. as n → ∞
Eprobn |
n
NOTE:
e.g.
a..s. Strong Law requires answer to Question B
Existence of prob∞
(s)n
→ 0) = 1
prob∞ (s ∈ ∂T :
n
a.s.
ANSWER A: Always non-ballistic (in weak sense)
regardless of disorder type.
(S)n
| → 0 a.s. as n → ∞
Eprobn |
n
NOTE:
e.g.,
a..s. Strong Law requires answer to Question B
Existence of prob∞
(s)n
→ 0) = 1
prob∞ (s ∈ ∂T :
n
a.s.
Similarly, law of iterated logarithm, and other a.s. , a.s. laws
ANSWER B. SPECIAL CASES
PROPOSITION B2 (Johnson, W.) In the case of weak disorder
prob∞ (ds) = lim probn (ds) exists a..s.
a.
n→∞
In the case of critical strong disorder, for each finite F ⊂ N
! (F ) = lim prob
! (F ) in probability
b. prob
∞
n
n→∞
an
Zn
→ A > 0,
an
a.s.
ANSWER B. SPECIAL CASES
PROPOSITION B2(Johnson, W. ) In the case of weak disorder
prob∞ (ds) = lim probn (ds) exists a..s.
a.
n→∞
In the case of critical strong disorder, for each finite F ⊂ N
! (F ) = lim prob
! (F ) in probability
b. prob
∞
n
n→∞
PROOF IDEA:
WD: Martingale Convergence
Theorem
!
Mn (f ) = Zn
f (s)probn (ds),
n ≥ 1.
∂T
an
Zn
→ A > 0,
an
a.s.
ANSWER B. SPECIAL CASES
PROPOSITION B2 (Johnson, W. ) In the case of weak disorder
a.
prob∞ (ds) = lim probn (ds) exists a..s.
n→∞
In the case of critical strong disorder, for each finite F ⊂ N
! (F ) = lim prob
! (F ) in probability
b. prob
∞
n
n→∞
PROOF IDEA:
WD: Martingale Convergence
Theorem
!
Mn (f ) = Zn
f (s)probn (ds),
n ≥ 1.
∂T
an
SD: Seneta-Heyde scaling & Derivative Martingale for BRW.
Aidekon and Shi (2011), Biggins and Kyprianou (2004)
Zn
!
→ A > 0, a.s.
If µ is a probability
anon group ∂T , µ̂(F ) =
is Fourier transform.
"
∂T j∈F
(sj )µ(ds)
PROPOSITION B2(Johnson, W. ) In the case of weak disorder
a.
prob∞ (ds) = lim probn (ds) exists a..s.
n→∞
In the case of critical strong disorder, for each finite F ⊂ N
! (F ) = lim prob
! (F ) in probability
b.
prob
∞
n
n→∞
PROOF IDEA:
WD: Martingale Convergence
Theorem
!
Mn (f ) = Zn
f (s)probn (ds),
n ≥ 1.
∂T
SD: Seneta-Heyde scaling & Derivative Martingale for BRW.
a
n
Aidekon and Shi (2011), Biggins and Kyprianou (2004)
Conventional Wisdom:
Zn General Nonexistence for SD.
→
A
>
0,
a.s.
Remark: Size-Biased
an Existence for SD (W. & Williams 1996)
Problem B. (Long-chain Diffusivity)
THEOREM (Bolthausen, 1989)
βW
Assume lognormally
distributed
weights:
X
=
e
√
√
Then for β <" 2 < βc#= 2 ln 2 one has the diffusive scaling
!
∂T
(S)n
√
n
2
probn (dS) → 1
Also a.s.asymptotically
normal distribution
!
(S)n
r2
r
√
r
Mn ( √ ) ≡
e n probn (dS) → e 2
n
∂T
a.s.
−δ <r <δ
NOTE: The full weak disorder regime is β < βc =
i.e., since weak disorder is:
X
X
β2
=E
ln
< ln 2
2
EX EX
√
2 ln 2
PROPOSITION C1 (Weak Disorder)W.&Williams (2010)
Under weak disorder the following limit exists a.s. :
ln Mn (r)
= ln cosh(r)
lim
n→∞
n
a.s.
THEOREM C1. (W&W 2010): Assume EX 1+! < ∞
For the tree polymer model weak disorder implies
diffusive scaling and asymptotic normality almost surely.
That is, a.s.
!
(S)n
! r
r2
r
√
r
Mn ( √ ) ≡
e n probn (dS) →! e 2
−δ <r <δ
M
Mn (s)ds
n
∂T n (r) = 1 +
0
REMARK: Comets and Yoshida (2006), AoP, Proved CLT 1
for d+1-dimensional lattice polymers, d ≥ 3 . Cov Σ = Id .
d
Proposition C3. (Strong Disorder) W.&Williams (2010)
Under strong disorder the following limit exists a.s. :
ln Mn (r)
F (r) = lim
"
!
n→∞
n h(r)
h(r)
h(r)
ln EX
+ ln pr (+) + pr (−)
= ln cosh(r) +
h(r)
where h = h(r) solves
! h
"
h
$
# h
X
X
h
(+),
(−)
=
!
E
ln
p
p
r
r
EX h EX h
h
p
r (±)
h
!(p, q) = −p ln p − q ln q ; pr (±) = h
pr (+) + phr (−)
SPECIAL CASE:
β2
βW − 2
X=e
2
F (r) = r tanh(rh(r)) + β h(r) − βc β
where
β 2 h2 (r) + 2rh(r) tanh(rh(r)) − 2 ln cosh(rh(r)) = βc2
Out[100]=
0.47
2
F (r) = r tanh(rh(r)) + β h(r) − ββc
0.46
β 2 h2 (r) + 2rh(r) tanh(rh(r)) − 2 ln cosh(rh(r)) = βc2
√
β = 2 2 ln 2 > βc
0.45
!1.0
In[101]:=
In[100]:=
0.0
0.5
1.0
xs ! Range!# 1, 1, 0.05";
gs ! g %. Table!FindRoot!With!#x ! xs!!n""$, G!x, g" $ 0",
G!x_, g_" :! g2 Log!16" " x g Tanh!x g" # Log!Cosh!x g"" # Log!2";
h(r)
In[99]:=
!0.5
ContourPlot!G!x, g" $ 0, #x, # 1, 1$, #g, 0.45, 0.5$"
In[103]:=
0.50
In[105]:=
F!x_, g_" :! x Tanh!x g" " Log!256" g # 4 Log!2";
ListLinePlot!Table!#xs!!n"", F!xs!!n"", gs!!n"""$, #n, Len
F (r)
0.49
0.20
0.48
0.15
Out[100]=
Out[105]=
0.47
0.10
0.46
r
0.05
0.45
!1.0
!0.5
0.0
0.5
1.0
!1.0
!0.5
r
In[101]:=
xs ! Range!# 1, 1, 0.05";
gs ! g %. Table!FindRoot!With!#x ! xs!!n""$, G!x, g" $ 0", #g, 0.25$", #n, Length!xs"$";
In[103]:=
F!x_, g_" :! x Tanh!x g" " Log!256" g # 4 Log!2";
In[105]:=
ListLinePlot!Table!#xs!!n"", F!xs!!n"", gs!!n"""$, #n, Length!xs"$ ", PlotRange % Automatic"
0.5
1.0
CONJECTURE (J. , W. 2011) Diffusive scaling and variance
σ = σ (β) under strong disorder
2
2
where
2ββc −
σ (β) =
β2
2
2
βc
,
β ≥ βc
CONJECTURE (J. , W. 2011) Diffusive scaling and variance
σ = σ (β) under strong disorder
2
2
where
2ββc −
σ (β) =
β2
2
2
βc
,
β ≥ βc
--SKETCHES OF PROOFS--
SIZE BIAS TOOL: (W., Williams 1994)
Define the size-bias probability on Ω × ∂T
Q(dω × ds) = Ps (dω)λ(ds)
where on σ(Xv : |v| ≤ n)
Ps (dω) =
n
!
j=1
Xs|j (ω)P (dω)
SIZE BIAS TOOL: (W., Williams 1996 TAMS)
Define the size-bias probability on Ω × ∂T
Q(dω × ds) = Ps (dω)λ(ds)
where on σ(Xv : |v| ≤ n)
Ps (dω) =
n
!
Xs|j (ω)P (dω)
j=1
Pω (ds) = probn (ω, ds)
SIZE BIAS TOOL: (W., Williams 1996 TAMS)
Define the size-bias probability on Ω × ∂T
Q(dω × ds) = Ps (dω)λ(ds)
where on σ(Xv : |v| ≤ n)
Ps (dω) =
n
!
Xs|j (ω)P (dω)
j=1
Pω (ds) = probn (ω, ds)
QΩ (dω) ≡ Q ◦
−1
πΩ (dω)
= Zn (ω)P (dω)
SIZE BIAS TOOL: (W., Williams 1996 TAMS)
Define the size-bias probability on Ω × ∂T
Q(dω × ds) = Ps (dω)λ(ds)
where on σ(Xv : |v| ≤ n)
Ps (dω) =
n
!
Xs|j (ω)P (dω)
j=1
Pω (ds) = probn (ω, ds)
−1
QΩ (dω) ≡ Q ◦ πΩ (dω)
−1
Q∂T (ds) ≡ Q ◦ π∂T (ds)
= Zn (ω)P (dω)
= λ(ds)
SIZE BIAS TOOL: (W., Williams 1996 TAMS)
Define the size-bias probability on Ω × ∂T
Q(dω × ds) = Ps (dω)λ(ds)
n
!
Xs|j (ω)P (dω) on σ(Xv : |v| ≤ n)
Ps (dω) =
j=1
LEMMA (W,W, `94): On F = σ(Xv : v ∈ T )
Weak disorder : QΩ (dω) << P (dω)
Strong disorder: QΩ (dω) ⊥ P (dω)
PROOF IDEAS:
WD:
dQΩ
(ω) = Z∞ (ω) < ∞
dP
PROOF IDEAS:
dQΩ
(ω) = Z∞ (ω) < ∞
dP
WD:
SD:
n
"
1
−n
Xs|j − ln 2)}
Xs|j 2 = exp{n(
Zn ≥
n
j=1
j=1
∼ exp{n(EX ln X − ln 2)}
n
!
→∞
if
EX ln X > ln 2
But, positive martingale property implies
P (Z∞ < ∞) = 1
i.e. QΩ (dω) ⊥ P (dω)
Calculate
ln Mn (r)
F (r) = lim
n→∞
n
WD:
Zn Mn (r)
coshn (r)
−δ <r <δ
Kahane’s T-Martingale
SD:
Via SIZE-BIASED Borel-Cantelli limsup and liminf calculations.
OR
Via a vector cascade T-martingale coding (Displ.,Weight)
PROPOSITION B2 (W., Williams 1996 TAMS)
In the case of strong disorder there is a random path
τ = τ (ω) such that
prob∞ (ds) = δτ (ds) QΩ − a.s.
PROOF IDEA:
Strong disorder QΩ − a.s. makes Z∞ = ∞
1. Fix a path s say s1 = −1
+1
s
Martingale
under Ps
+
(ω) < ∞
⇒ Z∞
2. So now fix ω ∈ [Z∞ = ∞]
+
(ω) < ∞ ⇒ τ1 (ω) = −1
Z∞
Theorem C1 (Proof Ideas)
For the tree polymer model weak disorder implies
diffusive scaling and asymptotic normality almost surely.
That is
r2
r
lim Mn ( √ ) = e
n→∞
n
2
|r| < δ a.s.
Sketch of Proof:
n
EZn Mn (r) = cosh (r)
Zn Mn (r)
coshn (r)
d Zn Mn (r)
n
dr cosh (r)
−δ <r <δ
Kahane’s T-MARTINGALE
− δ < r < δ (Signed) T-MARTINGALE
n
!
d Zn Mn (r)
=
mn,j (r)
=
n
dr cosh (r)
j=1
#
∞ "
q j
!
EX
q
!
E|mn (r)| ≤ C
q−1
2
j=1
!
mn (r)
WEAK DISORDER
∃
1<q<2
⇒
⇒
such that
Zn Mn (r)
→ Y (r)
n
cosh (r)
Zn Mn ( √rn )
EXq
<1
q−1
2
|r| < δ Y (0) = 1
1
1
r2
r2
r
n r
−1
−1
Mn ( √ ) = Zn
cosh ( √ ) → Z∞ Y (0)e 2 = e 2
n √r
cosh ( n )
n
n
a.s
THANK YOU