Intermediate Disorder for Directed Polymers
Tom Alberts
University of Toronto
Joint work with Kostya Khanin and Jeremy Quastel
Polymers
• Directed polymers in dimension 1 + 1
• Input is a field of i.i.d. random variables ω(i, x) on N × Z,
assume E [ω] = 0, Var (ω) = 1
3
2
1
0
−1
−2
−3
• Output is a Gibbs measure on nearest-neighbour paths
Polymers
• Polymer measures on paths
Pωn,β (S)
1
βHnω (S)
P(S)
:= ω e
Zn (β)
3
2
1
0
−1
−2
−3
• Hamiltonian
Hnω (S)
=
Pn
i=1 ω(i, Si)
Polymers
• Polymer measures on paths
Pωn,β (S)
1
βHnω (S)
P(S)
:= ω e
Zn (β)
3
2
1
0
−1
−2
−3
• Partition function
Znω (β)
=
P
paths e
βHnω (S)
βH ω (S)
P(S) = E e n
Polymers
• Polymer measures on paths
Pωn,β (S)
1
βHnω (S)
:= ω e
P(S)
Zn (β)
• β = 0 is exactly simple random walk
• β = ∞ is last passage percolation
• What happens for β in between?
Polymers
• Polymer measures on paths
Pωn,β (S)
1
βHnω (S)
P(S)
:= ω e
Zn (β)
• β = 0 is exactly simple random walk
• β = ∞ is last passage percolation
• What happens for β in between?
• Entropy vs. Energy
• For β small the walk is entropy dominated
• For β large the walk is energy dominated
Polymers
• β small is also called weak disorder
• β large is called strong disorder
• Precise separation between two regimes is defined in
terms of the martingale
Mnω (β)
Znω (β)
:=
E [Znω (β)]
• Definition:
lim Mnω (β) > 0 =⇒ β ∈ weak disorder regime
n→∞
lim Mnω (β) = 0 =⇒ β ∈ strong disorder regime
n→∞
Polymers
• Theorem: [CY06] There is a critical βc separating weak
disorder from strong disorder.
β < βc: weak disorder
β > βc: strong disorder
• Theorem: [CY06] βc = 0 for polymers in dimension 1 + 1
• Hence in dimension 1 + 1
weak disorder ⇔ simple random walk
Simple Random Walk
0 Pn ω(i,S )
ω
i
• Partition function: Zn (0) = E e i=1
=1
• Wandering Exponent:
• CLT:
|Sn| ∼
√
n
Sn (d)
√ −→ N (0, 1)
n
• Local Limit Theorem:
√
√ n→∞
1 −x2/2t
nP Sdnte = [x n] −−−→ P (Bt ∈ dx) = √ e
dx
2πt
Simple Random Walk: Local Limit Theorem
x 7→ P(Sn = x)
n=0
−30
−20
−10
0
10
20
30
Simple Random Walk: Local Limit Theorem
√
√
x 7→ nP(Sn = [x n])
n=0
−3
−2
−1
0
1
2
3
Strong Disorder Regime
• Conjectured behaviour: wandering exponent 2/3 plus
partition function fluctuations with exponent 1/3
Strong Disorder Regime
• Conjectured behaviour: wandering exponent 2/3 plus
partition function fluctuations with exponent 1/3
• Wandering Exponent:
|Sn| ∼ n2/3
Strong Disorder Regime
• Conjectured behaviour: wandering exponent 2/3 plus
partition function fluctuations with exponent 1/3
• Wandering Exponent:
|Sn| ∼ n2/3
• Localization: Law of Sn is concentrated in a window of
order 1 at distance n2/3 from the origin
Strong Disorder Regime
2/3 ω
2/3
x 7→ n Pn,1 Sn = [xn ]
−3
−2
−1
0
1
2
3
Strong Disorder Regime
• Conjectured behaviour: wandering exponent 2/3 plus
partition function fluctuations with exponent 1/3
• Wandering Exponent:
|Sn| ∼ n2/3
• Localization: Law of Sn is concentrated in a window of
order 1 at distance n2/3 from the origin
Strong Disorder Regime
• Conjectured behaviour: wandering exponent 2/3 plus
partition function fluctuations with exponent 1/3
• Wandering Exponent:
|Sn| ∼ n2/3
• Localization: Law of Sn is concentrated in a window of
order 1 at distance n2/3 from the origin
• Partition Function Behavior:
log Znω (β) = c(β)n + n1/3X
where X is something Tracy-Widomish
Strong Disorder Regime
• Conjectured behaviour: wandering exponent 2/3 plus
partition function fluctuations with exponent 1/3
• Wandering Exponent:
|Sn| ∼ n2/3
• Localization: Law of Sn is concentrated in a window of
order 1 at distance n2/3 from the origin
• Partition Function Behavior:
log Znω (β) = c(β)n + n1/3X
where X is something Tracy-Widomish
• All of this is conjectural, very few rigorous results
• Recent progress by [Sep10]
Intermediate Disorder Regime
• [AKQ10] introduces the intermediate disorder regime
and in this regime fully identifies:
• wandering exponent
• localization effects
• leading term behavior of log Znω (β)
• magnitude of fluctuations of log Znω (β)
• law of fluctuations of log Znω (β)
• limiting distribution of the rescaled polymer endpoint
Intermediate Disorder Regime
• So-called because it sits between weak and strong
disorder
• Accessed by scaling β to zero with n
• Replace β with βn−1/4
• Other scalings are of course possible, but we can’t prove
anything about them (but we have guesses!)
Intermediate Disorder Regime
• Theorem: [AKQ10]
Intermediate Disorder Regime
• Theorem: [AKQ10]
• Wandering exponent: |Sn| ∼ n1/2
Intermediate Disorder Regime
• Theorem: [AKQ10]
• Wandering exponent: |Sn| ∼ n1/2
• Localization: there is none
Intermediate Disorder Regime
√ √ ω
x 7→ nPn,n−1/4 Sn = [x n]
−3
−2
−1
0
1
2
3
Intermediate Disorder Regime
• Theorem: [AKQ10]
• Wandering exponent: |Sn| ∼ n1/2
• Localization: there is none
Intermediate Disorder Regime
• Theorem: [AKQ10]
• Wandering exponent: |Sn| ∼ n1/2
• Localization: there is none
• Partition Function:
e
2√
− β2
n
β2 √
n
2
log Znω (βn−1/4)
=
ω
−1/4 (d)
Zn (βn
) −→
Zβ
+ log Zβ
where Zβ has an explicit Wiener chaos representation
Intermediate Disorder Regime
• Theorem: [AKQ10]
• Wandering exponent: |Sn| ∼ n1/2
• Localization: there is none
• Partition Function:
e
2√
− β2
n
β2 √
n
2
log Znω (βn−1/4)
=
ω
−1/4 (d)
Zn (βn
) −→
Zβ
+ log Zβ
where Zβ has an explicit Wiener chaos representation
• Rescaled Polymer Endpoint:
√
nPωn,βn−1/4 (Sn
√
(d) 1 A (x) −x2/2
= [x n]) −→
e β e
dx
Zβ
where x 7→ Aβ (x) is a stationary process with the
crossover distribution as its one-point marginal
Intermediate Disorder Regime
• Theorem: [AKQ10] (short version)
Intermediate Disorder Regime
• Theorem: [AKQ10] (short version)
Under the scaling of intermediate disorder, we get random
walk fluctuation exponents but not Gaussian fluctuations.
The fluctuations still depend on the random environment.
Intermediate Disorder Regime
• Theorem: [AKQ10] (short version)
Under the scaling of intermediate disorder, we get random
walk fluctuation exponents but not Gaussian fluctuations.
The fluctuations still depend on the random environment.
• We can fully analyze βn−1/4 scaling because it almost
puts us back in the simple random walk regime
Intermediate Disorder Regime
• Theorem: [AKQ10] (short version)
Under the scaling of intermediate disorder, we get random
walk fluctuation exponents but not Gaussian fluctuations.
The fluctuations still depend on the random environment.
• We can fully analyze βn−1/4 scaling because it almost
puts us back in the simple random walk regime
• Theorem: [AKQ10] For any > 0
√
2
−x /2
√
e
(d)
nPωn,βn−(1/4+) (Sn = [x n]) −→ √
dx
2π
A More In Depth Look
√
nPωn,βn−1/4 (Sn
√
(d) 1 A (x) −x2/2
= [x n]) −→
e β e
dx
Zβ
• Why n−1/4 scaling?
• Why does it produce random walk fluctuation exponents?
• What are the methods used?
• What are the crossover distributions?
A More In Depth Look
√
nPωn,βn−1/4 (Sn
√
(d) 1 A (x) −x2/2
= [x n]) −→
e β e
dx
Zβ
• Why n−1/4 scaling?
• Why does it produce random walk fluctuation exponents?
• What are the methods used?
• What are the crossover distributions?
Sneak Peek: They’re already related to the Tracy-Widom
GUE distribution!
Why n−1/4 scaling?
• Simplest to explain via the partition function
h −1/4 ω i
Znω (βn−1/4) = E eβn Hn (S)
• Expand the exponential as a power series and keep only
terms up to order n−1/4:
Znω (βn−1/4)
−1/4 ω
≈ E 1 + βn
Hn (S)
n
X
= 1 + βn−1/4
E [ω(i, Si)]
= 1 + βn−1/4
i=1
n X
X
i=1 x∈Z
√ −→ 1 + N 0, 2β / π
(d)
2
ω(i, x)P (Si = x)
Why n−1/4 scaling?
• Another look:
βn−1/4
= βn
−1/4
n X
X
ω(i, x)P (Si = x)
i=1 x∈Z
Z 1X
n
0
−1/4
1/2
ω(dnte, x)P Sdnte = x dt
x∈Z
Z
1Z
√
√ ω(dnte, [x n])P Sdnte = [x n] dx dt
= βn
nn
0
R
Z 1Z
√
√ 3/4
1/2
= β
n ω(dnte, [x n])n P Sdnte = [x n] dx dt
Z0 1 ZR
(d)
−→ β
W (t, x)P(Bt ∈ dx) dx dt
0
R
• Here W (t, x) is a space-time white noise on [0, 1] × R
Why n−1/4 scaling?
• For higher order terms
" n
#
" n
#
Y −1/4
Y
ω
−1/4
βn
ω(i,Si)
−1/4
Zn (βn
)=E
e
≈E
1 + βn
ω(i, Si)
i=1
i=1
• Expand the product into a big series, summing all terms
where we pick exactly k of the ω gives
β k n−k/4
X


k
Y
E ω(ij , Sij )
1≤i1<...<ik ≤n
= β k n−k/4
X
j=1
k
Y
1≤i1<...<ik ≤n j=1
x1,...,xk ∈Zk
ω(ij , xj )P (Si1 = x1, . . . , Sik = xk )
Why n−1/4 scaling?
β k n−k/4
X
k
Y
ω(ij , xj )P (Si1 = x1, . . . , Sik = xk )
1≤i1<...<ik ≤n j=1
x1,...,xk ∈Zk
• Example of a discrete Wiener chaos
• Can get the explicit form of the limiting distribution for k =
1 and maybe k = 2, but not for any higher k
• However, can write down a continuous Wiener chaos
that the discrete one converges to
• Heuristic is to scale space and time diffusively
Why n−1/4 scaling?
β k n−k/4
k
Y
X
ω(ij , xj )P (Si1 = x1, . . . , Sik = xk )
1≤i1<...<ik ≤n j=1
x1,...,xk ∈Zk
converges to
Z 1Z
1
βk
Z
1
Z Z
...
...
0
t1
tk−1 R R
Z Y
k
R j=1
W (dtj , dxj )P (Bt1 ∈ dx1, . . . , Btk ∈ dxk )
precisely because of the n−k/4 factor in front
• Here W (t, x) is a space-time white noise on [0, 1] × R
• This type of convergence is the main focus of study in the
field of U-statistics [NP88, Jan97]
Why n−1/4 scaling?
"
E
n
Y
#
Z 1Z
(d)
(1 + βω(i, Si)) −→ 1 +
Z i=1
1Z 1Z
0
R
W (t, x)P(Bt ∈ dx)
Z
W (t, x)W (s, y)P(Bt ∈ dx, Bs ∈ dy)
Z0 1 Zt 1 ZR1 ZR Z Z
+
...
+
0
+ ...
t1
t2
R
R
R
• Right hand size is the continuum Wiener chaos
expansion of some random variable in L2(W )
#
" n
" n
#
Y −1/4
Y
βn
ω(i,Si)
ω
−1/4
−1/4
e
≈E
Zn (βn
)=E
1 + βn
ω(i, Si)
i=1
i=1
h
• Trivial modifications for E eβn
−1/4
Hnω (S)
i
The x 7→ Aβ (x) Process
√
nPωn,βn−1/4 (Sn
√
(d) 1 A (x) −x2/2
e β e
dx
= [x n]) −→
Zβ
• eAβ (x) also has a Wiener chaos expansion
"
E
n
Y
#
(d)
(1 + βω(i, Si)) −→ 1 +
Z i=1
1Z 1Z
Z 1Z
0
R
W (t, x)P(Xtx∈ dx)
Z
W (t, x)W (s, y)P(Xtx∈ dx, Xsx∈ dy)
Z0 1 Zt 1 ZR1 ZR Z Z
+
...
+
0
+ ...
t1
t2
R
R
R
where Xtx is a Brownian bridge going from 0 to x in one
unit of time
The x 7→ Aβ (x) Process
• This Wiener chaos is the limit of
#
" n
Y
√ −1/4
E
(1 + βn
ω(i, Si))1 Sn = x n
i=1
which is analyzed in the same way as
" n
#
Y
E
(1 + βn−1/4ω(i, Si))
i=1
• Only difference is that the kernels are replaced by finitedimensional distributions for random walk bridges, which
scale to finite-dimensional distributions for Brownian
bridges.
What are the crossover distributions?
Recall x 7→ Aβ (x) is a stationary process
• Theorem: [ACQ10](Amir, Corwin, Quastel) [SS10]
The one-point marginal of Aβ (x) is the crossover
distribution
4
Gβ (s) := P Aβ (x) + 2β /3 ≤ s
What are the crossover distributions?
Recall x 7→ Aβ (x) is a stationary process
• Theorem: [ACQ10](Amir, Corwin, Quastel) [SS10]
The one-point marginal of Aβ (x) is the crossover
distribution
4
Gβ (s) := P Aβ (x) + 2β /3 ≤ s
Z
=1−
e
−e−r
4
f s − log(32πβ )/2 − r dr
κ−1
β det(I
−1
f (r) =
− Kσβ )tr (I − Kσβ ) PAiry
PAiry (x, y) = Ai(x)ZAi(y),
Kσβ (x, y) = P.V. σβ (t) Ai(x + t) Ai(y + t) dt,
where κβ = 2β 4/3, σβ (t) = 1/(1 − e−κβ (t)) and PAiry , Kσβ are
operators acting on L2(κ−1
β s, ∞) given by the kernels above
What are the crossover distributions?
• Exact form of the distribution is not very important
• Gβ has more interesting asymptotic distributions
[ACQ10]
Z s −x2/2
1 1 e
β→0
2
4
√
dx
Gβ 2 π βs −−→
2π
−∞
4
3
4
3
β→∞
1
3
Gβ 2 β s −−−→ FGU E 2 s
where FGU E is the Tracy-Widom distribution for the
largest eigenvalue of a matrix from the Gaussian Unitary
Ensemble
What are the crossover distributions?
• Exact form of the distribution is not very important
• Gβ has more interesting asymptotic distributions
[ACQ10]
Z s −x2/2
1 1 e
β→0
2
4
√
dx
Gβ 2 π βs −−→
2π
−∞
4
3
4
3
β→∞
1
3
Gβ 2 β s −−−→ FGU E 2 s
where FGU E is the Tracy-Widom distribution for the
largest eigenvalue of a matrix from the Gaussian Unitary
Ensemble
• Name is because they cross over from Gaussian to TracyWidom as β varies from 0 to ∞
Origin of the Crossover Distributions
• [ACQ10] notices that the Wiener chaos
Z 1Z
W (t, x)P(Xt0 ∈ dx)
Z0 1 ZR1 Z Z
+
W (t, x)W (s, y)P(Xt0 ∈ dx, Xs0 ∈ dy)
Z0 1 Zt 1 ZR1 ZR Z Z
...
+
1 +
0
+ ...
t1
t2
R
R
R
is intimately connected to the stochastic heat equation
1
∂tZ = ∂xxZ + W Z
2
Z(t = 0, ·) = δ0(·)
Origin of the Crossover Distributions
Stochastic Heat Equation
Weakly Asymmetric Exclusion Process (WASEP)
Tracy-Widom Formula for ASEP
Steepest Descent Calculation
Crossover Distributions
Conclusion and Future Work
• Access the intermediate disorder regime by scaling β to
zero with n according to βn−1/4
• Polymers in the intermediate disorder regime are very
close to random walk but not completely decoupled from
the random environment
• We can fully analyze the wandering exponent, localization
effects, the partition function and the rescaled polymer
endpoint
• Work in progress:
polymer paths
construct the scaling limit of the
W 7→ measure PW
β on C[0, 1]
Brownian motion in a white-noise random environment
Slides Produced With
Asymptote: The Vector Graphics Language
symptote
http://asymptote.sf.net
(freely available under the GNU public license)
References
[ACQ10] Gideon Amir, Ivan Corwin, and Jeremy Quastel.
Probability distribution of the free energy of
the continuum directed random polymer in 1+1
dimensions. arXiv:1003.0443v2 [math-ph], 2010.
[AKQ10] Tom Alberts, Kostya Khanin, and Jeremy Quastel.
The endpoint distribution of the directed polymer in
the intermediate disorder regime. In preparation,
2010.
[CY06] Francis Comets and Nobuo Yoshida.
Directed
polymers in random environment are diffusive at weak
disorder. Ann. Probab., 34(5):1746–1770, 2006.
[Jan97] Svante Janson. Gaussian Hilbert spaces, volume
129 of Cambridge Tracts in Mathematics. Cambridge
University Press, Cambridge, 1997.
[NP88] Deborah Nolan and David Pollard.
Functional
limit theorems for U -processes.
Ann. Probab.,
16(3):1291–1298, 1988.
[Sep10] T. Seppäläinen.
Scaling for a one-dimensional
directed polymer with boundary conditions.
arXiv:0911.2446v2 [math.PR], 2010.
[SS10]
T. Sasamoto and H. Spohn. Exact height distributions
for the kpz equation with narrow wedge initial
condition. arXiv:1002.1879v2 [cond-mat.stat-mech],
2010.