White noise Continuum partition functions The continuum DPRE Pinning models
Polynomial Chaos andScaling Limits of Disordered Systems
2. Continuum model and free energy estimates
Francesco Caravenna
Universita degli Studi di Milano-Bicocca
YEP XIII, Eurandom ∼ March 7-11, 2016
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 1 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Overview
In the previous lecture we saw how to build continuum partition functions
Zωδ
d−−−→δ→0
ZW (scaling limits of discrete partition functions)
In this lecture we present two interesting applications of ZW
I Scaling limit of the full probability measure Pωδd−−−→
δ→0PW
constructing a continuum version of the disordered system
We will focus on the DPRE [Alberts, Khanin, Quastel 2014b]drawing inspiration from the Pinning [C., Sun, Zygouras 2016]
I Sharp asymptotics on the discrete model, in terms of free energy andcritical curve
For this we will focus on Pinning models (rather than DPRE)
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 2 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Outline
1. White noise and Wiener chaos
2. Continuum partition functions
3. The continuum DPRE
4. Pinning models
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 3 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Outline
1. White noise and Wiener chaos
2. Continuum partition functions
3. The continuum DPRE
4. Pinning models
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 4 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
White noise (1 dim.)
We are familiar with (1-dim.) Brownian motion B = (B(t))t≥0
We are interested in its derivative “W (t) := ddtB(t)” called white noise
Think of W as a stochastic process W = (W (·)) indexed by
Intervals I = [a, b] 7−→ W (I ) = B(b)− B(a) ∼ N (0, b − a)
Borel sets A ∈ B(R) 7−→ W (A) =
∫
R1A(t) dB(t) ∼ N (0, |A|)
W is a Gaussian process with
E[W (A)] = 0 Cov[W (A),W (B)] = |A ∩ B|
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 5 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
White noise
White noise on Rd
It is a Gaussian process W = (W (A))A∈B(Rd ) with
E[W (A)] = 0 Cov[W (A),W (B)] = |A ∩ B|
I ∀ (An)n∈N disjoint =⇒ W
( ⋃
n∈N
An
)a.s.=∑
n∈N
W (An)
Almost a random signed measure on Rd . . . (but not quite!)
We can define single stochastic integrals W (f ) :=∫f (x)W (dx)
E[W (f )] = 0 E[W (f )2] = ‖f ‖2L2(Rd )
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 6 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Multiple stochastic integrals
We can define
W⊗k(g) =
∫
(Rd )kg(x1, . . . , xk)W (dx1) · · ·W (dxk)
For d = 1 we can restrict x1 < x2 < . . . < xk iterated Ito integrals
For symmetric functions we have
E[W⊗k(g)] = 0 E[W⊗k(g)2] = k! ‖g‖2L2((Rd )k )
Cov[W⊗k(f ),W⊗k′(g)] = 0 ∀k 6= k ′
Wiener chaos expansion
Any r.v. X ∈ L2(ΩW ) measurable w.r.t. σ(W ) can be written as
X =∞∑
k=0
1
k!W⊗k(fk) with fk ∈ L2
sym((Rd)k)
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 7 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Discrete sums and stochastic integrals
Consider a lattice Tδ ⊆ Rd whose cells have volume vδ → 0
Take i.i.d. random variables (X z)z∈Tδ with zero mean and unit variance
Consider the “stochastic Riemann sum” (multi-linear polynomial)
Ψδ :=∑
(z1,...,zk )∈(Tδ)k
zi 6=zi ∀i 6=j
f (z1, . . . , zk)X z1 X z2 · · ·X zk
where f ∈ L2(Rd) is (say) continuous.
(√vδ)k Ψδ
d−−−→δ→0
∫
(Rd )kg(z1, . . . , zk)W (dz1) · · ·W (dzk)
(Check the variance!)
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 8 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Outline
1. White noise and Wiener chaos
2. Continuum partition functions
3. The continuum DPRE
4. Pinning models
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 9 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Continuum partition function for DPRE
1d rescaled RW Sδt :=√δSt/δ lives on Tδ =
([0, 1] ∩ δN0
)×√δZ
Zωδ = Eref
[exp
(Hω)]
= Eref
[exp
(N∑
n=1
(βω(n,Sn) − λ(β)
))]
= 1 +∑
(t,x)∈Tδ
Pref(Sδt = x)X t,x
+1
2
∑
(t,x)6=(t′,x′)∈Tδ
Pref(Sδt = x ,Sδt′ = x ′)X t,x X t′,x′ + . . .
Recall the LLT: Pref(Sn = x) ∼ 1√ng(
x√n
)with g(z) = e−
z2
2√2π
Pref(Sδt = x) = Pref(S tδ
= x√δ
) ∼√δ gt(x) gt(x) =
e−x2
2t√2πt
Replacing X t,x = e(βω(t,x)−λ(β)) − 1 ≈ βY t,x with Y t,x i.i.d. N (0, 1)
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 10 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Continuum partition function for DPRE
ZωN = 1 + β
√δ∑
(t,x)∈Tδ
gt(x)Y t,x
+1
2(β√δ)2
∑
(t,x)6=(t′,x′)∈Tδ
gt(x) gt′−t(x′ − x)Y t,x Y t′,x′ + . . .
Cells in Tδ have volume vδ = δ√δ = δ
32 “Stochastic Riemann sums”
converge to stochastic integrals if β√δ ≈ √vδ
β ∼ β δ 14 =
β
N14
ZωN
d−−−→δ→0
ZW = 1 + β
∫
[0,1]×Rgt(x)W (dtdx)
+β2
2
∫
([0,1]×R)2
gt(x) gt′−t(x′ − x)W (dtdx)W (dt ′dx ′)
+ . . .
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 11 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Constrained partition functions
We have constructed ZW = “free” partition function on [0, 1]× RRW paths starting at (0, 0) with no constraint on right endpoint
ZW = ZW((0, 0), (1, ?)
)E[ZW
]= 1
Consider now constrained partition functions: for (s, y), (t, x) ∈ [0, 1]× R
Discrete: Zωδ
((s, y), (t, x)
)= Eref
[exp
(Hω)1Sδt =x
∣∣∣Sδs = y]
Divided by√δ, they converge to a continuum limit:
ZW((s, y), (t, x)
)E[ZW
((s, y), (t, x)
)]= gt−s(x − y)
This is a function of white noise in the stripe W ([s, t]× R)
Four-parameter random process ZW((s, y), (t, x)
) regularity?
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 12 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Key properties
Key properties
For a.e. realization of W the following properties hold:
I Continuity: ZW ((s, y), (t, x)) is jointly continuous in (s, y , t, x)(on the domain s < t)
I Positivity: ZW ((s, y), (t, x)) > 0 for all (s, y , t, x) satisfying s < t
I Semigroup (Chapman-Kolmogorov): for all s < r < t and x , y ∈ R
ZW ((s, y), (t, x)) =
∫
RZW ((s, y), (r , z))ZW ((r , z), (t, x)) dz
(Inherited from discrete partition functions: drawing!)
How to prove these properties?
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 13 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
The 1d Stochastic Heat Equation
The four-parameter field ZW ((s, y), (t, x)) solves the 1d SHE∂tZW = 1
2 ∆xZW + βW ZW
limt↓s ZW ((s, y), (t, x)) = δ(y − x)
Checked directly from Wiener chaos expansion (mild solution)
It is known that solutions to the SHE satisfy the properties above
Alternative approach (to check, OK for pinning [C., Sun, Zygouras 2016])
I Prove continuity by Kolmogorov criterion, showing that
ZW ((s, y), (t, x))
gt−s(x − y)is continuous also for t = s
I Use continuity to prove semigroup for all times
I Use continuity to deduce positivity for close times, then bootstrap toarbitrary times using semigroup
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 14 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Outline
1. White noise and Wiener chaos
2. Continuum partition functions
3. The continuum DPRE
4. Pinning models
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 15 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
A naive approach
Consider DPRE in d = 1 (random walk + disorder)
Pω(S) ∝ e∑N
n=1 β ω(n,Sn) Pref(S)
Can we define its continuum analogue (BM + disorder)? Naively
PW (dB) ∝ e∫ 1
0βW (t,Bt) dt Pref(dB)
Pref = law of BM W (t, x) = white noise on R2 (space-time)
I∫ 1
0W (t,Bt) dt ill-defined. Regularization?
NO! The problem is more subtle (and interesting!)
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 16 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Partition functions and f.d.d.
Start from discrete: distribution of DPRE at two times 0 < t < t ′ < 1
Pωδ (Sδt = x ,Sδt′ = x ′) =Zωδ
((0, 0), (t, x)
)Zωδ
((t, x), (t ′, x ′)
)Zωδ
((t ′, x ′), (1, ?)
)
Zωδ
((0, 0), (1, ?)
)
(drawing!) Analogous formula for any finite number of times
Idea: Replace Zωδ ZW to define the law of continuum DPRE
Recall: to define a process (Xt)t∈[0,1] it is enough (Kolmogorov) to assignfinite-dimensional distributions (f.d.d.)
µt1,...,tk (A1, . . . ,Ak) “ = P(Xt1 ∈ A1, . . . ,Xtk ∈ Ak) ”
that are consistent
µt1,...,tj ,...,tk (A1, . . . ,R, . . . ,Ak) = µt1,...,tj−1,tj+1,...,tk (A1, . . . ,Aj−1,Aj+1, . . . ,Ak)
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 17 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
The continuum 1d DPRE
I Fix β ∈ (0,∞) (on which ZW depend) [recall that β ∼ βδ14 ]
I Fix space-time white noise W on [0, 1]× R and a realization ofcontinuum partition functions ZW satisfying the key properties(continuity, strict positivity, semigroup)
The Continuum DPRE is the process (Xt)t∈[0,1] with f.d.d.
PW (Xt ∈ dx ,Xt′ ∈ dx ′)
dx dx ′
:=ZW
((0, 0), (t, x)
)ZW
((t, x), (t ′, x ′)
)ZW
((t ′, x ′), (1, ?)
)
ZW((0, 0), (1, ?)
)
I Well-defined by strict positivity of ZW
I Consistent by semigroup property
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 18 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Relation with Wiener measure
The law of the continuum DPRE is a random probability
PW (X ∈ · ) (quenched law)
for the process X = (Xt)t∈[0,1] [ Probab. kernel S ′(R)→ R[0,1] ]
Define a new law P (mutually absolutely continuous) for disorder W by
dPdP
(W ) = ZW((0, 0), (1, ?)
)
Key Lemma
Pann(X ∈ ·) :=
∫
S′(R)
PW (X ∈ · ) P(dW ) = P(BM ∈ · )
Proof. The factor ZW in P cancels the denominator in the f.d.d. for PW
Since E[ZW
((s, y), (t, x)
)]= gt−s(x − y) one gets f.d.d. of BM
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 19 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Absolute continuity properties
Theorem
∀A ⊆ R[0,1] : P(BM ∈ A) = 1 ⇒ PW (X ∈ A) = 1 for P-a.e. W
Any given a.s. property of BM is an a.s. property of continuum DPRE,for a.e. realization of the disorder W
Corollary
PW (X has Holder paths with exp. 12−) = 1 for P-a.e. W
We can thus realize PW as a law on C ([0, 1],R), for P-a.e. W
(More precisely: PW admits a modification with Holder paths)
One is tempted to conclude that PW is absolutely continuous w.r.t.Wiener measure, for P-a.e. W . . .
NO ! “ ∀A ” and “ for P-a.e. W ” cannot be exchanged!
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 20 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Singularity properties
Theorem
The law PW is singular w.r.t. Wiener measure, for P-a.e. W .
for P-a.e. W ∃ A = AW ⊆ C ([0, 1],R) :
PW (X ∈ A) = 1 vs. P(BM ∈ A) = 0
Unlike discrete DPRE, there is no continuum Hamiltonian
PW (X ∈ · ) 6∝ eHW ( · )P(BM ∈ · )
Absolute continuity is lost in the scaling limit
In a sense, the laws PW are just barely not absolutely continuous w.r.t.Wiener measure (“stochastically absolutely continuous”)
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 21 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Proof of singularity
Let (Xt)t∈[0,1] be the canonical process on C ([0, 1],R) [ Xt(f ) = f (t) ]
Let Fn := σ(Xtni: tni = i
2n , 0 ≤ i ≤ 2n) be the dyadic filtration
Fix a typical realization of W . Setting Pref = Wiener measure
RWn (X ) :=
dPW |Fn
dPref |Fn
(X )
The process (RWn )n∈N is a martingale w.r.t. Pref (exercise!)
Since RWn ≥ 0, the martingale converges: RW
na.s.−−−→
n→∞RW∞
I PW Pref if and only if Eref [RW∞ ] = 1 (the martingale is UI)
I PW is singular w.r.t. Pref if and only if RW∞ = 0
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 22 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Proof of singularity
It suffices to show that RWn (X ) −−−→
n→∞0 in P⊗ Pref -probability
Fractional moment
For Pref -a.e. X EE[(RWn (X )
)γ] −−−→n→∞
0 for some γ ∈ (0, 1)
RWn (X ) =
1
ZW((0, 0), (1, ?)
)2n−1∏
i=0
ZW((tni ,Xtni
), (tni+1,Xtni+1))
g 12n
(Xtni+1− Xtni
)
I Switch from E to equivalent law E to cancel the denominator
I For fixed X , the ZW((tni ,Xtni
), (tni+1,Xtni+1))
’s are independent
We need to exploit translation and scale invariance of their laws
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 23 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Proof of singularity
Lemma 1 (Translation and scale invariance)
If we set ∆ni :=
Xtni+1− Xtni√
tni+1 − tniwe have
ZWβ
((tni ,Xtni
), (tni+1,Xtni+1))
g 12n
(Xtni+1− Xtni
)d=
ZWβ
2n/4
((0, 0), (1,∆n
i ))
g1(∆ni )
Lemma 2 (Expansion)
For z ∈ R and ε ∈ [0, 1] (say)
ZWε
((0, 0), (1, z)
)
g1(z)= 1 + εX z + ε2 Y ε,z
E[X z ] = 0 E[X ε,z ] = 0 E[X 2
z
]≤ C E
[Y 2ε,z
]≤ C unif. in ε, z
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 24 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Proof of singularity
By Taylor expansion, for fixed γ ∈ (0, 1)
E
[(ZWε
((0, 0), (1, z)
)
g1(z)
)γ]= E
[(1 + εX z + ε2Y ε,z
)γ]
= 1 + γεE[Xz ] + ε2E[Yε,z ]
+
γ(γ − 1)
2
ε2E[(X x)2] + . . .
+ . . .
= 1 − c ε2 ≤ e−c ε2
(?) First order terms vanish (?) γ(γ − 1) < 0 (?) For some c > 0
Estimate is uniform over z ∈ R We can set z = ∆ni and ε = 1
2n/4
E[(RWn (X )
)γ]=
2n−1∏
i=0
E
[(ZWε
((0, 0), (1,∆n
i ))
g1(∆ni )
)γ]≤ e−c ε
2 2n
= e−c 2n/2
which vanishes as n→∞
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 25 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Proof of Lemma 1
Introducing the dependence on β
ZWβ
((s, y), (t, x)
) d= ZW
β
((0, 0), (t − s, x − y)
)
ZWβ
((0, 0), (t, x)
) d=
1√tZW
βt14
((0, 0),
(1,
x√t
))
transl. invariance + diffusive rescaling (prefactor, new β) (drawing!)
ZW((0, 0), (t, x)
)= gt(x) + β
∫
[0,t]×Rgs(z) gt−s(x − z)W (dsdz) + . . .
=1√tg1( x√
t) +
1√t
(β t
34√t
)∫
[0,t]×Rg s
t( z√
t) g1− s
t( x−z√
t)W (dsdz)
t34
+ . . .
= OK !
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 26 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Convergence of discrete DPRE
I Pωδ = law of discrete DPRE (recall that Sδt :=√δSt/δ)
“Rescaled RW Sδ moving in an i.i.d. environment ω”
I PW = law of continuum DPRE
“BM moving in a white noise environment W ”
Both Pωδ and PW are random probability laws on E := C ([0, 1],R)i.e. RVs (defined on different probab. spaces) taking values in M1(E )
Does Pωδ converge in distribution toward PW as δ → 0 ?
∀ ψ ∈ Cb(M1(E )→ R) : E[ψ(Pωδ )
]−−−→δ→0
E[ψ(PW )
]
The answer is positive. . . almost surely ;-)
Statement for Pinning model proved in [C., Sun, Zygouras 2016]
Details need to be checked for DPRE (stronger assumptions on RW ?)
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 27 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Universality
The convergence of Pωδ toward PW is an instance of universality
There are many discrete DPRE:
I any RW S (zero mean, finite variance + technical assumptions)
I any (i.i.d.) disorder ω (finite exponential moments)
In the continuum (δ → 0) and weak disorder (β → 0) regime, all thesemicroscopic models Pωδ give rise to a unique macroscopic model PW
Tomorrow we will see how the continuum model PW can tellquantitative information on discrete models Pωδ (free energy estimates)
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 28 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Convergence
How to prove convergence in distribution Pωδd−−−→
δ→0PW ?
Prove a.s. convergence through a suitable coupling of (ω,W )
Assume we have convergence in distribution of discrete partition functionsto continuum ones, in the space of continuum functions of (s, y), (t, x)
Zωδ
((s, y), (t, x)
) d−−−→δ→0
ZW((s, y), (t, x)
)
By Skorokhod representation theorem, there is a coupling of (ω,W )under which this convergence holds a.s.
Fix such a coupling: for a.e. (ω,W ) the f.d.d. of Pωδ converge weakly tothose of PW . It only remains to prove tightness of Pωδ (·).
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 29 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Outline
1. White noise and Wiener chaos
2. Continuum partition functions
3. The continuum DPRE
4. Pinning models
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 30 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Ingredients: renewal process & disorder
0 = τ0 τ1 τ2 τ3 τ4 τ5 τ6
Discrete renewal process τ = 0 = τ0 < τ1 < τ2 < . . . ⊆ N0
Gaps (τi+1 − τi )i≥0 are i.i.d. with polynomial-tail distribution:
Pref(τ1 = n) ∼ cKn1+α
, cK > 0, α ∈ (0, 1)
τ = n ∈ N0 : Sn = 0 zero level set of a Markov chain S = (Sn)n≥0
Disorder ω = (ωi )i∈N: i.i.d. real random variables with law P
λ(β) := log E[eβω1 ] <∞ E[ω1] = 0 Var[ω1] = 1
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 31 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Bessel random walks
For α ∈ (0, 1) the α-Bessel random walk is defined as follows:
0prob. 1
2
xprob. 1
2
(1 + cα
x
)Sn
prob. 12
prob. 12
(1− cα
x
) cα := 12 − α
I (α = 12 ) no drift (cα = 0) simple random walk
I (α < 12 ) drift away from the origin (cα > 0)
I (α > 12 ) drift toward the origin (cα < 0)
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 32 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Disordered pinning model
0 N
Pinning model rewards ωn > 0 penalties ωn < 0
N ∈ N (system size) β ≥ 0, h ∈ R (disorder strength, bias)
The pinning model
Gibbs change of measure PωN = PωN,β,h of the renewal distribution Pref
dPωNdPref
(τ) :=1
ZωNexp
(N∑
n=1
(βωn + h − λ(β))1n∈τ 1Sn=0
)
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 33 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
The phase transition
How are the typical paths τ of the pinning model PωN?
Contact number CN :=∣∣τ ∩ (0,N]
∣∣ =∑N
n=1 1n∈τ =∑N
n=1 1Sn=0
Theorem (phase transition)
∃ continuous, non decreasing, deterministic critical curve hc(β):
I Localized regime: for h > hc(β) one has CN ≈ N
∃µ = µβ,h > 0 : PωN
(∣∣∣∣CNN− µ
∣∣∣∣ > ε
)−−−−→N→∞
0 ω–a.s.
I Deocalized regime: for h < hc(β) one has CN = O(logN)
∃A = Aβ,h > 0 : PωN
( CNlogN
> A
)−−−−→N→∞
0 ω–a.s.
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 34 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Estimates on the critical curve
For β = 0 (homogeneous pinning, no disorder) one has hc(0) = 0
What is the behavior of hc(β) for β > 0 small ?
Theorem ( P(τ1 = n) ∼ cKn1+α )
I (α < 12 ) disorder is irrelevant: hc(β) = 0 for β > 0 small
[Alexander] [Toninelli] [Lacoin] [Cheliotis, den Hollander]
I (α ≥ 12 ) disorder is relevant: hc(β) > 0 for all β > 0
I (α > 1) hc(β) ∼ C β2 with explicit C = α1+α
12E(τ1)
[Berger, C., Poisat, Sun, Zygouras]
I ( 12< α < 1) C1 β
2α2α−1 ≤ hc(β) ≤ C2 β
2α2α−1 hc(β) ∼ C β
2α2α−1
using continuum part. funct.![Derrida, Giacomin, Lacoin, Toninelli] [Alexander, Zygouras] [C., Torri, Toninelli]
I (α = 12) hc(β) = e
− c+o(1)
β2 [Giacomin, Lacoin, Toninelli] [Berger, Lacoin]
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 35 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Discrete free energy and critical curve
Partition function ZωN := E
[eHN (τ)
]= E
[e∑N
n=1(h+βωn−Λ(β))1n∈τ]
Consider first the regime of N →∞ with fixed β, h
I Free energy F(β, h) := limN→∞
1N log Zω
N ≥ 0 P(dω)-a.s.
ZωN ≥ E
[eHN (τ) 1τ∩(0,N]=∅
]= P(τ ∩ (0,N] = ∅) ∼ (const.)
Nα
I Critical curve hc(β) = suph ∈ R : F(β, h) = 0 non analiticity!
(convexity)∂F(β, h)
∂h= lim
N→∞EωN
[CNN
] > 0 if h > hc(β)
= 0 if h < hc(β)
F(β, h) and hc(β) depend on the law of τ and ω
Universality as β, h→ 0 ? YES, connected to continuum model
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 36 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Continuum partition functions
Build continuum partition functions for Pinning Model with α ∈ ( 12 , 1)
(disorder relevant) following “usual” approach [C, Sun, Zygouras 2015+]
We need to rescale
β = βN =β
Nα−1/2h = hN =
h
Nα
One has ZωN
d−−−−→N→∞
ZW with
ZW := 1 + C
∫
0<t<1
dW β,ht
t1−α + C 2
∫
0<t<t′<1
dW β,ht dW β,h
t′
t1−α(t ′ − t)1−α + . . .
where W β,ht := βW t + h t and C = α sin(απ)
π cK
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 37 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Continuum free energy
In analogy with the discrete model, define
Continuum free energy F(β, h) := limt→∞
1
tlogZW
β,h(0, t) a.s.
(existence and self-averaging need some work)
Again F(β, h) ≥ 0 and define
Continuum critical curve Hc(β) := suph ∈ R : F(β, h) = 0
Scaling relations
∀c > 0 : ZWβ,h
(c t)d= ZW
cα−12 β,cαh
(t) (Wiener chaos exp.)
F(cα−
12 β, cαh
)= c F(β, h) Hc(β) = Hc(1) β
2α2α−1
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 38 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Interchanging the limits
Can we relate continuum free energy to the discrete one?
By construction of continuum partition functions
ZWβ,h
(t)d= lim
N→∞ZωβN ,hN (Nt)
Assuming uniform integrability of log Zω (OK)
F(β, h) = limt→∞
1
tE[
logZWβ,h
(t)]
= limt→∞
1
tlim
N→∞E[
log ZωβN ,hN (Nt)
]
Assuming we can interchange the limits N →∞ and t →∞
F(β, h) = limN→∞
N limt→∞
1
Nt tE[
log ZωβN ,hN (Nt)
]= lim
N→∞N F(βN , hN)
Setting δ = 1N for clarity, we arrive at. . .
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 39 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Interchanging the limits
Conjecture
F(β, h) = limδ→0
F(βδα−
12 , hδα
)
δ
Theorem [C., Toninelli, Torri 2015]
For all β > 0, h ∈ R and η > 0 there is δ0 > 0 such that ∀δ < δ0
F(β, h − η) ≤ F(βδα−
12 , hδα
)
δ≤ F(β, h + η)
This implies Conj. and hc(β) ∼ Hc(β) ∼ Hc(1)β2α
2α−1
For any discrete Pinning model with α ∈ ( 12 , 1), the free energy F (β, h)
and the critical curve hc(β) have a universal shape in the regime β, h→ 0
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 40 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
Interchanging the limits
Very delicate result. How to prove it?
I Assume that there is a continuum Hamiltonian:
Zω = E[eHωNt
]ZW = E
[eH
Wt]
I Couple HωNt and HW
t on the same probability space in such a waythat the difference ∆N,t := Hω
Nt −HWt is “small”
I Deduce that
E[
log Zω]≤ E
[logZW
]+ log EE
[e∆N,t
]
and show that the last term is “negligible”
Problem: there is no continuum Hamiltonian!
Solution: perform coarse-graining and define an “effective” Hamiltonian
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 41 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
The DPRE case
What about the DPRE?
We can still define discrete F(β) and continuum F(β) free energy
Since F(β) ∼ F(1)β4 we can hope that
F(β) ∼ F(1)β4 as β → 0
provided the “interchanging of limits” is justified
N. Torri is currently working on this problem. A finer coarse-graining isneeded, together with sharper estimates on continuum partition functions
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 42 / 43
White noise Continuum partition functions The continuum DPRE Pinning models
References
I T. Alberts, K. Khanin, J. QuastelThe intermediate disorder regime for directed polymers in dimension1 + 1Ann. Probab. 42 (2014), 1212–1256
I T. Alberts, K. Khanin, J. QuastelThe Continuum Directed Random PolymerJ. Stat. Phys. 154 (2014), 305–326
I F. Caravenna, R. Sun, N. ZygourasPolynomial chaos and scaling limits of disordered systemsJ. Eur. Math. Soc. (JEMS), to appear
I F. Caravenna, R. Sun, N. ZygourasThe continuum disordered pinning modelProbab. Theory Related Fields 164 (2016), 17-59.
Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 43 / 43