EQUALITY OF CRITICAL POINTS FOR POLYMER DEPINNING
TRANSITIONS WITH LOOP EXPONENT ONE
KENNETH S. ALEXANDER AND NIKOS ZYGOURAS
Abstract. We consider a polymer with configuration modeled by the trajectory
of a Markov chain, interacting with a potential of form u + Vn when it visits a
particular state 0 at time n, with {Vn } representing i.i.d. quenched disorder. There
is a critical value of u above which the polymer is pinned by the potential. A
particular case not covered in a number of previous studies is that of loop exponent
one, in which the probability of an excursion of length n takes the form ϕ(n)/n for
some slowly varying ϕ; this includes simple random walk in two dimensions. We
show that in this case, at all temperatures, the critical values of u in the quenched
and annealed models are equal, in contrast to all other loop exponents, for which
these critical values are known to differ at least at low temperatures.
1. Introduction
A polymer pinning model is described by a Markov chain (Xn )n≥0 on a state space
containing a special point 0 where the polymer interacts with a potential. The spacetime trajectory of the Markov chain represents the physical configuration of the polymer, with the nth monomer of the polymer chain located at (n, Xn ) (or just at Xn ,
for an undirected model.) When the chain visits 0 at some time n, it encounters a
potential of form u + Vn , with the random values Vn typically modeling variation in
monomer species. We study the phase transition in which the polymer depins from
the potential when u goes below a critical value. We denote the distribution of the
Markov chain (started from 0) in the absence of the potential by P X and we assume
that it is recurrent. This recurrence assumption is merely a convenience and does not
change the essential mathematics; see [Al08], [GT05]. Of greatest interest is the case
when the excursion length distribution decays as a power law:
(1.1)
P X (E = n) = n−c ϕ(n),
n ≥ 1.
Here the loop exponent is c ≥ 1, E denotes the length of an excursion from 0 and ϕ
is a slowly varying function, that is, a function satisfying ϕ(κn)/ϕ(n) → 1 as n tends
to infinity, for all κ > 0.
Date: November 9, 2008.
1991 Mathematics Subject Classification. Primary: 82B44; Secondary: 82D60, 60K35.
Key words and phrases. pinning, polymer, disorder, random potential, quenched critical point.
The research of the first author was supported by NSF grants DMS-0405915 and DMS-0804934.
1
2
KENNETH S. ALEXANDER AND NIKOS ZYGOURAS
A large part of the existing rigorous literature on such models omits the case c = 1,
because it is often technically different and not covered by the methods that apply to
c > 1; see e.g. [Al08], [GT07], [To08c], [To08a]. That omission is partially remedied
in this paper, and we will see that the behavior for c = 1 can be quite different from
c > 1. The case c = 1 includes symmetric simple random walk in two dimensions, for
which ϕ(n) ∼ π/(log n)2 [JP72]. The essential feature of c = 1 is that P X (E > n) is a
slowly varying function of n, so that for example the longest of the first m excursions
typically has length greater than any power of m. This in effect enables the polymer
to (at low cost) bypass stretches of disorder in which the values Vn are insufficiently
favorable, and make returns to 0 in more-favorable stretches.
The quenched version of the pinning model is described by the Gibbs measure
(1.2)
dµβ,u,V
(x) =
N
1 βHNu (x,V) X
e
dP (x)
ZN
where x = (xn )n≥0 is a path, V = (Vn )n≥0 is a realization of the disorder, and
(1.3)
HNu (x, V)
=
N
X
(u + Vn )δ0 (xn ).
n=0
The normalization
£
¤
u
ZN = ZN (β, u, V) = E X eβHN (x,V)
is the partition function. The disorder V is a sequence of i.i.d. random variables with
mean zero, variance one. We denote the distribution of this sequence by P V . We
assume that V1 has exponential moments of all orders and we denote by MV (β) the
moment generating function of P V .
Let
N
X
X
LX
=
L
(x)
:=
δ0 (xn )
N
N
n=1
denote the local time at 0, and define the quenched free energy
fq (β, u) :=
1
1
lim
log ZN (β, u, V).
β N →∞ N
The fact that this limit exists and does not depend on V (except on a null set) is
standard; see [Gi07]. The parameter u ∈ R can be thought of as the mean value
of the potential, while the parameter β > 0 is the inverse temperature. It is known
that the phase space in (β, u) is divided by a critical line u = uqc (β) into two regions:
the localized and the delocalized one. In the delocalized region u < uqc (β) we have
fq (β, u) = 0 P V -a.s., while in the localized region u > uqc β) we have fq (β, u) > 0 P V a.s. It is proved in [GT06] that fq (β, ·) is infinitely differentiable for all u > uc (β). An
LOOP EXPONENT ONE
3
alternate, more phenomenological, characterization of the two regions is as follows.
From convexity we have for fixed β that
¿
Àβ,u,{Vi }
µ
¶
LN
1 ∂
1
∂
=
log ZN (β, u, V) →
fq (β, u) for all u,
N [0,N ]
β ∂u N
∂u
P V -a.s. This limiting value is called the contact fraction, denoted Cq (β, u), and it is
positive in the localized region and zero in the delocalized region. When the contact
fraction is positive we say the polymer is pinned.
The effect of the quenched disorder on the phase transition is quantified by comparing the quenched model to the corresponding annealed model, which is obtained
by averaging the Gibbs weight over the disorder to give
¡
¢
u
E V eβHN (x,V) = eβ∆LN (x) ,
where ∆ = u + β −1 log MV (β), so the corresponding annealed partition function is
¡
¢
ZNa = ZNa (β, u) := E X eβ∆LN
and the Gibbs measure is
(1.4)
dµβ,u
N (x) =
1 β∆LN (x) X
e
dP (x).
ZNa
The corresponding annealed free energy is denoted fa (β, u). The annealed critical
point is readily shown to be uac (β) = −β −1 log MV (β) for all β > 0 (see [AS06]), so
∆ = u − uac (β). It is a standard consequence of Jensen’s Inequality that fa (β, u) ≥
fq (β, u), so uac (β) ≤ uqc (β). The effect, or lack of effect, of the disorder on the
depinning transition may be seen in whether these two critical points actually differ,
and whether the specific heat exponent (describing the behavior of the free energy as
u decreases to the critical point) is different in the quenched case.
Though most mathematically rigorous work is relatively recent, there is an extensive physics literature on polymer pinning models—see the recent book [Gi07] and
surveys [Gi08], [To08b] and the references therein. In [Al08] (see also [To08a] for a
slightly weaker statement
with simpler proof) it was proved that for 1 < c < 3/2, and
P∞
for c = 3/2 with n=1 1/n(ϕ(n))2 < ∞, for sufficiently small β, one has uqc (β) = uac (β)
and the specific heat exponents are the same. Both works considered Gaussian disorder, though the method in [Al08] can be extended to accommodate more general
disorder having a finite exponential moment.
By contrast, it follows straightforwardly from the sufficient condition ([To08c],
equation (3.6)) that for c > 1, if V1 is unbounded, or if V1 is bounded and its essential supremum v satisfies P V (V1 = v) = 0, then for sufficiently large β one has
uqc (β) > uac (β); the method is based on fractional moment estimates. These results
together with [Al08] suggest that for 1 < c < 3/2 there should be a transition from
4
KENNETH S. ALEXANDER AND NIKOS ZYGOURAS
weak to strong disorder, i.e. there should exist a value β0 > 0 below which the annealed and quenched critical curves coincide, i.e. uqc (β) = uac (β), for β < β0 , while for
β > β0 , one has uac (β) < uqc (β), but this has not been proved.
For 1 < c < 3/2 and certain choices of bounded V1 (necessarily with P (V1 = v) >
0), it is known that the quenched and annealed critical points are equal for all β > 0
[DGLT08]. But in these examples Var(eβV1 )/[E(eβV1 )]2 stays bounded as β → ∞, so
there is no true “strong disorder” regime.
For c > 3/2 it follows from [GT05] that the quenched and annealed specific heat
exponents are different, and it was proved in [DGLT07] that the critical points are
strictly different for all β > 0, that is, β0 = 0. In [AZy08] the distinctness of critical
points at high temperature was extended to include c = 3/2 with ϕ(n) → 0 as n → ∞,
and the asymptotic order of the gap uqc (β) − uac (β) was given. Recently in [GLT08]
the critical points were shown to be distinct for all β > 0 for the case of c = 3/2
and ϕ(n) asymptotically a positive constant, a case about which physicists had long
disagreed ([DHV92], [FLNO88].)
Here we show that even with true strong disorder, the critical points remain the
same in the case c = 1.
Theorem 1.1. Consider the quenched model (1.2) and suppose E(etV1 ) < ∞ for all
t ∈ R and (1.1) holds with c = 1. For all β > 0 and all u > uac (β), the quenched free
energy fq (β, u) > 0, and thus uqc (β) = uac (β) for all β > 0.
2. Notation and Idea of the Proof.
Denote the local time at zero over a time interval I by
X
(2.1)
LX
δ0 (xn ),
I =
n∈I
X
0
so LX
N = L[0,N ] . The overlap between two paths X, X in an interval I is defined as
X
0
(2.2)
BIX,X =
δ0 (xn )δ0 (x0n ).
n∈I
0
We denote by P X,X the measure corresponding to two independent copies X, X 0 of
the Markov chain. The “energy gained over an interval I” is
X
(2.3)
HIu (x, V ) =
(u + Vn )δ0 (xn ).
n∈I
The annealed correlation length is defined to be M = M (β, u) := 1/βfa (β, u).
From (1.4), both βfa (β, u) and M are functions of only the product β∆. Using
Laplace aymptotics and the large deviations for the local time LN , one can deduce
LOOP EXPONENT ONE
5
the asymptotics of M and Ca (β, u), for β∆ ∼ 0. Specifically, letting
Z ∞
Ψ(t) =
ϕ(es ) ds,
t
we obtain
β∆ ∼ Ψ(log M ) and Ca (β, u) ∼
1
M ϕ(M )
For example, if ϕ(n) ∼ K(log n)−α for some α > 1 then
µ
¶−1/(α−1)
α−1
1
log M = log
∼
β∆
βfa (β, u)
K
as β∆ → 0.
as β∆ → 0,
so fa (β, ·) is C ∞ even at u = uac (β). The details are similar to the case c > 1 carried
out in [Al08], but we do not include them here as they are not required for our
analysis.
We use length scales K1 (β, M ), K2 (β, M ) related as follows, for β, M > 0. Let
ΛV (β) := log MV (2β)−2 log MV (β). For c = 1, summability of (1.1) means ϕ(x) → 0
as x → ∞. For a slowly varying function ϕ(x) → 0, we have
log x
→ ∞ as x → ∞.
1
log ϕ(x)
Therefore we can choose K1 , K2 satisfying
32K2 < eΛV (β)K2
(2.4)
and
(2.5)
4(M ∨ 1) log
1
K1
1
< K2 <
log
.
ϕ(K1 )
2ΛV (β)
2
For fixed β, as ∆ → 0 (i.e. M → ∞) we then have M ¿ K2 ¿ K1 . We assume
henceforth that K1 , K2 are even integers.
Define intervals
Ii = [iK1 , (i + 1)K1 ],
Iiγ = [iK1 , (i + γ)K1 ],
for 0 < γ < 1. For an interval I, let τI = inf{n ∈ I : xn = 0} and σI = sup{n ∈
I : xn = 0}. We set τI = σI = ∞ if the path does not visit the interval I. We denote
by ΞN K1 the set of all paths of length N K1 which have the following property : if
1/2
τIi < ∞ for some i ≤ N , then τIi ∈ Ii and σIi − τIi < K2 .
Idea of the proof. We will look at a scale N K1 and restrict the partition function
ZN K1 (u, β, V) to paths that belong to the set ΞN K1 . Further we will restrict attention
to paths within ΞN K1 which bypass bad blocks of length K1 . Roughly speaking a bad
block is defined to be a block for which the quenched partition function of a path
starting at a random point in the block, and not spending more than K2 time in
6
KENNETH S. ALEXANDER AND NIKOS ZYGOURAS
this block, is on average less than half the corresponding annealed partition function.
In Lemma 3.2 we control the probability of having a bad block. What is left is to
make an energy-entropy balancing of the paths that belong into ΞN K1 and bypass
bad blocks, and show that for β > 0 and ∆ = u + β −1 log MV (β) > 0 this balance
is uniformly (in N ) bounded away from zero. For this we will use the fact that in
a good block the free energy gained is of the order K2 /M (this is essentially Lemma
3.1), and the fact that because P X (E > k) is a slowly varying function of k the cost
of bypassing bad blocks is small.
3. Proof of the Theorem
Lemma 3.1. Let β > 0, u ∈ R, ∆ = u + β −1 log MV (β) and M = M (β, u). Then for
all N > 2β∆M ,
£
¤ 1N
log E V eβ∆LN ≥
(3.1)
.
2M
£
¤
Proof. It is observed in [Al08] that aN := β∆ + log E X eβ∆LN is subadditive in N .
Since aN /N → βfa (β, u), it follows that
£
¤
N
,
β∆ + log E X eβ∆LN ≥ N βfa (β, ∆) =
M
and the result is immediate.
¤
The block Ii is called good if it satisfies
h
i
i
X
u
1 X V X h βH[b,b+K
βH u
(x,V) ¯¯
(x,V) ¯¯
2]
E X e [b,b+K2 ]
xb = 0 >
E E e
xb = 0
2 1/2
1/2
b∈Ii
b∈Ii
1/2
=
|Ii | X £ β∆LK ¤
2 ,
E e
2
and bad otherwise. Let pVgood := P V (Ii is good) and pVbad := P V (Ii is bad).
Proposition 3.2. For K1 , K2 satisfying (2.4), (2.5) we have pVgood > 1/2.
Proof. By Chebyshev’s inequality,
i´
³P
h
u
βH[b,b+K
(x,V) ¯¯
X
2]
x
=
0
VarV
e
1/2 E
b
b∈Ii
pVbad ≤ 4 ³P
h
i´2
¯
u
V X eβH[b,b+K2 ] (x,V) ¯x = 0
1/2 E E
b
b∈Ii
i
h
¯
u
0
u
P
(x,V)+βH[b
βH[b,b+K
0 ,b0 +K ] (x ,V) ¯
0
V X,X 0
2]
2
0
x
=
x
=
0
e
1
E
E
1/2
b
|b−b |≤K2
b0
b,b0 ∈I
h
i .
< 4 P i
¯
u
u
0
0
βH[b,b+K ] (x,V)(X)+βH[b0 ,b0 +K ] (x ,V ) ¯
0
0
0
V,V
X,X
2
2
E
e
xb = xb0 = 0
1/2 E
b,b0 ∈I
i
LOOP EXPONENT ONE
7
Here we used the fact that whenever the two independent paths x, x0 visit zero at
u
points b, b0 such that |b−b0 | > K2 , then the energies H[b,b+K
(x, V) and H[bu0 ,b0 +K2 ] (x, V)
2]
are independent.
An easy calculation shows that the above is equal to
·
P
1/2
4
b,b0 ∈Ii
1|b−b0 |≤K2 E
X,X 0
0
0
X,X
β∆(LX
+LX
) ΛV (β)B[b,b+K
[b,b+K ]
[b0 ,b0 +K ]
e
2
2
e
0 0
2 ]∩[b ,b +K2 ]
¯
¯ xb = x0 0 = 0
b
¸
i
h
0
β∆(LX
+LX
)¯
[b,b+K2 ]
[b0 ,b0 +K2 ] ¯
xb = x0b0 = 0
E X,X 0 e
¸
·
0
¯
P
0
+LX
β∆(LX
0 ,b0 +K ] ) ¯
0
Λ
(β)K
X,X
[b,b+K2 ]
2
[b
V
2
E
e
xb = xb0 = 0
1/2 1|b−b0 |≤K2 e
b,b0 ∈Ii
≤4
h
i
¯
X
X0
P
X,X 0 eβ∆(L[b,b+K2 ] +L[b0 ,b0 +K2 ] ) ¯ x = x0 = 0
E
1/2
0
b
b
b,b0 ∈Ii
X
4
= 1/2
1|b−b0 |≤K2 eΛV (β)K2
2
|Ii ∩ Z| 0 1/2
P
1/2
b,b0 ∈Ii
b,b ∈Ii
32K2 ΛV (β)K2
e
K1
1 2ΛV (β)K2
e
<
K1
1
< ,
2
<
for K1 , K2 satsifying (2.4), (2.5). In the third line we have used the fact that the
expectations in the second line do not depend on b and b0 .
¤
We now return to the proof of Theorem 1.1. Let
JN := {i ≤ N : Ii is good} ∪ {1} = {i1 < · · · < i|JN | }.
By convention we let I|JN |+1 := {N }. Under P V , the sequence (ij − ij−1 )j≥2 is an
i.i.d. sequence of geometric random variables with parameter pVgood .
We denote by ΞJNNK1 = ΞJNNK1 (V) the set of paths that belong to ΞN K1 and make no
returns to 0 in bad blocks after the first block. In the following computation aj and
bj are the starting and ending points, respectively, of the excursion from Iij to Iij+1 .
8
KENNETH S. ALEXANDER AND NIKOS ZYGOURAS
By convention we set b0 := 0 and b|JN | := N . We then have
X X
X
X
X
¡ JN ¢
ZN K1 ΞN
=
·
·
·
K1
a1 <K2 b ∈I 1/2 a2 −b1 <K2
1
i
1/2
b|JN |−1 ∈Ii
2
|JN |
Y
X
≥
|JN |
h βH u
i
¯
(x,V)
E X e [bj−1 ,aj ]
δ0 (xaj ) ¯ xbj−1 = 0 pbj −aj
j=1
X
X
1/2
b|JN |−1 ∈Ii
2
|JN |
E
X
h
X
X
···
a1 <K2 b ∈I 1/2 a2 −b1 <K2
1
i
Y
a|JN | −b|JN |−1<K2
|JN |
u
βH[b
e
j−1 ,aj ]
(x,V)
a|JN | −b|JN |−1<K2
i
¯
¯
= aj xbj−1 = 0 pbj −aj .
; σ[bj−1 ,bj−1 +K2 ]
j=1
Moreover, on the set {σ[bj−1 ,bj−1 +K2 ] = aj } we have that H[buj−1 ,aj ] (x, V) = H[buj−1 ,bj−1 +K2 ] (x, V),
and therefore for some C the above is bounded below by
X X
X
X
X
···
a1 <K2 b ∈I 1/2 a2 −b1 <K2
1
i
2
|JN |
Y
1/2
b|JN |−1 ∈I|J
N|
a|JN | −b|JN |−1<K2
h βH u
i
¯
(x,V)
[bj−1 ,bj−1 +K2 ]
¯
E e
; σ[bj−1 ,bj−1 +K2 ] = aj xbj−1 = 0
X
j=1
h
i
u
βH[b
(x,V)
,b0 +K2 ]
0
=E e
X
min
3/4
1/2
, b1 ∈Ii
1
2
a1 ∈Ii
|JN |
·
Y
X
1/2
j=2 b
j−1 ∈Ii
j
h
i
u
βH[b
(x,V)
X
,b0 +K2 ]
0
≥E e
Ã
min
3/4
1/2
, b∈Ii
1
2
a∈Ii
pb−a
pbj −aj
h βH u
i
(x,V)
[bj−1 ,bj−1 +K2 ]
E e
| xbj−1 = 0
X
min
1/2
3/4
, b∈Ii
j+1
j
a∈Ii
pb−a
! |JN | 1/2
Y |Iij |
j=2
2
min
1/2
3/4
, b∈Ii
j+1
j
pb−a
a∈Ii
£
¤
E X eβ∆LK2
min
3/4
1/2
, b∈Ii
j
j+1
a∈Ii
pb−a
h
i ϕ¡(i − i + 1)K ¢
u
2
1
1
βH
(x,V)
≥ E X e [b0 ,b0 +K2 ]
C
(i2 − i1 + 1)K1
!
Ã
¢ 1/2
¡
|JN |
Y
ϕ (ij+1 − ij + 1)K1 |Iij | X £ β∆LK ¤
2
·
C
E e
(i
−
i
+
1)K
2
j+1
j
1
j=2

¢
¡
|JN |
i
h
Y
¤ ¢|JN |−1
£
¡
ϕ (ij+1 − ij + 1)K1
1 X βH[bu ,b +K ] (x,V) 
 E X eβ∆LK2
C
.
=
E e 0 0 2
K1
2(ij+1 − ij + 1)
j=1
LOOP EXPONENT ONE
9
In the second inequality we used the fact that the interval Iij is good, while the last
equality makes essential use of c = 1 in the cancellation of factors K1 . We then have
that
¡
¢
1
1
log ZN K1 ≥
log ZN K1 ΞJNNK1
N K1
N K1
¶
µ
£
¤
|JN | − 1
1 X h βH[bu ,b +K ] (x,V) i
1
+
log
E e 0 0 2
log E X eβ∆LK2
≥
N K1
K1
N K1
¡
¢
|JN |
Cϕ (ij+1 − ij + 1)K1
1 X
+
log
N K1 j=1
(ij+1 − ij + 1)
Letting N → ∞ we get that the left hand side converges to the quenched free energy
fq (β, u), while the right hand side converges to
¡
¢
£
¤
Cϕ
i
K
1 V
1
2
1
p
log E X eβ∆LK2 +
pV E V log
.
K1 good
K1 good
i2
Recall that i2 − 1 is a geometric random variable under P V with parameter pVgood . For
K sufficiently large we have
½
¾
xϕ(kx)
Cϕ := inf
: x ≥ 1, k ≥ K > 0,
ϕ(k)
and we may assume K1 ≥ K. We then have
Ã
¡ ¢!
£
¤
CC
ϕ
K1
1 V
ϕ
fq (β, u) ≥
pgood log E X eβ∆LK2 + E V log
2
K1
i2
³
´
£
¤
¡ ¢´
1 V ³
=
pgood log E X eβ∆LK2 + log CCϕ ϕ K1 − 2E V [log i2 ]
K1
Ã
Ã
!!
³
£ β∆L ¤
¡ ¢´
1 V
1
X
K2
≥
p
log E e
+ log CCϕ ϕ K1 − 2 log
+1
K1 good
pVgood
and by Lemma 3.1 and Proposition 3.2 this is bounded below by
µ
¶
³
´
K2
1
(3.2)
+ log CCϕ ϕ(K1 ) − 2 log 3 .
2K1 2M
Then using (2.4), (2.5) we get that provided M is sufficiently large, i.e. ∆ is small,
µ
¶
1
K2
CCϕ
fq (β, u) >
+ log
> 0.
2K1 4M
9
10
KENNETH S. ALEXANDER AND NIKOS ZYGOURAS
References
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Math. Phys. 279 117–146. MR2377630
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Department of Mathematics KAP 108, University of Southern California, Los
Angeles, CA 90089-2532 USA
E-mail address: alexandr@usc.edu
Department of Statistics, University of Warwick, Coventry CV4 7AL, UK
E-mail address: N.Zygouras@warwick.ac.uk