NEW CHARACTERIZATION OF THE WEAK DISORDER PHASE OF
DIRECTED POLYMERS IN BOUNDED RANDOM ENVIRONMENTS

STEFAN JUNK
University of Tsukuba. (Current address: AIMR, Tohoku University)
Abstract. We show that the weak disorder phase for the directed polymer model ina random environment is characterized by the integrability of the running supremum
supn∈N Wnβ of the associated martingale (Wnβ )n∈N . If the environment is bounded, we
also show that (Wnβ )n∈N is Lp -bounded in the whole weak disorder phase, for somep > 1. The argument generalizes to non-negative martingales with a certain productstructure.

1. The directed polymer model
1.1. Overview. The directed polymer model was introduced in the physics literature todescribe the folding of long molecule chains in a solution with random impurities. Mathematically, it is a model for random paths, called polymers, that are attracted or repulsed
by a space-time random environment with a parameter β ≥ 0, called inverse temperature,
governing the strength of the interaction. In recent years, the model has attracted muchinterest because it is conjectured that in a certain low temperature regime it belongs to
the KPZ (Kardar-Parisi-Zhang) universality class of randomly growing surfaces. In contrast, we focus on the high temperature phase, where it is known that the influence ofthe disorder disappears asymptotically and that the long-term behavior is diffusive. This
weak disorder phase is characterized by whether an associated martingale, (Wnβ )n∈N , isuniformly integrable, which is known to hold for small β if the spatial dimension is atleast three.
There is no closed-form characterization for the critical inverse temperature βcr and, inpractice, the uniform integrability is not easy to analyze. Several important features of
the weak disorder phase have been established, see for example [8, 9, 2], but many more
papers have focused on a different, very high temperature phase, which is characterizedby L2 -boundedness of the associated martingale. This condition is known to be strictly
stronger than uniform integrability [4], which naturally raises the question of whether
the L2 -regime is relevant mainly because of its computational convenience, or whether
the model undergoes a true phase transition within the weak disorder phase with somequantifiable change in behavior.
Theorem 1.1 below is an indicatation that no such transition occurs. Namely, we showthat if the environment is bounded, then (Wnβ )n∈N is Lp -bounded in the whole weakdisorder phase, for some p > 1 depending on β. We believe that our result will be
useful for extending results from the L2 -phase to the whole weak disorder phase. A
first step in this direction will be presented in the follow-up paper [14], see Section 1.5.
E-mail address: sjunk@tohoku.ac.jp.
2010 Mathematics Subject Classification. 60K37.
Key words and phrases. directed polymers and random environment.
1Moreover, Theorem 1.1 suggests that the weak disorder phase is better characterized
by the integrability of supn∈N Wnβ , which is a strictly stronger condition than uniform
integrability, see Section 2.1.
1.2. Definition. We introduce the directed polymer with site disorder, i.e. with disorderattached to the sites of the directed graph N × Zd . Let (ω0 , P0 ) be a probability measureon (Ω0 , F0 ) = (R, B(R)), where we assume
for all β ≥ 0,(1)
E eβ|ω0 | < ∞Ndand let Ω = ΩN×Z
and ω = (ωt,x )t∈N,x∈Zd , P = (t,x)∈N×Zd P0 . The energy of a path0x = (xt )t∈N up to time n in environment ω isP
Hn (ω, x) := nt=1 ωt,xt .
Let (X = (Xn )n∈N , P SRW ) denote the simple random walk on Zd and define the polymer
measure byµβω,n (dX) := (Znβ (ω))−1 eβHn (ω,X) P SRW (dX),
where Znβ is the normalizing constant, called the partition function of the model. Underµβω , paths are attracted to areas of space-time where the environment is positive and repelled by areas where it is negative. Note that this is not a consistent family of probability
measures, i.e., there is no infinite volume probability measure “µβω,∞ ” whose projection
to time n agrees with µβω,n , simultaneously for all n ∈ N. The associated martingale
mentioned in the previous section is defined byWnβ (ω) := Znβ (ω)e−nλ(β) ,βwhere λ(β) := log E[eβω0 ]. As a non-negative martingale, it is clear that the limit W∞≥0exists almost surely. We say that weak disorder (WD), resp. strong disorder (SD), holds
if
β
P(W∞
> 0) > 0.

(WD)

β
P(W∞
= 0) = 1.

(SD)βsatisfies a zero-one law, i.e., for all
Using assumption (1), it is not hard to see that W∞
β ≥ 0,βP(W∞> 0) ∈ {0, 1}.

(2)

Finally, we say that the environment is upper bounded (U-Bd.) or lower bounded (L-Bd.)
if there exists K > 1 such that, respectively,
P0 ([K, ∞)) = 0,
P0 ((−∞, −K]) = 0.

(U-Bd.)(L-Bd.)

1.3. Known results. We highlight a few key results and refer to [7] for a detailed surveyof the model.
It is known [8, Theorem 1.1] that there exists a critical temperature βcr = βcr (d) such thatweak disorder (WD) holds for β < βcr and strong disorder (SD) holds for β > βcr , withβcr (d) > 0 if and only if d ≥ 3. Moreover, (WD) is equivalent to uniform integrability of(Wnβ )n∈N [9, Proposition 3.1].
As mentioned in Section 1.1, many papers consider a stronger condition than uniform
integrability, namely L2 -boundedness. The second moment of Wnβ has a nice representation involving two independent random walks, called replicas in this context, and in
L2
particular one can explicitly compute the critical temperature βcr
for L2 -boundedness.
2

2L< βcr .
In contrast, no closed-form expression is known for βcr , but [4] has shown that βcr
Several important results have been obtained for the whole weak disorder phase, mostnotably a central limit theorem in probability [9, Theorem 1.2] for (µβω,n )n∈N , but our
understanding of the L2 -phase is much more complete.
To give an example, it is known that convergence in probability in the central limit
theorem can be replaced by almost sure convergence [13, 5, 18] in the L2 -phase, and one
would expect that this extends to the whole weak disorder phase. Based in part on the
results of this paper, some progress towards this extension for related models is obtainedin the follow-up paper [14]. Further results that are at present only known in L2 -weakdisorder are [9, Theorem 6.2], [10] and [17, 19]. We believe that Theorem 1.1 will be auseful tool towards extending them to the whole weak disorder phase.
We also mention that much research has focused on the strong disorder phase (SD).
In particular, the one-dimensional case is a very active field of research because of its
conjectured relation to the KPZ universality class and because exactly solvable models
are known. The behavior of µβω,n in strong disorder is radically different from the weakdisorder phase that is the focus of the present article.

1.4. Lp -boundedness in weak disorder. We state the main result of the paper.
Theorem 1.1. (i) If (WD) holds, thenβE sup Wn < ∞.

(3)

n∈N

(ii) Assume (U-Bd.). If (WD) holds, then there exists p > 1 such thatsup ∥Wnβ ∥p < ∞.

(4)

n∈N

The interval of p > 1 satisfying (4) is open. In contrast, if (SD) holds then, for allt > 1,1βP sup Wn > t ≥.
(5)
4K 2 tn∈N(iii) Assume (L-Bd.). If (WD) holds, then there exists p > 0 such that
sup E (Wnβ )−p < ∞.

(6)

n∈N

The set of p > 0 satisfying (6) is open.
In Section 2.2, we state a general result about non-negative martingales that implies
Theorem 1.1, and the proof is therefore postponed until Section 3. We make a few
comments to put the result into perspective.
Remark 1.2. (i) It is an interesting question whether (U-Bd.) is necessary in part (ii).
Theorem 2.1 below is certainly not valid without (10), so a proof of (4) without(U-Bd.) would have to be more specific to the directed polymer model.
(ii) The result can be taken as an indication that strong disorder (SD) holds at βcr .
Indeed, otherwise Theorem 1.1(ii) implies that supn ∥Wnβcr ∥p < ∞ for some p > 1
while (Wnβ )n∈N is not uniformly integrable for any β > βcr , which seems unlikely.
Lp(iii) In [11], a lower bound for βcr was obtained by estimating βcr, the critical temperaturefor Lp -boundedness, for p ∈ (1, 2). Theorem 1.1(ii) shows that such a lower boundcan potentially be sharp.
3

2L, there exists(iv) One consequence of Theorem 1.1(ii) is that in the L2 -phase, β < βcr
ε > 0 such that
sup E (Wnβ )2+ε < ∞.
n∈N

The same result has been obtained with the help of hypercontractivity in [6, display(3.12)] in dimension d = 2 in the so-called intermediate weak disorder phase.
(v) We note that the zero-one law (2) is only necessary for part (iii). In particular, theconclusions of parts (i)–(ii) remain valid if assumption (1) is replaced by E eβω0 < ∞for all β ≥ 0. Using the convention e−∞ = 0, we can therefore include degeneratemodels with P0 (ω0 = −∞) > 0. A natural example for such a degenerate model isoriented site or bond percolation, see [20].
1.5. Extensions and further research. We only present the result for the most commonly studied variation of this model, in discrete time and with the disorder attached tothe sites of N × Zd , but Theorem 2.1 below applies more generally:• The simple random walk P SRW can be replaced by other random walks.
• We can replace site disorder by bond disorder.
• We may consider a model in continuous space-time with Poissonian disorder and
P SRW replaced by Brownian motion.
All of these generalizations are discussed in the follow-up paper [14]. That paper also
introduces a different tool for the directed polymer model, which is related to the socalled noise operator appearing, for example, in the context of the hypercontractive
inequality. For bond disorder, we give an alternative proof for the central limit theorem
in probability from [9] and we identify the large deviation rate function with that of theunderlying random walk. For the Brownian polymer model in continuous space-time, we
improve the convergence in the central limit theorem to almost sure convergence. To
prove these results, we repeatedly use Theorem 1.1 (i) to justify interchanging limit andexpectation.
2. Lp -boundedness for martingales with product structure
2.1. Background. Consider a non-negative martingale (Mn )n∈N with M0 = 1 and let
M∞ := limn→∞ Mn denote its almost sure limit. We have in mind a situation where(Mn )n∈N is associated with a model undergoing a phase transition characterized bywhether M∞ > 0 or M∞ = 0, similar to the directed polymer model. One can usually show that M∞ > 0 implies that (Mn )n∈N is uniformly integrable, hence
sup E[φ(Mn )] < ∞(7)n= ∞. In practice, (7) is a rather weak
for some convex function φ with limx→∞ φ(x)xintegrability condition since φ is not explicit and φ(x)may grow extremely slowly. Wexshow that, under certain assumptions, P(M∞ > 0) > 0 implies(a) L1 -boundedness of the running maximum Mn∗ := supk=0,...,n Mk(b) and Lp -boundedness of (Mn )n∈N for some p > 1.
Of course, (b) implies (a) by Doob’s maximal inequality, but we need (a) to prove (b).
Note that, without any assumptions, E[M∞ ] = 1 only yields a tail bound of the form
P(M∞ > t) = o(t−1 ), regardless of whether P(M∞ > 0) is positive or not. We think it
is surprising that in our setup, we obtain an upper bound on Mn that relies on a lower
bound on M∞ .
4

We briefly give some context for both steps. For the implication (a), we note that∗] < ∞ is a stronger condition than uniform integrability, see [15, Chapter II, ExerE[M∞cise 3.15]. The structure of that counterexample is similar to the classical gambler’s ruin
problem, in that the martingale grows larger and larger before it is eventually absorbed
in a much smaller value. To prove (a), we assume that (Mn )n∈N has a certain “productstructure” which excludes such behavior. Namely, under this assumption, the probability that Mk+l is much smaller than Mk can be compared with the probability that Ml ismuch smaller than M0 = 1, and hence with the probability that M∞ is small. The latterprobability is bounded away from zero if P(M∞ > 0) > 0.
For the implication (b), we recall from Doob’s maximal inequality that∥Mn∗ ∥1 ≤ C(1 + E[Mn log+Mn ]).
In [12], it was shown that if Mn+1 /Mn is bounded, then the converse is also true, i.e.,∗E[M∞] < ∞ implies supn E[Mn log+Mn ] < ∞. Note that, together with implication (a),this is already an improvement on (7). The proof of the implication (b) is inspired bythis result, in the sense that we use integrability of the running maximum Mn∗ to get amoment bound for Mn itself. We obtain the stronger conclusion of Lp -boundedness byusing the product structure introduced for implication (a).
2.2. Lp -boundedness for martingales. We now state the main result of this section.
The “product structure” mentioned above is (8).
Theorem 2.1. Let ((Fn )n∈N , (Mn )n∈N ) be a non-negative martingale with M0 = 1. Assume that for every k, l ∈ N and f : R+ → R convex, almost surely on Mk > 0, Mk+l 
E f
Fk ≤ E[f (Ml )].
(8)Mk
Let Mn∗ := supk=0,...,n Mk and M∞ := limn→∞ Mn . Then the following hold.
(i) IfP (M∞ > 0) > 0(9)∗] < ∞.
then E[M∞(ii) If (9) holds and if there exists K > 1 such that

P(Mn+1 ≤ KMn ) = 1

for all n ∈ N,

(10)then there exists p > 1 such thatsup ∥Mn ∥p < ∞.

(11)

n

Moreover, the interval of p > 1 satisfying (11) is open.
(iii) If P(M∞ = 0) = 1 and if (10) holds, then, for all t > 1,
1∗
P(M∞> t) >.
4K 2 t(iv) If P(M∞ > 0) = 1 and if there exists K > 1 such that
P(Mn+1 ≥ Mn /K) = 1

for all n ∈ N,

(12)

(13)

then there exists p > 0 such that
sup E[Mn−p ] < ∞.
nMoreover, the interval of p > 0 satisfying (14) is open.
5(14)

We give two examples to demonstrate that the assumption (8) is natural, at least in
the context of branching processes. See [1] and [16] for background on Galton-Watsonprocesses and on branching random walks.
Example 2.2. Let (Zn )n∈N be a Galton-Watson process with Z0 = 1 and associated martingale Mn := Zn m−n , where m := E[Z1 ] is the expected number of offspring. Then, on
Mk > 0,
"
#
Zk
 1 X

h
h M  i
i
k+l
(i)
(1) 
Fk = E f
Ml
Fk ≤ E f Ml
E f
Fk = E[f (Ml )],Mk
Zk i=1(1)(2)

where Mn , Mn , . . . are independent copies of Mn . Therefore (8) holds.
Example 2.3. We follow the notation from [3] for a one-dimensional branching random
walk. For θ ∈ R, a martingale (W (n) (θ))n∈N is defined by
P (n)
(n)
W (n) (θ) = (m(θ))−n ri=1 e−θzi ,
(n)

(n)

where r(n) denotes the number of particles at time n and z1 , . . . , zr(n) their positions. To
f (l,1) (θ), W
f (l,2) (θ), . . . denote independent copies of W (l) (θ). On {W (k) (θ) >verify (8), let W(k)0} = {r ≥ 1}, we havePr(k) −θz(k) f (l,i)
r(k)
i W
(θ) X
W (k+l) (θ) di=1 ef (l,i) (θ),=µ({i})W(15)=
Pr(k) −θz(k)W (k) (θ)
i
e
i=1
i=1
where µ is a probability measure on {1, . . . , r(k) }. Now (8) follows as in Example 2.2.
The conclusion (11) is not new for Examples 2.2 and 2.3, because (10) implies that themartingales converge in Lp for some explicit p > 1, see [1, Part I.B, Theorem 2] and [3, p.
26]. Roughly speaking, the branching structure provides a lot of independence between
particles in generation n, so the pth moment cannot be large if the influence from theearly generations is small. This is not true for the directed polymer model, where themartingale has much larger correlations. The main observation in this paper is that adecomposition similar to (15) is almost sufficient for Lp -boundedness.
3. Proofs
Proof of Theorem 2.1. Part (i): We will find ε, η > 0 such that, for all n ∈ N and t > 1,
P(Mn∗ > t) ≤ P(Mn > tε)/η,(16)

so thatE[Mn∗ ] ≤1
1
E[Mn ] + 1 =
+ 1.
εη
εη

Now, the right-hand side does not depend on n while the left-hand side converges to∗E[M∞] for n → ∞, so the claim follows. To prove (16), we set, for some δ, ε > 0 to bechosen later,xfδ,ε (x) := δ
− 1 ∧ 1.
εThe important feature is that fδ,ε is concave and that, for all x ≥ 0,1(ε,∞) (x) ≥ fδ,ε (x) ≥ 1[(δ−1 +1)ε,∞) (x) − δ 1[0,ε) (x).
6(17)

Let τ := inf{n ∈ N : Mn > t} and note that Mτ > 0 on {τ < ∞}. We get


Mn
>ε
P(Mn > tε) ≥ P τ ≤ n,
Mτ
n
X 


=
E 1τ =k E 1 Mn >ε |Fk
Mk

k=1
n
X

h
h  M  iin
E 1τ =k E fδ,ε
Fk
M
k
k=1
n
h
X

i
≥
E 1τ =k E fδ,ε (Mn−k )
≥

(18)

k=1

≥ P(τ ≤ n) inf E[fδ,ε (Mk )],k

We have used (17) in the second inequality and (8) for the third inequality. For any
δ > 0,

inf E[fδ,ε (Mk )] ≥ E inf fδ,ε (Mk )
k∈N
k∈N




−1
≥ P inf Mk ≥ (δ + 1)ε − δP inf Mk < ε
k∈N

k∈N

ε↓0

−−→ P(M∞ > 0) − δP(M∞ = 0).
The second inequality is due to (17). In the last line, we used that 0 is an absorbing statefor non-negative martingales, hence{M∞ > 0} = inf Mn > 0 .
(19)
n∈N

Using (9), we find ε, δ > 0 such thatinf E[fδ,ε (Mk )] =: η > 0,k∈Nwhich together with (18) implies (16).
Part (ii): We again use τ := inf{n : Mn > t}, where t > 1 will be chosen later.
non {τ ≤ n}. Therefore, for any ε > 0,
Assumption (10) implies that Mn ≤ tK M
MτE[Mn1+ε ] ≤ t1+ε + E[1τ ≤n Mn1+ε ]n M 1+ε i
h
X
n
1+ε
1+ε
E 1τ =k≤ t + (Kt)
Mk
k=1
n
h
h M 1+εii
Xn= t1+ε + (Kt)1+ε
E 1τ =k E
Fk
Mk
k=1
1+ε

≤t

+ (Kt)

1+ε

n
X

(20)




E 1τ =k E (Mn−k )1+ε

k=1

≤ t1+ε + (Kt)1+ε P(τ ≤ n)E[Mn1+ε ].
The second-to-last inequality is (8) and in the final inequality we used that n 7→ E[f (Mn )]is increasing for any convex function f . By part (i),Z ∞∗∗E[M∞ ] = 1 +P(M∞> t)dt < ∞,17

so we find t such that, for all n ∈ N,
1.
4K 2 tOnce t is fixed, we choose ε ∈ (0, 1) such that tε ≤ 2, which by the above calculation
implies1
E[Mn1+ε ] ≤ t1+ε + E[Mn1+ε ].
2
1+ε
1+ε
Since n is arbitrary we get supn E[Mn ] ≤ 2t . Next, suppose that supn ∥Mn ∥p < ∞∗for some p > 1. Doob’s maximal inequality implies that ∥M∞∥p < ∞ and, in particular,
there exists t > 1 such that1∗> t) ≤
P(M∞
.
p+1
4K tp
Hence, using (20) with q ∈ [p, p + 1] in place of 1 + ε, for any n ∈ N,∗> t) ≤P(τ ≤ n) ≤ P(M∞E[Mnq ] ≤ tq + (Kt)q P(τ ≤ n)E[Mnq ]tq−p≤ tq +
E[Mnq ].
4
Now choose q ∈ (p, p + 1) such that tq−p ≤ 2, which shows supn ∥Mn ∥q < ∞.
Part (iii): If (12) fails for some t > 1, the argument from part (ii) shows supn E[Mn1+ε ] ≤2t1+ε < ∞ for some ε > 0. Hence (Mn )n∈N is uniformly integrable and E[M∞ ] = 1,
contradicting P(M∞ = 0) = 1.
Part (iv): Using τ := inf{n : Mn ≤ 1/t}, similar calculations as in part (ii) giveE[Mn−p ] ≤ tp + (Kt)p P(τ ≤ n)E[Mn−p ].
Using P(M∞ > 0) = 1 and (19), we havet→∞P(τ ≤ n) ≤ P (inf n Mn ≤ 1/t) −−−→ 0,
so we can choose t large enough that, for all n ∈ N,
1
P(τ ≤ n) ≤.
4K
Once t is fixed, we choose p ∈ (0, 1) such that tp ≤ 2 and get
sup E[Mn−p ] < 2tp .
n

It remains to prove the final claim, that the set of p satisfying (14) is open. Assume−p/2
that (Mn )n∈N satisfies (14) and note that (Mn )n∈N is a submartingale. Hence, Doob’s
maximal inequality implies
h
h
i
−p i
−p/2 2
E inf k=1,...,n Mn= E supk=1,...,n Mn≤ 4E[Mn−p ].
Since the right-hand side is bounded in n, we find t > 1 such that1.
P inf n Mn < 1/t ≤4K p+1 tp
From here on, the argument is identical to the corresponding claim in part (ii).
Proof of Theorem 1.1. Note that (U-Bd.) implies (10) since, almost surely,
β
X
Wn+1
=
µβω,n (Xn+1 = x)eβωn+1,x −λ(β) ≤ eβK−λ(β) ,
Wnβ
x∈Zd
8

□

and that (L-Bd.) similarly implies (13). In view of Theorem 2.1, it remains to verify that
(Wnβ )n∈N satisfies (8). For k, l ∈ N, we computeh W β  i
h X
 i
β
β
k+l
E f
Fk = E fµω,k (Xk = y)Wl ◦ θk,y Fk
Wkβ
y∈Zd
hX
 i
≤E
µβω,k (Xk = y)f Wlβ ◦ θk,y Fk
y∈Zd

=

X

 



µβω,k (Xk = y)E f Wlβ ◦ θk,y Fk = E f (Wlβ ) ,y∈Zdwhere θk,y denotes the space-time shift and where we have applied Jensen’s inequality. □Acknowledgements
This work was supported by the JSPS Postdoctoral Fellowship for Research in Japan,
Grant-in-Aid for JSPS Fellows 19F19814. The author is grateful to Ryoki Fukushima formany inspiring discussions and suggestions and to Shuta Nakajima for helpful suggestions.
We also thank an anonymous referee for their helpful comments.
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