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Moderate deviations for the energy of charged polymer
Statistics and Probability Letters 83 (2013) 1078–1082
Contents lists available at SciVerse ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Moderate deviations for the energy of charged polymer✩ Yanqing Wang ∗ School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, PR China LMBA, UMR 6205, Université of Bretagne-Sud, Campus de Tohannic, BP 573, 56017 Vannes, France
article
abstract
info
Article history: Received 19 October 2012 Received in revised form 8 December 2012 Accepted 8 December 2012 Available online 25 December 2012
In this paper, we study the energy of the polymer {S1 , · · · , Sn } equipped with random electrical charges {ω1 , · · · , ωn }: Hn =
ωi ωj I{Si =Sj }
1≤i
where {Sn } is a symmetric random walk in Z in the domain of attraction of the symmetric α -stable process. Based on the large deviation result of the local time of α -stable random
MSC: 60F15
walk and the Gärtner–Ellis theorem, we get the moderate deviations for Hn . © 2012 Elsevier B.V. All rights reserved.
Keywords: Charged polymer Stable random walk Local time Moderate deviations
1. Introduction Consider a sequence {ωi }i≥1 of independent, identically-distributed mean-zero random variables, and let {Sn }n≥1 denote an independent random walk on Z d . Kantor and Kardar (1991) consider a model of polymers with random electrical charges associated with the Hamiltonian: Hn =
ωi ωj I{Si =Sj } .
(1.1)
1≤i
Hn is called the energy of the polymer. Roughly speaking, ω1 , ω2 , . . . are random charges that are placed on a polymer path modeled by the trajectories of S; and one can construct a Gibbs-type polymer measure from the Hamiltonian Hn . Hn is a very rich topic of researches, we only mention some recent references (see Asselah, 2010, Asselah, 2011, Chen and Khoshnevisan, 2009, Chen, 2008, Hu, 2010, Hu et al., 2011). Our work is partially inspired by two papers. One is Chen (2008) in which the author study the limit laws of Hn when Sn is a symmetric random walk with finite second moment. The key technique is the comparison between the moments of Hn and the self-intersection local time of random walk. Another is the large deviation result for the local time of α -stable random walk (see Chen et al., 2005). Naturally, one will ask whether the results in Chen (2008) can be extended to the case that Sn belong to the domain of attraction of the symmetric α -stable distribution. In this paper, we restrict our attention to the one-dimensional charged energy in this case. Throughout, {ωk }k≥1 is an i.i.d sequence of symmetric random variables with
Eω12 = 1 and Eeλ0 ω1 < ∞ for some λ0 > 0. 2
✩ Research supported by Million Project of Zhongnan University of Economics and Law.
∗
Correspondence to: School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, PR China. E-mail addresses: [email protected], [email protected].
0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.12.013
(1.2)
Y. Wang / Statistics and Probability Letters 83 (2013) 1078–1082
1079
Let now Sn be a symmetric random walk in Z in the domain of attraction of the symmetric stable process Xt of index α , i.e. Sn /g (n) → X1 in law with g (x) a function of regular variation of index 1/α and α ∈ (1, 2]. For simplicity, we assume further that our random walk is strongly aperiodic with support Z , and as x → ∞,
σ := lim x−1/α g (x) > 0. x→∞
For convenience, n we call {Sn } the symmetric α -stable random walk. Let Ln (x) = i=1 I{Si =x} . It is pointed out (Theorem 4 in Chen et al., 2005) that lim
n→∞
1 bn
log P
L2n
(x) ≥ λn
1 2− α
1
bnα
= −λ
x∈Z
ασ
α
2α
2α − 1
2α−1
2α Mα,2
(1.3)
where Mα,2 = sup
g ∈Fα
∥g ∥24 −
∞
|λ|α |ˆg (λ)|2 dλ < ∞
−∞
and gˆ is the Fourier transform of g,
∞
Fα = f ∈ L2 (R) : ∥f ∥2 = 1 and
|λ|α |fˆ (λ)|2 dλ < ∞ .
−∞
Based on the above result and the Gärtner–Ellis theorem, we study the moderate deviations for the energy Hn when Sn is symmetric α -stable random walk. Some ideas of the proof come from Chen (2008). The main result is the following theorem. Theorem 1.1. Let bn be a positive sequence satisfying
1
bn = o n α+1
bn → ∞,
,
n → ∞.
(1.4)
Then, for any λ > 0, lim
n→∞
1 bn
1
1
log P ±Hn ≥ λn1− 2α bn2
+ 21α
=−
α + 1 2α α+1 1 λ α+1 Cα 2α
where Cα := σ
α
2α − 1
2α−1
2α Mα,2
.
2. Preliminaries In this section, we will present some useful lemmas. Lemma 2.1 (See Theorem 3.1 in Li, 2012). Let {bn } be a positive sequence satisfying bn → ∞ and
bn = o(n), n → ∞.
For any λ > 0, lim sup n→∞
1 bn
1 1− α bn log E exp λ sup Ln (x) < ∞.
n
x
Lemma 2.2. Let bn be a positive sequence satisfying (1.4) and Mn be a positive sequence satisfying
Mn → ∞ and
Mn
1 bα+ n
n
21α
→ 0, n → ∞.
Then for any λ > 0, lim
n→∞
1 bn
1
1
log P sup Ln (x) ≥ Mn n1− α bnα x
= −∞.
(1.5)
1080
Y. Wang / Statistics and Probability Letters 83 (2013) 1078–1082
Proof. Using Chebyshev’s inequality and Lemma 2.1, we have 1
lim
bn
n→∞
log P sup Ln (x) ≥ Mn n
1 1− α
1
bnα
x
1
≤ lim
bn
n→∞
1 1 1− α 1− α 1 bn b 1 n log exp − Mn n1− α bnα E exp sup Ln (x)
n
n
x
= −∞. Lemma 2.3 (See Lemma 2.1 in Chen, 2008). Assume (1.2). Then
E
n
2 ωi
−
n
i =1
ωi2
= 2n(n − 1).
i=1
More generally, there is a constant C > 0 such that for any integers n ≥ 1 and m ≥ 2,
m 2 n n 2 ωi − E ωi ≤ m!(Cn(n − 1))m/2 . i=1 i =1 3. Proof of Theorem 1.1 1
1
ˆ n = Hn I{sup Ln (x)≤Kn } . Let Kn = Mn n1− α bnα and H x∈Z ˆ ˆ n to We will prove Hn satisfies the moderate deviations given in Theorem 1.1. Then the moderate deviations pass from H Hn through the exponential equivalence given by lim sup n→∞
1 bn
1
ˆ n ̸= Hn } = lim log P{H
n→∞
bn
log P sup Ln (x) > Kn
= −∞
x
where the last step follows from Lemma 2.1. According to the Gärtner–Ellis theorem, we need only to prove that for any θ > 0,
lim
n→∞
1 bn
log E exp
1
±θ
bn2
− 21α 1
n1− 2α
ˆn = α − 1 H 2α
θ 2α
α−1 1
Cα
.
(3.1)
Notice that
Hn =
n 1
2 x∈Z
2 ωi I{Si =x}
−
n
i=1
ωi2 I{Si =x}
.
i=1
For each x ∈ Z , write D(x) = {1 ≤ k ≤ n; Sk = x}. It is easy to see that D(x) random walk {Sk }, the variables
n
2 ωi I{Si =x}
−
n
i=1
ωi2 I{Si =x} ,
x∈Z
i =1
form an independent family and for each fixed x ∈ Z ,
n
2 ωi I{Si =x}
−
n
i =1
Ln (x)
2
ω
d 2 i I{Si =x}
=
i =1
i=1
ωi
Ln (x)
−
ωi2 .
i =1
For convenience, we write 1
λ(n, θ , α, x) = θ
bn2
− 21α 1
n1 − 2 α ω
I{sup Ln (x)≤Kn } . x
We adopt the notation E for the expectation with respect to {ωk }k≥1 .
D(y) = Φ if x ̸= y. So conditioned on the
Y. Wang / Statistics and Probability Letters 83 (2013) 1078–1082 1− 1 bn2 2α 1− 1 n 2α
Notice that
1081
Kn = o(1), by Taylor’s expansion and Lemma 2.3, we have
b 12 − 21α b 12 − 21α n ˆ n = E Eω exp ±θ n ˆn E exp ±θ H H 1 1− 21α n n1− 2α 2 L L n (x) n (x) ωi − ωi2 = E Eω exp ±λ(n, θ , α, x) x∈Z i =1 i=1 2 L L n (x) n (x) ω 2 = E E exp ±λ(n, θ , α, x) ωi − ωi x∈Z i =1 i =1 1 2 =E 1 + λ (n, θ , α, x)Ln (x)(Ln (x) − 1)(1 + o(1)) 2
x∈Z
= E exp
log 1 +
x∈Z
= E exp
1 x∈Z
Because
2
1 2
λ2 (n, θ , α, x)Ln (x)(Ln (x) − 1)(1 + o(1))
λ (n, θ , α, x)Ln (x)(Ln (x) − 1)(1 + o(1)) . 2
Ln (x) = n, it is easy to see
x∈Z
b 12 − 21α 1 1 2 n 2 ˆ Ln (x) log E exp ±θ Hn = lim log E exp λ (n, θ , α, x) lim 1 n→∞ bn n→∞ bn 2 n1− 2α x∈Z 1 2 1− α 1 θ bn = lim L2n (x)I{sup Ln (x)≤Kn } . log E exp 1 n→∞ bn 2 n2− α x∈Z x 1
Write Lˆ n = lim sup n→∞
x∈Z
1 bn
L2n (x)I{supx Ln (x)≤Kn } and Ln =
log P{Lˆ n ̸= Ln } = lim
n→∞
1 bn
x∈Z
L2n (x). From (1.3) and the exponential equivalence given by
log P{sup Ln (x) > Kn } = −∞, x
we have lim
n→∞
1 bn
log P
x∈Z
L2n
(x)I{sup Ln (x)≤Kn } ≥ λn
1 2− α
1
bnα
x
= −λα
σα 2α
2α − 1 2α Mα,2
2α−1 =−
λα Cα . 2α
(3.2)
2 x∈Z Ln (x) ≤ n supx Ln (x) and Lemma 2.1, we have 1 2 1− α 1 θ bn lim log E exp L2n (x)I{sup Ln (x)≤Kn } < ∞ 1 n→∞ bn 2 n2− α x∈Z x
Using the fact that
According to the Varadhan’s integral lemma, we have
2 1 θ 2 b1− α1 θ λα Cα α − 1 θ 2α α−1 n 2 lim log E exp L (x)I = sup λ− = . Ln (x)≤Kn } n→∞ bn 2 n2− α1 x∈Z n {sup x 2 2α 2α Cα λ 1
The proof is complete. Acknowledgments The author is grateful to an anonymous referee for useful comments and remarks. The work has been partially supported by National Natural Science Foundation of China, Tian Yuan Foundation (No. 11226212) and the Foundation of Ministry of Education of China (No. 12YJCGJW015).
1082
Y. Wang / Statistics and Probability Letters 83 (2013) 1078–1082
References Asselah, A., 2010. Annealed lower tails for the energy of a charged polymer. J. Stat. Phys. 138, 619–644. Asselah, A., 2011. Annealed upper tails for the energy of a charged polymer. Ann. Inst. H. Poincaré Probab. Statist. 47, 80–110. Chen, X., 2008. Limit laws for the energy of a charged polymer. Ann. Inst. H. Poincaré Probab. Statist. 44, 638–672. Chen, X., Khoshnevisan, D., 2009. From charged polymer to random walk in random scenery. In: Optimality: The 3rd E.L. Lehmann Symposium. In: IMS Lecture Notes, vol. 57. pp. 237–251. Chen, X., Li, W., Rosen, J., 2005. Large deviations for local times of stable processes and stable random walks in 1 dimension. Electron. J. Probab. 16, 577–608. Hu, Y., 2010. Strong approximations in a charged-polymer model. Period. Math. Hungar. 61, 213–224. Hu, Y., Khoshnevisan, D., Wouts, M., 2011. Charged polymers in the attractive regime: a first transition from Brownian scaling to four-point localization. J. Stat. Phys. 144, 948–977. Kantor, Y., Kardar, M., 1991. Polymers with random self-interactions. Europhys. Lett. 14, 421–426. Li, Y., 2012. Moderate deviations for stable random walks in random scenery. J. Appl. Probab. 49, 280–294.
Contents lists available at SciVerse ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Moderate deviations for the energy of charged polymer✩ Yanqing Wang ∗ School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, PR China LMBA, UMR 6205, Université of Bretagne-Sud, Campus de Tohannic, BP 573, 56017 Vannes, France
article
abstract
info
Article history: Received 19 October 2012 Received in revised form 8 December 2012 Accepted 8 December 2012 Available online 25 December 2012
In this paper, we study the energy of the polymer {S1 , · · · , Sn } equipped with random electrical charges {ω1 , · · · , ωn }: Hn =
ωi ωj I{Si =Sj }
1≤i
where {Sn } is a symmetric random walk in Z in the domain of attraction of the symmetric α -stable process. Based on the large deviation result of the local time of α -stable random
MSC: 60F15
walk and the Gärtner–Ellis theorem, we get the moderate deviations for Hn . © 2012 Elsevier B.V. All rights reserved.
Keywords: Charged polymer Stable random walk Local time Moderate deviations
1. Introduction Consider a sequence {ωi }i≥1 of independent, identically-distributed mean-zero random variables, and let {Sn }n≥1 denote an independent random walk on Z d . Kantor and Kardar (1991) consider a model of polymers with random electrical charges associated with the Hamiltonian: Hn =
ωi ωj I{Si =Sj } .
(1.1)
1≤i
Hn is called the energy of the polymer. Roughly speaking, ω1 , ω2 , . . . are random charges that are placed on a polymer path modeled by the trajectories of S; and one can construct a Gibbs-type polymer measure from the Hamiltonian Hn . Hn is a very rich topic of researches, we only mention some recent references (see Asselah, 2010, Asselah, 2011, Chen and Khoshnevisan, 2009, Chen, 2008, Hu, 2010, Hu et al., 2011). Our work is partially inspired by two papers. One is Chen (2008) in which the author study the limit laws of Hn when Sn is a symmetric random walk with finite second moment. The key technique is the comparison between the moments of Hn and the self-intersection local time of random walk. Another is the large deviation result for the local time of α -stable random walk (see Chen et al., 2005). Naturally, one will ask whether the results in Chen (2008) can be extended to the case that Sn belong to the domain of attraction of the symmetric α -stable distribution. In this paper, we restrict our attention to the one-dimensional charged energy in this case. Throughout, {ωk }k≥1 is an i.i.d sequence of symmetric random variables with
Eω12 = 1 and Eeλ0 ω1 < ∞ for some λ0 > 0. 2
✩ Research supported by Million Project of Zhongnan University of Economics and Law.
∗
Correspondence to: School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, PR China. E-mail addresses: [email protected], [email protected].
0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.12.013
(1.2)
Y. Wang / Statistics and Probability Letters 83 (2013) 1078–1082
1079
Let now Sn be a symmetric random walk in Z in the domain of attraction of the symmetric stable process Xt of index α , i.e. Sn /g (n) → X1 in law with g (x) a function of regular variation of index 1/α and α ∈ (1, 2]. For simplicity, we assume further that our random walk is strongly aperiodic with support Z , and as x → ∞,
σ := lim x−1/α g (x) > 0. x→∞
For convenience, n we call {Sn } the symmetric α -stable random walk. Let Ln (x) = i=1 I{Si =x} . It is pointed out (Theorem 4 in Chen et al., 2005) that lim
n→∞
1 bn
log P
L2n
(x) ≥ λn
1 2− α
1
bnα
= −λ
x∈Z
ασ
α
2α
2α − 1
2α−1
2α Mα,2
(1.3)
where Mα,2 = sup
g ∈Fα
∥g ∥24 −
∞
|λ|α |ˆg (λ)|2 dλ < ∞
−∞
and gˆ is the Fourier transform of g,
∞
Fα = f ∈ L2 (R) : ∥f ∥2 = 1 and
|λ|α |fˆ (λ)|2 dλ < ∞ .
−∞
Based on the above result and the Gärtner–Ellis theorem, we study the moderate deviations for the energy Hn when Sn is symmetric α -stable random walk. Some ideas of the proof come from Chen (2008). The main result is the following theorem. Theorem 1.1. Let bn be a positive sequence satisfying
1
bn = o n α+1
bn → ∞,
,
n → ∞.
(1.4)
Then, for any λ > 0, lim
n→∞
1 bn
1
1
log P ±Hn ≥ λn1− 2α bn2
+ 21α
=−
α + 1 2α α+1 1 λ α+1 Cα 2α
where Cα := σ
α
2α − 1
2α−1
2α Mα,2
.
2. Preliminaries In this section, we will present some useful lemmas. Lemma 2.1 (See Theorem 3.1 in Li, 2012). Let {bn } be a positive sequence satisfying bn → ∞ and
bn = o(n), n → ∞.
For any λ > 0, lim sup n→∞
1 bn
1 1− α bn log E exp λ sup Ln (x) < ∞.
n
x
Lemma 2.2. Let bn be a positive sequence satisfying (1.4) and Mn be a positive sequence satisfying
Mn → ∞ and
Mn
1 bα+ n
n
21α
→ 0, n → ∞.
Then for any λ > 0, lim
n→∞
1 bn
1
1
log P sup Ln (x) ≥ Mn n1− α bnα x
= −∞.
(1.5)
1080
Y. Wang / Statistics and Probability Letters 83 (2013) 1078–1082
Proof. Using Chebyshev’s inequality and Lemma 2.1, we have 1
lim
bn
n→∞
log P sup Ln (x) ≥ Mn n
1 1− α
1
bnα
x
1
≤ lim
bn
n→∞
1 1 1− α 1− α 1 bn b 1 n log exp − Mn n1− α bnα E exp sup Ln (x)
n
n
x
= −∞. Lemma 2.3 (See Lemma 2.1 in Chen, 2008). Assume (1.2). Then
E
n
2 ωi
−
n
i =1
ωi2
= 2n(n − 1).
i=1
More generally, there is a constant C > 0 such that for any integers n ≥ 1 and m ≥ 2,
m 2 n n 2 ωi − E ωi ≤ m!(Cn(n − 1))m/2 . i=1 i =1 3. Proof of Theorem 1.1 1
1
ˆ n = Hn I{sup Ln (x)≤Kn } . Let Kn = Mn n1− α bnα and H x∈Z ˆ ˆ n to We will prove Hn satisfies the moderate deviations given in Theorem 1.1. Then the moderate deviations pass from H Hn through the exponential equivalence given by lim sup n→∞
1 bn
1
ˆ n ̸= Hn } = lim log P{H
n→∞
bn
log P sup Ln (x) > Kn
= −∞
x
where the last step follows from Lemma 2.1. According to the Gärtner–Ellis theorem, we need only to prove that for any θ > 0,
lim
n→∞
1 bn
log E exp
1
±θ
bn2
− 21α 1
n1− 2α
ˆn = α − 1 H 2α
θ 2α
α−1 1
Cα
.
(3.1)
Notice that
Hn =
n 1
2 x∈Z
2 ωi I{Si =x}
−
n
i=1
ωi2 I{Si =x}
.
i=1
For each x ∈ Z , write D(x) = {1 ≤ k ≤ n; Sk = x}. It is easy to see that D(x) random walk {Sk }, the variables
n
2 ωi I{Si =x}
−
n
i=1
ωi2 I{Si =x} ,
x∈Z
i =1
form an independent family and for each fixed x ∈ Z ,
n
2 ωi I{Si =x}
−
n
i =1
Ln (x)
2
ω
d 2 i I{Si =x}
=
i =1
i=1
ωi
Ln (x)
−
ωi2 .
i =1
For convenience, we write 1
λ(n, θ , α, x) = θ
bn2
− 21α 1
n1 − 2 α ω
I{sup Ln (x)≤Kn } . x
We adopt the notation E for the expectation with respect to {ωk }k≥1 .
D(y) = Φ if x ̸= y. So conditioned on the
Y. Wang / Statistics and Probability Letters 83 (2013) 1078–1082 1− 1 bn2 2α 1− 1 n 2α
Notice that
1081
Kn = o(1), by Taylor’s expansion and Lemma 2.3, we have
b 12 − 21α b 12 − 21α n ˆ n = E Eω exp ±θ n ˆn E exp ±θ H H 1 1− 21α n n1− 2α 2 L L n (x) n (x) ωi − ωi2 = E Eω exp ±λ(n, θ , α, x) x∈Z i =1 i=1 2 L L n (x) n (x) ω 2 = E E exp ±λ(n, θ , α, x) ωi − ωi x∈Z i =1 i =1 1 2 =E 1 + λ (n, θ , α, x)Ln (x)(Ln (x) − 1)(1 + o(1)) 2
x∈Z
= E exp
log 1 +
x∈Z
= E exp
1 x∈Z
Because
2
1 2
λ2 (n, θ , α, x)Ln (x)(Ln (x) − 1)(1 + o(1))
λ (n, θ , α, x)Ln (x)(Ln (x) − 1)(1 + o(1)) . 2
Ln (x) = n, it is easy to see
x∈Z
b 12 − 21α 1 1 2 n 2 ˆ Ln (x) log E exp ±θ Hn = lim log E exp λ (n, θ , α, x) lim 1 n→∞ bn n→∞ bn 2 n1− 2α x∈Z 1 2 1− α 1 θ bn = lim L2n (x)I{sup Ln (x)≤Kn } . log E exp 1 n→∞ bn 2 n2− α x∈Z x 1
Write Lˆ n = lim sup n→∞
x∈Z
1 bn
L2n (x)I{supx Ln (x)≤Kn } and Ln =
log P{Lˆ n ̸= Ln } = lim
n→∞
1 bn
x∈Z
L2n (x). From (1.3) and the exponential equivalence given by
log P{sup Ln (x) > Kn } = −∞, x
we have lim
n→∞
1 bn
log P
x∈Z
L2n
(x)I{sup Ln (x)≤Kn } ≥ λn
1 2− α
1
bnα
x
= −λα
σα 2α
2α − 1 2α Mα,2
2α−1 =−
λα Cα . 2α
(3.2)
2 x∈Z Ln (x) ≤ n supx Ln (x) and Lemma 2.1, we have 1 2 1− α 1 θ bn lim log E exp L2n (x)I{sup Ln (x)≤Kn } < ∞ 1 n→∞ bn 2 n2− α x∈Z x
Using the fact that
According to the Varadhan’s integral lemma, we have
2 1 θ 2 b1− α1 θ λα Cα α − 1 θ 2α α−1 n 2 lim log E exp L (x)I = sup λ− = . Ln (x)≤Kn } n→∞ bn 2 n2− α1 x∈Z n {sup x 2 2α 2α Cα λ 1
The proof is complete. Acknowledgments The author is grateful to an anonymous referee for useful comments and remarks. The work has been partially supported by National Natural Science Foundation of China, Tian Yuan Foundation (No. 11226212) and the Foundation of Ministry of Education of China (No. 12YJCGJW015).
1082
Y. Wang / Statistics and Probability Letters 83 (2013) 1078–1082
References Asselah, A., 2010. Annealed lower tails for the energy of a charged polymer. J. Stat. Phys. 138, 619–644. Asselah, A., 2011. Annealed upper tails for the energy of a charged polymer. Ann. Inst. H. Poincaré Probab. Statist. 47, 80–110. Chen, X., 2008. Limit laws for the energy of a charged polymer. Ann. Inst. H. Poincaré Probab. Statist. 44, 638–672. Chen, X., Khoshnevisan, D., 2009. From charged polymer to random walk in random scenery. In: Optimality: The 3rd E.L. Lehmann Symposium. In: IMS Lecture Notes, vol. 57. pp. 237–251. Chen, X., Li, W., Rosen, J., 2005. Large deviations for local times of stable processes and stable random walks in 1 dimension. Electron. J. Probab. 16, 577–608. Hu, Y., 2010. Strong approximations in a charged-polymer model. Period. Math. Hungar. 61, 213–224. Hu, Y., Khoshnevisan, D., Wouts, M., 2011. Charged polymers in the attractive regime: a first transition from Brownian scaling to four-point localization. J. Stat. Phys. 144, 948–977. Kantor, Y., Kardar, M., 1991. Polymers with random self-interactions. Europhys. Lett. 14, 421–426. Li, Y., 2012. Moderate deviations for stable random walks in random scenery. J. Appl. Probab. 49, 280–294.