Directed Polymers in Random Environment: Path Localization and Strong Disorder

Francis COMETS

Université Paris 7, Mathématiques, Case 7012 2 place Jussieu, 75251 Paris, France email: comets@math.jussieu.fr

Tokuzo SHIGA

Tokyo Institute of Technology Oh-okayama, Meguroku, Tokyo 152-8551 Japan email tshiga@math.titech.ac.jp

Nobuo YOSHIDA

Division of Mathematics Graduate School of Science Kyoto University, Kyoto 606-8502, Japan. email: nobuo@math.kyoto-u.ac.jp

July 24, 2002

Abstract

We consider directed polymers in random environment. Under some mild assumptions on the environment, we prove here: (i) equivalence between the decay rate of the partition function and some natural localization properties of the path, (ii) some quantitative estimates of the decay of the partition function in dimensions one or two, or at sufficiently low temperature, (iii) the existence of quenched free energy. In particular, we generalize to general environments, the results recently obtained by P. Carmona and Y. Hu for a Gaussian environment. Our approach is based on martingale decomposition and martingale analysis. It leads to a natural, asymptotic relation between the partition function, and the probability that two polymers in the same environment, but independent otherwise, end up at the same point.

Short Title. Directed Polymers in Random Environment Key words and phrases. Directed polymers, martingales, random environment AMS 1991 subject classifications. Primary 60K35; secondary 60G42, 82A51, 82D30

1 Introduction and Main Results

1.1 Directed Polymers in Random Environment

The models we consider in this paper are defined in terms of a random walk and of a random environment, which we introduce now:

• The random walk: ({S_{n}}_{n≥0}, {P^{x}}_{x∈ℤ^{d}}) is a simple random walk on the d-dimensional integer lattice ℤ^{d}. More precisely, let Ω be the path space Ω = {ω = (ω_{n})_{n≥0}; ω_{n} ∈ ℤ^{d}, n ≥ 0}, let F be the cylindric σ-field on Ω, and, for all n ≥ 0, S_{n} : ω ↦ ω_{n} the projection map. For any x ∈ ℤ^{d} we consider the unique probability measure P^{x} on (Ω, F) such that S_{1} − S_{0}, …, S_{n} − S_{n−1} are independent and

P^{x}{S_{0} = x} = 1, P^{x}{S_{n}−S_{n−1} = ±δ_{j}} = (2d)^{−1}, j = 1, 2, …, d,

where δ_{j} = (δ_{kj})_{k=1}^{d} is the j-th vector of the canonical basis of ℤ^{d}. For x = 0 we will write simply P by P^{0}.

• The random environment: ξ = {ξ(x, n) : x ∈ ℤ^{d}, n ≥ 1} is a sequence of random variables which are real valued, non-constant, and i.i.d. defined on a probability space (Ξ, E, Q) such that

Q[exp(βξ(x, n))] < ∞ for all β ∈ ℝ.  (1.1)

(Throughout, Q[Y] denotes the Q-expectation of a r.v. Y.) Let λ(β) be the logarithmic moment generating function of ξ(x, n),

λ(β) = ln Q[exp(βξ(x, n))], β ∈ ℝ.  (1.2)

For any n > 0, define the probability measure μ_{n} on the path space (Ω, F)

μ_{n}(dω) = P [e_{n}]^{−1}e_{n} P (dω),  (1.3)

where

e_{n} = e_{n}(ξ, S) = exp ( ∑_{1≤j≤n}(βξ(S_{j}, j) − λ(β)))  (1.4)

with a parameter β ∈ ℝ. Here, the graph {(S_{j}, j)}_{j≥0} may be interpreted as a polymer chain living in the (d + 1)-dimensional space, constrained to stretch in the (d + 1)-th direction, and governed by the Hamiltonian

−β ∑_{j≥1} ξ(S_{j}, j) ,

i.e. the so-called directed polymer in the environment ξ. Note that the term λ(β), from the exponent in (1.4), cancels out in definition (1.3) of μ_{n}. The reason why to add it in (1.4), is to normalize P [e_{n}], which has now expectation equal to 1. If β > 0, then the parameter β > 0 plays the role of the inverse temperature in this interpretation. Since this Hamiltonian is parametrized by ξ, the polymer measure μ_{n} is random. The polymer is attracted to sites where the random environment is large and positive, and repelled by sites where the environment is large and negative. Here are two standard choices for ξ.

Example 1.1 Gaussian environment (Carmona and Hu, 2001) This is the case in which ξ(x, n) is a standard normal random variable, so that λ(β) = \frac{1}{2}β^{2}.

Example 1.2 Bernoulli environment (Bolthausen 1989, Imbrie and Spencer 1988, Song and Zhou 1996): This is the case in which ξ(x, n) takes two different values a and b with probability p > 0 and 1 − p > 0, respectively, so that λ(β) = ln(pe^{βa} + (1 − p)e^{βb}). As discussed by Johansson (2000, Remark 1.8), directed percolation can be understood as the case of 0 = a > b and zero-temperature (β → ∞), which however is outside the scope of this paper.

We are interested in the large time behavior of the path {S_{k}}_{k=1}^{n} under the (sequence of) polymer measures μ_{n}. As is the case in many other models in statistical mechanics, one of the fundamental questions is the asymptotic behavior of the partition function

Z_{n} = Z_{n}(ξ) = P [e_{n}] .  (1.5)

Since Z_{n} is a positive martingale on (Ξ, E, Q), the following limit exists Q-a.s.:

Z_{∞} =^{def.} lim_{n↗∞} Z_{n} .  (1.6)

The event {Z_{∞} = 0} is measurable with respect to the tail σ-field

⋂_{n≥1} σ[ξ(x, j) ; j ≥ n, x ∈ ℤ^{d}]

and therefore by Kolmogorov’s 0-1 law

Q{Z_{∞} = 0} = 0 or 1.  (1.7)

We refer to the former case as weak disorder and the latter as strong disorder. It is known (e.g., Song and Zhou 1996) that for d ≥ 3,

Q{Z_{∞} = 0} = 0 if γ_{1}(β) =^{def.} λ(2β) − 2λ(β) < − ln(1 − q)  (1.8)

where q = P {S_{n} ≠ 0 for all n ≥ 1}; similar results for weak disorder were obtained by Bolthausen (1989) and Sinai (1995). Note that γ_{1}(β) is decreasing on (−∞, 0], increasing on [0, ∞) and γ_{1}(0) = 0 so that the condition in (1.8) does hold if |β| is small. In dimension d ≥ 3, this condition amounts to L^{2}-convergence in (1.6), and it allows using the so-called second moment method: for small β and d ≥ 3, Imbrie and Spencer (1988) first, then Bolthausen (1989) with martingales techniques, proved that the polymer is diffusive, i.e., μ_{n}[S_{n}^{2}] ∼ n as n ↗ ∞; more recently Albeverio and Zhou (1996) showed that the invariance principle holds for almost every environment. On the other hand, for the strong disorder, it can be seen that

Q{Z_{∞} = 0} = 1 if γ_{2}(β) =^{def.} βλ′(β) − λ(β) ≥ ln(2d).  (1.9)

This was shown by Kahane and Peyrière (1976) for a different model called Mandelbrot martingale (or, equivalently, multiplicative chaos), where graphs {(S_{j}, j)}_{j≥0} are replaced by infinite paths, without loops and starting from the root, on the d-ary tree. Although the directed polymer we are considering here is more intricate due to correlations, the same argument applies as far as to deduce (1.9). Note that γ_{2}(β) is decreasing on (−∞, 0], increasing on [0, ∞) and γ_{2}(0) = 0 so that the condition in (1.9) roughly says that |β| is large enough. Recently, P. Carmona and Y. Hu (2001) proved for Gaussian environment that for all β ≠ 0,

Q{Z_{∞} = 0} = 1, d = 1, 2,  (1.10)

which, together with (1.8) and (1.9), displays a non-trivial dependence on the dimension.

In the present paper, we consider general environments and present some results mainly for the strong disorder case: Q{Z_{∞} = 0} = 1, including the extension of (1.10) to non-Gaussian case. Using martingale analysis, we also obtain natural localization properties which characterize the strong disorder regime. More precisely, the decay of the partition function is equivalent to concentration of the path on favourite sites. We present the proofs in a selfcontained way, except for that of Proposition 1.4(b).

Among other interesting subjects related to the directed polymer are superdiffusivity and critical exponents. Although we will not discuss about them here, we refer to Johansson (2000), Licea et al (1996), Petermann (2000) and Piza (1997) for rigorous results in this direction.

1.2 Results

On the product space (Ω^{2}, F^{⊗2}), we consider the probability measure μ_{n}^{⊗2} = μ_{n}^{⊗2}(dω, dω̃), that we will view as the distribution of the couple (S, S̃) with S̃ = {S̃_{k}}_{k≥0} an independent copy of S = {S_{k}}_{k≥0} with law μ_{n}. An important role in the analysis is played by the random sequence

I_{n} = μ_{n−1}^{⊗2}(S_{n} = S̃_{n}) ,  (1.11)

which conveys some information on the localization of paths under μ_{n}, see (1.18) below. Roughly, large values of I_{n} ∈ (0, 1] indicate that the polymer concentrates, at time n, on a few significant sites, though small values indicate that it spreads out on a large number of sites. Our basic result relates the partition function Z_{n} and the expected intersection time ∑_{j≤n} I_{j} of two independent polymers in the same environment.

Theorem 1.1 Let β ≠ 0. Then,

{Z_{∞} = 0} = {∑_{n≥1} I_{n} = ∞} , Q-a.s.  (1.12)

Moreover, if Q{Z_{∞} = 0} = 1, then there exist c_{1}, c_{2} ∈ (0, ∞) such that

−c_{1} ln Z_{n} ≤ ∑_{1≤j≤n} I_{j} ≤ −c_{2} ln Z_{n}  for large enough n’s, Q-a.s.  (1.13)

We make a brief comment on the result. On the one hand, we recall the definition – below (1.7) – of weak and strong disorder, which is natural in view of the high temperature behavior (in high dimension) (1.8) and the low temperature behavior (1.9). On the other hand, when the polymer strongly feels the environment, it is strongly attracted to sites with favourable environment and it follows from the definition (1.11) that I_{n} takes large values. Our result is a rigourous statement of equivalence of these two properties. The decay property of Z_{n} is reflected in some specific localization property of the path {S_{n}}_{n≥1} under the random measure (1.3). The proof of Theorem 1.1 is based on a general estimate for the summation of i.i.d.random variable (Lemma 2.1 below) and martingale analysis.

The most interesting case relative to the following, straightforward corollary is a_{n} = n, n ≥ 1.

Corollary 1.2 For β ≠ 0 and a sequence a_{n} ↗ ∞ of positive numbers, the following properties are equivalent:

(Z1) There exists c > 0 such that

Q \{ \underline{lim}_{n↗∞} −\frac{1}{a_{n}} ln Z_{n} ≥ c\} = 1.  (1.14)

(I1) There exists c > 0 such that

Q \{ \underline{lim}_{n↗∞} \frac{1}{a_{n}} ∑_{1≤j≤n} I_{j} ≥ c\} = 1.  (1.15)

Remark 1.1 The equivalence presented in Theorem 1.1 was shown first by Carmona and Hu (2001, Theorem 1.1 and Proposition 5.1) in the Gaussian case.

Some sufficient conditions for (Z1) and (I1) in Corollary 1.2 are provided by the following result.

Theorem 1.3 (a) (Z1) in Corollary 1.2 holds for a_{n} = n if γ_{2}(β) > ln(2d) (cf. (1.9)).

(b) If β ≠ 0, then for Q-a.s.,

Z_{n} \{ \begin{matrix}= O (exp(−c_{1}n^{1/3})) , & as n ↗ ∞ if d = 1 \\ ⟶ 0, & as n ↗ ∞ if d = 2, \end{matrix}

where c_{1} is a positive constant.

The proof of Theorem 1.3 is carried out by estimating fractional moment: Q[Z_{n}^{θ}], 0 < θ < 1; see Lemma 3.1 below. Beside the quantitative bound for the rate of decay for d = 1 presented above, we also give quantitative bound for the fractional moment for d = 1, 2 in the course of the proof.

Remark 1.2 Theorem 1.3(b) generalizes Theorem 1.1 in Carmona and Hu (2001) to nongaussian environments. Moreover, the proof in this paper shed more light on the decay rate.

We now go on to discuss sufficient conditions for another localization property of the polymer chain, described in terms of I_{n};

Proposition 1.4 Consider the following property:

(I2) There is a constant c ∈ (0, ∞) such that

\overline{lim}_{n↗∞} I_{n} ≥ c, Q-a.s.  (1.16)

Then,

(a) (I2) holds true if Property (Z1) in Corollary 1.2 holds with a_{n} = n, in particular if γ_{2}(β) > ln(2d) (cf. (1.9) and Theorem 1.3(a)).

(b) (I2) holds true if d = 1, 2.

(c) If Q{Z_{∞} > 0} = 1, then, in contrast with (I2),

lim_{n↗∞} I_{n} = 0, Q-a.s.

Assume moreover that γ_{1}(β) < − ln(1 − q) (cf.(1.8)). Then, there is a constant c > 0 such that

I_{n} = O(n^{−c}) in Q-probability.  (1.17)

A natural quantity of interest here, related to localization phenomenon, is the favorite site for the path at time n. First observe that

max_{x∈ℤ^{d}}μ_{n−1}(S_{n} = x)^{2} ≤ I_{n} ≤ max_{x∈ℤ^{d}}μ_{n−1}(S_{n} = x) .  (1.18)

Therefore, all statements we obtained for I_{n} can be translated into those for max_{x∈ℤ^{d}} μ_{n−1}(S_{n} = x). In particular, we showed in Proposition 1.4 that the probability of the favorite site vanishes for weak disorder, but does not vanish for strong disorder. In the latter case the polymer localizes (in a set of lattice points depending on the evironment), though in the former it spreads out somewhat similarly to the usual simple random walk.

Remark 1.3 Proposition 1.4(b) generalizes Theorem 1.2 in Carmona and Hu (2001) to non-gaussian environments. To prove this, and only there, we refer the readers to some of the arguments in Carmona and Hu (2001).

Finally, we remark that the “quenched free energy”

lim_{n↗∞} \frac{1}{n} ln Z_{n}

exists Q-a.s. under our assumption (1.1).

Proposition 1.5 The limit

ψ(β) = lim_{n↗∞} \frac{1}{n}Q[ln Z_{n}] ∈ (−∞, 0]

exists. Moreover, for any ε > 0, there is an n_{0} = n_{0}(β, ε) < ∞ such that

Q \{|\frac{1}{n} ln Z_{n} − Q [\frac{1}{n} ln Z_{n}]| > ε\} ≤ exp (−\frac{ε^{2/3}n^{1/3}}{4} ) , n ≥ n_{0}.  (1.19)

As a consequence,

lim_{n↗∞} \frac{1}{n} ln Z_{n} = ψ(β), Q-a.s.

Remark 1.4 The inequality (1.19) is a concentration inequality with the stretched exponential decay rate. An inspection of our proof reveals that an exponential concentration result can be obtained by a slightly stronger assumption. In fact, if we assume that there is δ > 0 such that

Q [exp(δ|ξ(x, n)|^{2})] < ∞,  (1.20)

then, we obtain the following; for any ε > 0, there is an n_{0} = n_{0}(β, ε) < ∞ such that

Q \{|\frac{1}{n} ln Z_{n} − Q [\frac{1}{n} ln Z_{n}]| > ε\} ≤ exp (−\frac{ε^{2}n}{c}) , n ≥ n_{0}.  (1.21)

where c = c(β) > 0. See Remark 5.1 below for the proof. Note also that (1.20) is true if ξ(x, n) is a Gaussian or Bernoulli r.v. as in Example 1.1 or Example 1.2.

Remark 1.5 We can define a similar model by considering a Markov chain ({S_{n}}_{n≥0}, {P^{x}}_{x∈Γ}) on a certain state space Γ instead of the random walk on ℤ^{d}. The proofs presented in this paper apply without change to this generalization.

2 Proof of Theorem 1.1

We first state some technical estimates.

Lemma 2.1 Let η_{i}, 1 ≤ i ≤ m be positive, non-constant i.i.d. random variables on a probability space (Ξ, E, Q) such that

Q[η_{1}] = 1, Q[η_{1}^{3} + ln^{2} η_{1}] < ∞.

For {α_{i}}_{1≤i≤m} ⊂ [0, ∞) such that ∑_{1≤i≤m} α_{i} = 1, define a centered random variable U > −1 by U = ∑_{1≤i≤m} α_{i}η_{i} − 1. Then, there exists a constant c ∈ (0, ∞), independent of {α_{i}}_{1≤i≤m} such that

\frac{1}{c} ∑_{1≤i≤m} α_{i}^{2} ≤ Q [\frac{U^{2}}{2 + U} ] ,  (2.1)

\frac{1}{c} ∑_{1≤i≤m} α_{i}^{2} ≤ −Q [ln(1 + U)] ≤ c ∑_{1≤i≤m} α_{i}^{2},  (2.2)
Q [ln^{2}(1 + U)] ≤ c ∑_{1≤i≤m} α_{i}^{2}.  (2.3)

Remark 2.1 These estimates are proved in Carmona and Hu (2001) for the Gaussian case with the help of Brownian motion and making use of Itô’s formula. Here, we present a simple argument which works in the general case.

We postpone the proof of Lemma 2.1 to the end of the section, and, assuming the lemma, we start to prove Theorem 1.1. To conclude (1.12) and (1.13), it is enough to show the following (2.4) and (2.5):

{Z_{∞} = 0} ⊂ {∑_{n≥1} I_{n} = ∞} , Q-a.s.  (2.4)

There are c_{1}, c_{2} ∈ (0, ∞) such that

{∑_{n≥1} I_{n} = ∞} ⊂ {−c_{1} ln Z_{n} ≤ ∑_{1≤j≤n} I_{j} ≤ −c_{2} ln Z_{n} for large enough n’s.} , Q-a.s. (2.5)

The proof of (2.4) and (2.5) is based on Doob’s decomposition for the process − ln Z_{n}. It is convenient to introduce some more notations. For a sequence (a_{n})_{n≥0} (random or nonrandom), we set Δa_{n} = a_{n} − a_{n−1} for n ≥ 1. We denote by E_{n} the σ-field generated by {ξ(x, j) ; 1 ≤ j ≤ n, x ∈ ℤ^{d}}, and we denote by Q_{n}^{ξ} the conditional expectation with respect to Q given E_{n}. Let us now recall Doob’s decomposition in this context; any (E_{n})-adapted process X = {X_{n}}_{n≥0} ⊂ L^{1}(Q) can be decomposed in a unique way as

X_{n} = M_{n}(X) + A_{n}(X), n ≥ 1,

where M(X) is an (E_{n})-martingale and

A_{0} = 0, ΔA_{n} = Q_{n−1}^{ξ}[ΔX_{n}], n ≥ 1.

M_{n}(X) and A_{n}(X) are called respectively, the martingale part and compensator of the process X. If X is a square integrable martingale, then the compensator A_{n}(X^{2}) of the process X^{2} = {(X_{n})^{2}}_{n≥0} ⊂ L^{1}(Q) is denoted by ⟨ X ⟩_{n} and is given by the following formula;

Δ⟨ X ⟩_{n} = Q_{n−1}^{ξ}[(ΔX_{n})^{2}]

Here, we are interested in the Doob’s decomposition of X_{n} = − ln Z_{n}, whose martingale part and the compensator will be henceforth denoted M_{n} and A_{n} respectively;

− ln Z_{n} = M_{n} + A_{n}.  (2.6)

To compute M_{n} and A_{n}, we introduce

U_{n} = μ_{n−1}[exp(βξ(S_{n}, n)−λ(β))] − 1 .

It is then clear that

Z_{n}/Z_{n−1} = 1 + U_{n}  (2.7)

and hence that

ΔA_{n} = −Q_{n−1}^{ξ} ln(1 + U_{n}), ΔM_{n} = − ln(1 + U_{n}) + Q_{n−1}^{ξ} ln(1 + U_{n}).  (2.8)

In particular,

Δ⟨ M ⟩_{n} ≤ Q_{n−1}^{ξ} ln^{2}(1 + U_{n}).  (2.9)

On the other hand, we have that

I_{n} = ∑_{|z|_{1}≤n} μ_{n−1}(S_{n} = z)^{2}.

We now claim that there is a constant c ∈ (0, ∞) such that

\frac{1}{c}I_{n} ≤ ΔA_{n} ≤ cI_{n},  (2.10)
Δ⟨ M ⟩_{n} ≤ cI_{n}.  (2.11)

Indeed, both follow from (2.8), (2.9) and Lemma 2.1; {η_{i}}, {α_{i}} and Q in the lemma play the roles of {exp(βξ(z, n) − λ(β))}_{|z|_{1}≤n}, {μ_{n−1}(S_{n} = z)}_{|z|_{1}≤n} and Q_{n−1}^{ξ}.

We now conclude (2.4) from (2.10), (2.11) as follows (the equalities and the inclusions here being understood as Q-a.s.):

{∑_{n≥1} I_{n} < ∞} = {A_{∞} < ∞}
= {A_{∞} < ∞, ⟨ M ⟩_{∞} < ∞}
⊂ {A_{∞} < ∞, lim_{n↗∞} M_{n} exists and is finite}
⊂ {Z_{∞} > 0}.

Here, on the third line, we have used a well-known property for martingales, e.g. (4.9) page 255 in Durrett (1995) or Neveu (1975).

Finally we prove (2.5). By (2.10), it is enough to show that

\{A_{∞} = ∞\} ⊂ \{ lim_{n↗∞} −\frac{ln Z_{n}}{A_{n}} = 1\} , Q-a.s.  (2.12)

Thus, let us suppose that A_{∞} = ∞. If ⟨ M ⟩_{∞} < ∞, then again by (4.9) page 255 in Durrett (1995) or Neveu (1975). lim_{n↗∞} M_{n} exists and is finite and therefore (2.12) holds. If, on the

contrary, ⟨ M ⟩_{∞} = ∞, then

−\frac{ln Z_{n}}{A_{n}} = \frac{M_{n}}{⟨ M ⟩_{n}} \frac{⟨ M ⟩_{n}}{A_{n}} + 1 → 1 Q-a.s.

by (2.10), (2.11) and the law of large numbers for martingales, see (4.10) page 255 in Durrett (1995) or Neveu (1975). This completes the proof of Theorem 1.1. □

Proof of Lemma 2.1: In this proof, we let c_{1}, c_{2}, … stand for constants which are independent of {α_{i}}_{1≤i≤m}. We have by direct computations that

Q[U^{2}] = c_{1} ∑_{1≤i≤m} α_{i}^{2}, Q[U^{3}] ≤ c_{2} ∑_{1≤i≤m} α_{i}^{2}.

Then, (2.1) is obtained as follows;

c_{1} ∑_{1≤i≤m} α_{i}^{2} = Q [\frac{U}{\sqrt{2 + U}} U\sqrt{2 + U}]

≤ Q [\frac{U^{2}}{2 + U} ]^{1/2} Q [2U^{2} + U^{3}]^{1/2}

≤ c_{3}Q [\frac{U^{2}}{2 + U} ]^{1/2} ( ∑_{1≤i≤m} α_{i}^{2})^{1/2} .

To prove the other inequalities, it is convenient to define a function φ : (−1, ∞) → [0, ∞) by φ(u) = u − ln(1 + u), so that

−Q [ln(1 + U)] = Q [φ(U)] .

Since \frac{1}{4} \frac{u^{2}}{2+u} ≤ φ(u), u > −1, the left-hand-side inequality of (2.2) follows from (2.1). The right-hand-side inequality can be seen as follows. We have for any ε ∈ (0, 1),

Q [φ(U)] = Q[φ(U) : 1 + U ≥ ε] + Q[φ(U) : 1 + U ≤ ε]
≤ Q[φ(U) : 1 + U ≥ ε] − Q[ln(1 + U) : 1 + U ≤ ε].

Since φ(u) ≤ \frac{1}{2}(u/ε)^{2} if 1 + u ≥ ε,

Q[φ(U) : 1 + U ≥ ε] ≤ \frac{1}{2}ε^{−2}Q[U^{2}]
= \frac{1}{2}ε^{−2}c_{1} ∑_{1≤i≤m} α_{i}^{2}.  (2.13)

We now set γ = −Q[ln η_{1}] ≥ 0 and choose ε > 0 so small that ln(1/ε) − γ ≥ 1. We introduce another centered random variable V = ∑_{1≤i≤m} α_{i}(ln η_{i} + γ). We then see from Jensen’s inequality that

{1 + U ≤ ε} = {V − γ ≤ ln(1 + U) ≤ ln ε}
⊂ {− ln(1 + U) ≤ −V + γ} ∩ {1 ≤ −V }.

Hence we have

−Q[ln(1 + U) : 1 + U ≤ ε] ≤ Q[−V : 1 ≤ −V] + γQ{1 ≤ −V }
≤ (1 + γ)Q[V^{2}]
= c_{4} ∑_{1≤i≤m} α_{i}^{2}.

This, together with (2.13) proves the right-hand-side inequality of (2.2). The proof of (2.3) is similar. Indeed, since | ln(1 + u)| ≤ ε^{−1} ln(ε^{−1})|u| if ε ≤ 1 + u, we have that

Q[ln^{2}(1 + U) : ε ≤ 1 + U] ≤ ε^{−2} ln^{2}(ε^{−1})Q[U^{2}].

We see on the other hand that

{1 + U ≤ ε} = {V − γ ≤ ln(1 + U) ≤ ln ε}
⊂ {ln^{2}(1 + U) ≤ 2V^{2} + 2γ^{2}} ∩ {1 ≤ −V }.

Therefore, we obtain

Q[ln^{2}(1 + U) : 1 + U ≤ ε] ≤ 2Q[V^{2}] + 2γ^{2}Q{1 ≤ −V }
≤ c_{5} ∑_{1≤i≤m} α_{i}^{2}.

□

3 Proof of Theorem 1.3

3.1 A sufficient condition for (Z1) via fractional moment

Lemma 3.1 Suppose that there exist constants c ∈ (0, ∞), θ ∈ (0, 1) and a sequence a_{n} ↗ ∞ such that

Q[Z_{n}^{θ}] ≤ c exp(−a_{n}), n ≥ 1.  (3.1)

Then Q{Z_{∞} = 0} = 1. If moreover

∑_{n≥1} exp(−δa_{n}) < ∞ for some δ ∈ (0, 1),

then (Z1) in Corollary 1.2 holds true.

Proof: The first statement follows from Fatou’s lemma and the second from the Borel-Cantelli lemma. □

3.2 Proof of part (a)

We will check (3.1) with a_{n} = cn for some c > 0. Set η(x, j) = exp(βξ(x, j) − λ(β)) and

Z_{n,m}^{x} = P^{x} [exp ( ∑_{1≤j≤m}(βξ(S_{j}, j + n) − λ(β)))] , n, m ≥ 1.  (3.2)

For θ ∈ (0, 1), by the subadditive estimate (u + v)^{θ} ≤ u^{θ} + v^{θ}, u, v > 0, we get

Z_{n}^{θ} ≤ (2d)^{−θ} ∑_{x,|x|_{1}=1}(η(x, 1)Z_{1,n−1}^{x})^{θ}.

Since Z_{1,n−1}^{x} has the same law as Z_{n−1},

Q[Z_{n}^{θ}] ≤ r(θ)Q[Z_{n−1}^{θ}],

where r(θ) = (2d)^{1−θ}Q[η(x, 1)^{θ}]. Note that θ ↦ ln r(θ) is convex, continuously differentiable, and that ln(2d) = ln r(0) > ln r(1) = 0. Therefore r(θ) < 1 for some θ ∈ (0, 1) if and only if 0 < \frac{d ln r(θ)}{dθ} |_{θ=1}, which is equivalent to γ_{2}(β) > ln(2d). □

3.3 Proof of part (b)

We will check (3.1) with

a_{n} = \{ \begin{matrix}c_{1}n^{1/3} & if d = 1 \\ c_{2}\sqrt{ln n} & if d = 2 \end{matrix}  (3.3)

where c_{1}, c_{2} ∈ (0, ∞) are some constants. In this respect, we first prove an auxiliary lemma.

Lemma 3.2 For θ ∈ [0, 1] and Λ ⊂ ℤ^{d},

Q [Z_{n−1}^{θ}I_{n}] ≥ \frac{1}{|Λ|}Q [Z_{n−1}^{θ}] − \frac{2}{|Λ|}P (S_{n} ∉ Λ)^{θ}.  (3.4)

Proof: Repeating the argument in Liggett (1985, page 453), we see that

I_{n} ≥ ∑_{z∈Λ} μ_{n−1}(S_{n} = z)^{2}
≥ \frac{1}{|Λ|}μ_{n−1}(S_{n} ∈ Λ)^{2}
= \frac{1}{|Λ|} (1 − μ_{n−1}(S_{n} ∉ Λ))^{2}
≥ \frac{1}{|Λ|} (1 − 2μ_{n−1}(S_{n} ∉ Λ))
≥ \frac{1}{|Λ|} (1 − 2μ_{n−1}(S_{n} ∉ Λ)^{θ}) .

Note also that

Q [Z_{n−1}^{θ}μ_{n−1}(S_{n} ∉ Λ)^{θ}] ≤ Q [Z_{n−1}μ_{n−1}(S_{n} ∉ Λ)]^{θ}
= P (S_{n} ∉ Λ)^{θ}.

We therefore see that

Q [Z_{n−1}^{θ}I_{n}] ≥ \frac{1}{|Λ|}Q [Z_{n−1}^{θ}] − \frac{2}{|Λ|}Q [Z_{n−1}^{θ}μ_{n−1}(S_{n} ∉ Λ)^{θ}]
≥ \frac{1}{|Λ|}Q [Z_{n−1}^{θ}] − \frac{2}{|Λ|}P (S_{n} ∉ Λ)^{θ}.

□ Assume now that θ ∈ (0, 1), and define a function f : (−1, ∞) → [0, ∞) by

f(u) = 1 + θu − (1 + u)^{θ}.

It is then clear that there are constants c_{1}, c_{2} ∈ (0, ∞) such that

\frac{c_{1}u^{2}}{2 + u} ≤ f(u) ≤ c_{2}u^{2} for all u ∈ (−1, ∞).  (3.5)

We see from (2.7), (3.5) and (2.1) that

Q_{n−1}^{ξ}ΔZ_{n}^{θ} = Z_{n−1}^{θ}Q_{n−1}^{ξ} ((1 + U_{n})^{θ} − 1)
= −Z_{n−1}^{θ}Q_{n−1}^{ξ}f(U_{n})
≤ −c_{3}Z_{n−1}^{θ}I_{n}.

We therefore have by (3.4) that

QZ_{n}^{θ} ≤ (1 − \frac{c_{3}}{|Λ|}) Q [Z_{n−1}^{θ}] + \frac{2c_{3}}{|Λ|} P (S_{n} ∉ Λ)^{θ}.  (3.6)

For d = 1, set Λ = (−n^{2/3}, n^{2/3}]. Then,

P (S_{n} ∉ Λ) = P (|\frac{S_{n}}{n^{1/2}} | ≥ n^{1/6}) ≤ 2 exp(−\frac{n^{1/3}}{2} ),

so that (3.6) reads,

QZ_{n}^{θ} ≤ (1 − \frac{c_{3}}{2n^{2/3}} ) Q [Z_{n−1}^{θ}] + 4c_{3} exp(−\frac{n^{1/3}}{2} ).

It is not difficult to conclude (3.1) with a_{n} = c_{1}n^{1/3} from the above.

For d = 2, we set

Λ = (−n^{1/2} ln^{1/4} n, n^{1/2} ln^{1/4} n]^{2}

to get (3.1) with a_{n} = c_{2}\sqrt{ln n} in a similar way as above. □

4 Proof of Proposition 1.4

4.1 Proof of part (a) This follows directly from (1.15). □

4.2 Proof of part (b)

We now state the following lemma which corresponds to Lemma 2.2 in Carmona and Hu (2001).

Lemma 4.1 Let η_{i}, 1 ≤ i ≤ m be positive, non-constant i.i.d. random variables on a probability space (Ξ, E, Q) such that

m_{θ} =^{def} Q[η_{1}^{θ}] < ∞ for θ = ±4 and m_{1} = 1.

For {α_{i}}_{1≤i≤m} ⊂ [0, ∞)^{m} such that ∑_{1≤i≤m} α_{i} = 1, define a centered random variable U > −1 by U = ∑_{1≤i≤m} α_{i}η_{i} − 1. Then,

1 − 2(m_{2} − 1)(α_{1} + α_{2}) + \frac{1}{C} ∑_{1≤i≤m} α_{i}^{2} ≤ Q [\frac{η_{1}η_{2}}{(1 + U)^{2}} ] ≤ m_{2}\sqrt{m_{−4}},  (4.1)

m_{2} − 2(m_{3} − m_{2})α_{1} + \frac{1}{C} ∑_{1≤i≤m} α_{i}^{2} ≤ Q [\frac{η_{1}^{2}}{(1 + U)^{2}} ] ≤ \sqrt{m_{4}m_{−4}},  (4.2)

where C > 0 is a constant which depends only on m_{4}.

Proof: Since the proofs of (4.1) and (4.2) are similar, we present that of (4.1) only.

Q [η_{1}η_{2}(1 + U)^{−2}]^{2} ≤ m_{2}^{2}Q [(1 + U)^{−4}]
≤ m_{2}^{2}Q [ ∑_{1≤i≤m} α_{i}η_{i}^{−4}]
= m_{2}^{2}m_{−4},

where, on the second line, we have used the Jensen inequality for the measure {α_{i}}.

To prove the other inequalities, it is convenient to define a function φ : (−1, ∞) → [0, ∞) by φ(u) = (1 + u)^{−2} − 1 + 2u. By an elementary inequality: \frac{cu^{2}}{2+u} ≤ φ(u), u > −1, we have

Q [η_{1}η_{2}(1 + U)^{−2}] ≥ 1 − 2Q[η_{1}η_{2}U] + c_{1}Q [\frac{η_{1}η_{2}U^{2}}{2 + U} ] .  (4.3)

On the other hand, we have by direct computations that

Q[η_{1}η_{2}U] = (m_{2} − 1)(α_{1} + α_{2}),
\frac{1}{c} ∑_{1≤i≤m} α_{i}^{2} ≤ Q[η_{1}η_{2}U^{2}] ≤ c ∑_{1≤i≤m} α_{i}^{2},
Q[η_{1}η_{2}U^{3}] ≤ c ∑_{1≤i≤m} α_{i}^{2}.

Therefore,

\frac{1}{c} ∑_{1≤i≤m} α_{i}^{2} ≤ Q[η_{1}η_{2}U^{2}]

= Q [\frac{\sqrt{η_{1}η_{2}}U}{\sqrt{2 + U}} \sqrt{η_{1}η_{2}}U\sqrt{2 + U}]
≤ Q [\frac{η_{1}η_{2}U^{2}}{2 + U} ]^{1/2} Q [2η_{1}η_{2}U^{2} + η_{1}η_{2}U^{3}]^{1/2}

≤ cQ [\frac{η_{1}η_{2}U^{2}}{2 + U} ]^{1/2} ( ∑_{1≤i≤m} α_{i}^{2})^{1/2} .

Putting things together, we get (4.1) □

With Lemma 4.1 established, Proposition 1.4(b) is obtained merely by following the argument in Carmona and Hu (2001, Section 6). In doing so, we use Lemma 4.1 above in place of Lemma 2.2 in that paper. In fact, Carmona and Hu used the specific properties of the Gaussian random variable only in the proof of Lemma 2.2. □

4.3 Proof of part (c)

The first statement is derived using the convergence of I_{n} to 0. We now prove (1.17). Since Q{Z_{∞} > 0} = 1 in the present case -see (1.8)-,it is enough to show that

Z_{n−1}^{2}I_{n} = O(n^{−c})  (4.4)

in Q-probability. With γ = λ(2β) − 2λ(β) < − ln(1 − q), we compute

Q [Z_{n−1}^{2}I_{n}] = Q [P^{⊗2}(e_{n−1}(ξ, S)e_{n−1}(ξ, S̃) : S_{n} = S̃_{n})]
= P^{⊗2}(Q[e_{n−1}(ξ, S)e_{n−1}(ξ, S̃)] : S_{n} = S̃_{n})
= P^{⊗2} (exp {γ ∑_{j=1}^{n−1} 1_{S_{j}=S̃_{j}}} : S_{n} = S̃_{n})

≤ P^{⊗2} (exp {αγ ∑_{j=1}^{n−1} 1_{S_{j}=S̃_{j}}})^{1/α} P^{⊗2}(S_{n} = S̃_{n})^{1/α′} ,

using Hölder’s inequality with the conjugate exponents α, α′. Since ∑_{j≥1} 1_{S_{j}=S̃_{j}} is geometrically distributed with failure probability 1 − q ∈ (0, 1) with q as in (1.8), the first factor on the right-hand side is bounded for αγ < − ln(1 − q). The second factor is O(n^{−d/(2α′)}). From this we obtain (1.17) for arbitrary c < d[1 + γ/ ln(1 − q)]/2. □

5 Proof of Proposition 1.5

Though the first statement is well known, we give a proof here for completeness. Recall the notation Z_{n,m}^{x} introduced by (3.2) and note that for m, n ≥ 1,

Z_{n+m} = Z_{n} ∑_{x} μ_{n}{S_{n} = x}Z_{n,m}^{x} ,

Since Z_{n,m}^{x} has the same law as Z_{m}, we have by Jensen’s inequality that

ln Z_{n+m} ≥ ln Z_{n} + ∑_{x} μ_{n}{S_{n} = x} ln Z_{n,m}^{x} .

Recall also the notation E_{n} and Q_{n}^{ξ} introduced in the proof of Theorem 1.1. Taking expectation and using independence, we obtain

Q[ln Z_{n+m}] ≥ Q[ln Z_{n}] + Q[∑_{x} μ_{n}{S_{n} = x}Q_{n}^{ξ}[ln Z_{n,m}^{x}]]
= Q[ln Z_{n}] + Q[ln Z_{m}] ,

i.e., Q[ln Z_{n}] is super-additive. From the super-additive Lemma we see that

lim_{n↗∞} \frac{1}{n}Q[ln Z_{n}] = sup_{n} \frac{1}{n}Q[ln Z_{n}] = ψ(β).

In order to prove the second statement (1.19), we write ln Z_{n} − Q[ln Z_{n}] as a sum of (E_{j})_{1≤j≤n}- martingale differences,

ln Z_{n} − Q[ln Z_{n}] = ∑_{j=1}^{n} V_{n,j}

with V_{n,j} = Q_{j}^{ξ}[ln Z_{n}] − Q_{j−1}^{ξ}[ln Z_{n}]. Set

ê_{n,j} = exp ( ∑_{1≤k≤n,k≠j}(βξ(S_{k}, k) − λ(β))) , Ẑ_{n,j} = P [ê_{n,j}].

Clearly Q_{j}^{ξ}[ln Ẑ_{n,j}] = Q_{j−1}^{ξ}[ln Ẑ_{n,j}], and hence,

V_{n,j} = Q_{j}^{ξ} [ln \frac{Z_{n}}{Ẑ_{n,j}} ] − Q_{j−1}^{ξ} [ln \frac{Z_{n}}{Ẑ_{n,j}} ] .

By (2.2) in Lemma 2.1 with η_{·} = η(·, j) = exp(βξ(·, j) − λ(β)) and α_{·} = \frac{P [ê_{n,j}:S_{j}=·]}{Ẑ_{n,j}}, we see that

−Q_{j−1}^{ξ} [ln \frac{Z_{n}}{Ẑ_{n,j}} ] = −Q_{j−1}^{ξ} [Q [ln (∑_{x} α_{x}η(x, j))| E_{n,j}]]
∈ [0, c] ,

where E_{n,j} = σ[ξ(·, k) ; 1 ≤ k ≤ n, k ≠ j]; note that η(x, j) is independent of E_{n,j} and that α_{·} is E_{n,j}-measurable. Therefore, for θ ∈ ℝ, we have by Jensen’s inequality that

Q[exp θV_{n,j}| E_{n,j}] ≤ e^{cθ^{+}}Q [exp θQ_{j}^{ξ} [ln \frac{Z_{n}}{Ẑ_{n,j}} ]| E_{n,j}]

≤ e^{cθ^{+}}Q [(\frac{Z_{n}}{Ẑ_{n,j}} )^{θ}| E_{n,j}]

= e^{cθ^{+}}Q [(∑_{x} α_{x}η(x, j))^{θ}| E_{n,j}] ,

where θ^{+} = max{0, θ}. For θ ∈ (0, 1), the function x ↦ x^{θ} is concave on (0, ∞) and hence

Q[exp θV_{n,j}| E_{n,j}] ≤ e^{cθ}Q [∑_{x} α_{x}η(x, j)| E_{n,j}]^{θ} = e^{cθ}.

For θ ∉ (0, 1), the function x ↦ x^{θ} is convex on (0, ∞) and hence

Q[exp θV_{n,j}| E_{n,j}] ≤ e^{cθ^{+}} ∑_{x} α_{x}Q [η(x, j)^{θ}| E_{n,j}]
= e^{cθ^{+}} ∑_{x} α_{x}Q[η(x, 1)^{θ}]
= exp{cθ^{+} + λ(θβ) − θλ(β)},

Finally we conclude that

Q[exp |V_{n,j}|] ≤ Q[exp(V_{n,j}) + exp(−V_{n,j})] ≤ c_{1} < ∞ .

Therefore, the large deviation estimate for sum of martingale-differences of Lesigne and Volný (2001, Theorem 3.2) applies to our case, yielding (1.19). The final statement in Proposition 1.5 is now a simple consequence of the Borel-Cantelli lemma. □

Remark 5.1 Let us remark that the stronger assumption (1.20) implies the exponential concentration (1.21). In what follows, c_{i} = c_{i}(δ) (i = 1, 2) and c_{i} = c_{i}(β, δ) (i = 3, 4, …) are positive constants. Note first that (1.20) implies that λ(β) ≤ c_{1} + c_{2}β^{2} for all β ∈ ℝ and hence that

Q [exp(θV_{n,j})|E_{n,j}] ≤ exp (c_{3} + c_{4}θ^{2}) , for all θ ∈ ℝ,  (5.1)

by the computations in our proof of Proposition 1.5. By expanding the exponential and using the fact Q[V_{n,j}|E_{n,j}] = 0, we can improve (5.1) into the following stronger form;

Q [exp(θV_{n,j})|E_{n,j}] ≤ exp (c_{5}θ^{2}) , for all θ ∈ ℝ.  (5.2)

It is then not difficult to conclude (1.21) by (5.2) and a standard Gaussian estimate for a martingale.

Acknowledgements: We would like to thank Philippe Carmona and Yueyun Hu for sending us their preprint (2001) prior to its publication. FC thanks the Graduate School of Science of Kyoto University for its hospitality. FC is partially supported by CNRS, UMR 7599 “Probabilités et Modèles Aléatoires”. TS is partially supported by JSPS Grant-in-Aid for Scientific Research, Kiban(C) 13640103. NY is partially supported by JSPS Grant-in-Aid for Scientific Research, Shorei(A) 40240303.

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