Moderate deviations and laws of the iterated logarithm for the renormalized self-intersection

local times of planar random walks

Richard F. Bass ^{∗} Xia Chen ^{†} Jay Rosen^{‡}

March 27, 2006

Abstract

We study moderate deviations for the renormalized self-intersection local time of planar random walks. We also prove laws of the iterated logarithm for such local times.

1 Introduction

Let {S_{n}} be a symmetric random walk on ℤ^{2} with covariance matrix Γ. Let

B_{n} = ∑_{1≤j<k≤n} δ(S_{j}, S_{k}) (1.1)

where (1.2) δ(x, y) = \{ \begin{matrix}1 & if x = y \\ 0 & otherwise \end{matrix}

is the usual Kronecker delta. We refer to B_{n} as the self-intersection local time up to time n. We call B_{n} − 𝔼 B_{n} the renormalized self-intersection local time of the random walk up to time n.

∗Research partially supported by NSF grant DMS-0244737.

†Research partially supported by NSF grant DMS-0405188.

‡This research was supported, in part, by grants from the National Science Foundation and PSC-CUNY.

In [5] it was shown that B_{n} − 𝔼 B_{n}, appropriately scaled, converges to the renormalized self-intersection local time of planar Brownian motion. Renormalized self-intersection local time for Brownian motion was originally studied by Varadhan [18] for its role in quantum field theory. Renormalized self-intersection local time turns out to be the right tool for the solution of certain “classical” problems such as the asymptotic expansion of the area of the Wiener sausage in the plane and the range of random walks, [4], [14], [13].

One of the applications of self-intersection local time is to polymer growth. If S_{n} is a planar random walk and ℙ is its law, one can construct self-repelling and self-attracting random walks by defining

dℚ_{n}/dℙ = c_{n}e^{ζB_{n}/n},

where ζ is a parameter and c_{n} is chosen to make ℚ_{n} a probability measure. When ζ < 0, more weight is given to those paths with a small number of selfintersections, hence ℚ_{n} is a model for a self-repelling random walk. When ζ > 0, more weight is given to paths with a large number of self-intersections, leading to a self-attracting random walk. Since 𝔼 B_{n} is deterministic, by modifying c_{n}, we can write

dℚ_{n}/dℙ = c_{n}e^{ζ(B_{n}−𝔼B_{n})/n}.

It is known that for small positive ζ the self-attracting random walk grows with n while for large ζ it “collapses,” and its diameter remains bounded in mean square. It has been an open problem to determine the critical value of ζ at which the phase transition takes place. The work [2] suggested that the critical value ζ_{c} could be expressed in terms of the best constant of a certain Gagliardo-Nirenberg inequality, but that work was for planar Brownian motion, not for random walks. In the current paper we obtain moderate deviations estimates for B_{n}−𝔼 B_{n} and these are in terms of the best constant of the Gagliardo-Nirenberg inequality; see Theorem 1.1. However the critical constant ζ_{c} is different (see Remark 1.4) and it is still an open problem to determine it. See [6] and [7] for details and further information on these models.

In the present paper we study moderate deviations of B_{n}−𝔼 B_{n}. Before stating our main theorem we recall one of the Gagliardo-Nirenberg inequalities:

∥f∥_{4} ≤ C∥∇f∥_{2}^{1/2}∥f∥_{2}^{1/2},

which is valid for f ∈ C^{1} with compact support, and can then be extended to more general f’s. We define κ(2, 2) to be the infimum of those values of C for which the above inequality holds. In particular, 0 < κ(2, 2) < ∞. For further details, see [8].

In this paper we will always assume that the smallest group which supports {S_{n}} is ℤ^{2}. For simplicity we assume further that our random walk is strongly aperiodic.

Theorem 1.1 Let {b_{n}} be a positive sequence satisfying

lim_{n→∞} b_{n} = ∞ and b_{n} = o(n). (1.3)

For any λ > 0,

(1.4)  lim_{n→∞} \frac{1}{b_{n}} log ℙ\{B_{n} − 𝔼 B_{n} ≥ λnb_{n}\} = −λ\sqrt{det Γ} κ(2, 2)^{−4}.

We call Theorem 1.1 a moderate deviations theorem rather than a large deviations result for two reasons: The first reason is the second restriction in (1.3). Our techniques do not apply when this restriction is not present, and in fact it is not hard to show that the value on the right hand side of (1.4) should be different when b_{n} ≈ n; see Remark 1.4. The second reason is that Theorem 1.1 is closely related to the following weak law (see Le Gall [13] and Rosen [16])

\frac{1}{n}(B_{n} − 𝔼 B_{n}) ⟶^{d} (det Γ)^{−1/2}γ_{1}

where γ_{t} is the renormalized self-intersection local time of a planar Brownian motion W_{t}; this can be formally written as

γ_{t} = ∫ ∫_{0≤r<s≤t} δ_{0}(W_{r} − W_{s})dr ds − 𝔼 ∫ ∫_{0≤r<s≤t} δ_{0}(W_{r} − W_{s})dr ds.

Recently, it has been proved by the authors ([2] and [3]) that

lim_{t→∞} \frac{1}{t} log ℙ\{γ_{1} ≥ t\} = −κ(2, 2)^{−4}

lim_{t→∞} t^{−2π} log ℙ{−γ_{1} ≥ log t} = −L

where 0 < L < ∞. Theorem 1.1 could perhaps be regarded as an invariance principle linking the moderate deviations of B_{n}−𝔼 B_{n} to the large deviations for γ_{1}.

Moderate deviations linked to the large deviations for −γ_{1} are more subtle. In the next theorem we obtain the correct rate, but not the precise constant.

Theorem 1.2 Suppose 𝔼 |S_{1}|^{2+δ} < ∞ for some δ > 0. There exist C_{1}, C_{2} > 0 such that if b_{n} → ∞ with b_{n} = o(n^{1/θ}) for some θ > 0, then

−C_{1} ≤ lim inf_{n→∞} b_{n}^{−θ} log ℙ{𝔼 B_{n} − B_{n} ≥ θ(2π)^{−1} det(Γ)^{−1/2}n log b_{n}}
≤ lim sup_{n→∞} b_{n}^{−θ} log ℙ{𝔼 B_{n} − B_{n} ≥ θ(2π)^{−1} det(Γ)^{−1/2}n log b_{n}}
(1.5)  ≤ −C_{2}.

Here are the corresponding laws of the iterated logarithm for B_{n}−𝔼 B_{n}.

Theorem 1.3

(1.6)  lim sup_{n→∞} \frac{B_{n} − 𝔼 B_{n}}{n log log n} = det(Γ)^{−1/2}κ(2, 2)^{4} a.s.

and if 𝔼 |S_{1}|^{2+δ} < ∞ for some δ > 0,

(1.7)  lim inf_{n→∞} \frac{B_{n} − 𝔼 B_{n}}{n log log log n} = −(2π)^{−1} det(Γ)^{−1/2} a.s.

In this paper we deal exclusively with the case where the dimension d is 2. We note that in dimension 1 no renormalization is needed, which makes the results much simpler. See [15, 9]. When d ≥ 3, the renormalized intersection local time is in the domain of attraction of a centered normal random variable. Consequently the tails of the weak limit are expected to be of Gaussian type, and in particular, the tails are symmetric; see [13].

Theorems 1.1-1.3 are the analogues of the theorems proved in [2] for the renormalized self-intersection local time of planar Brownian motion. Although the proofs for the random walk case have some elements in common with those for Brownian motion, the random walk case is considerably more difficult. The major difficulty is the fact that we do not have Gaussian random variables. Consequently, the argument for the lower bound of Theorem 1.1 needs to be very different from the one given in [2, Lemma 3.4]. This requires several new tools, such as Theorem 4.1, which we expect will have applications beyond the specific needs of this paper.

Remark 1.4 We remark that without the restriction that b_{n} = o(n), Theorem 1.1 is not true. To see this, let N be an arbitrarily large integer, let ε = 2/N^{2}, and let X_{i} be be an i.i.d. sequence of random vectors in ℤ^{2} that take the values (N, 0), (−N, 0), (0, N), and (0, −N) with probability ε/4 and ℙ(X_{1} = (0, 0)) = 1 − ε. The covariance matrix of the X_{i} will be the identity. Let b_{n} = (1 − ε)n. Then the event that S_{i} = S_{0} for all i ≤ n will have probability at least (1 − ε)^{n}, and on this event B_{n} = n(n − 1)/2. This shows that

log ℙ(B_{n} − 𝔼 B_{n} > nb_{n}/2) ≥ n log(1 − ε),

which would contradict (1.4).

The same example shows that the critical constant in the polymer model is different than the one in [2]. Then

𝔼 exp \{C \frac{B_{n} − 𝔼 B_{n}}{n} \} ≥ exp \{ − C \frac{𝔼 B_{n}}{n} \}(1 − ε)^{n} exp \{C \frac{n − 1}{2}\}.

This shows that the critical constant is no more than 2 log \frac{1}{1−ε}.

2 Integrability

Let {S′_{n}} be an independent copy of the random walk {S_{n}}. Let

I_{m,n} = ∑_{j=1}^{m} ∑_{k=1}^{n} δ(S_{j}, S′_{k}) (2.1)

and set I_{n} = I_{n,n}. Thus

I_{n} = #{(j, k) ∈ [1, n]^{2}; S_{j} = S′_{k}}. (2.2)

Lemma 2.1

(2.3) 𝔼 I_{m,n} ≤ c ((m + n) log(m + n) − m log m − n log n) .

In particular 𝔼 (I_{n}) ≤ cn. (2.4)

We also have 𝔼 I_{m,n} ≤ c\sqrt{mn}. (2.5)

Proof Using symmetry and independence

(2.6)  𝔼 I_{m,n} = ∑_{j=1}^{m} ∑_{k=1}^{n} 𝔼 δ(S_{j}, S′_{k})
= ∑_{j=1}^{m} ∑_{k=1}^{n} 𝔼 δ(S_{j} − S′_{k}, 0)
= ∑_{j=1}^{m} ∑_{k=1}^{n} 𝔼 δ(S_{j+k}, 0) = ∑_{j=1}^{m} ∑_{k=1}^{n} p_{j+k}(0)

where p_{n}(a) = P (S_{n} = a). By [17, p. 75],

p_{m}(0) = \frac{1}{2π\sqrt{det Γ}} \frac{1}{m} + o (\frac{1}{m}) (2.7)

so that

𝔼 I_{m,n} ≤ c ∑_{j=1}^{m} ∑_{k=1}^{n} \frac{1}{j + k} ≤ c ∫_{r=0}^{m} ∫_{s=0}^{n} \frac{1}{r + s} dr ds (2.8)

and (2.3) follows. (2.4) is then immediate. (2.5) follows from (2.8) and the bound (r + s)^{−1} ≤ (\sqrt{rs})^{−1}.

It follows from the proof of [8, Lemma 5.2] that for any integer k ≥ 1 𝔼 (I_{n}^{k}) ≤ (k!)^{2}2^{k}(1 + 𝔼 (I_{n}))^{k}. (2.9) Furthermore, by [13, (5.k)] we have that I_{n}/n converges in distribution to a random variable with finite moments. Hence for any integer k ≥ 1

lim_{n→∞} \frac{𝔼 (I_{n}^{k})}{n^{k}} = c_{k} < ∞. (2.10)

Lemma 2.2 There is a constant c > 0 such that

sup_{n} 𝔼 exp \{\frac{c}{n}I_{n}\} < ∞. (2.11)

Proof. We need only to show that there is a C > 0 such that 𝔼 I_{n}^{m} ≤ C^{m}m!n^{m} m, n ≥ 1.

We first consider the case m ≤ n and write l(m, n) = [n/m] + 1. Using [8, Theorem 5.1] with p = 2 and a = m, and then (2.4), (2.9) and (2.10), we obtain

(𝔼 I_{n}^{m})^{1/2} ≤ ∑_{k_{1},···,k_{m}≥0^{k_{1}+···+km=m}} \frac{m!}{k_{1}! ⋯ k_{m}!}(𝔼 I_{l(m,n)}^{k_{1}})^{1/2} ⋯ (𝔼 I_{l(m,n)}^{k_{m}})^{1/2}

(2.12)  ≤ ∑_{k_{1},···,k_{m}≥0^{k_{1}+···+km=m}} \frac{C^{m}m!}{k_{1}! ⋯ k_{m}!}k_{1}! ⋯ k_{m}!(𝔼 I_{l(m,n)})^{k_{1}/2} ⋯ (𝔼 I_{l(m,n)})^{k_{m}/2}

≤ \binom{2m − 1}{m}m!C^{m}(\frac{n}{m})^{m/2} ≤ \binom{2m}{m}m!C^{m}(\frac{n}{m})^{m/2}

where C > 0 can be chosen independently of m and n. Hence

𝔼 I_{n}^{m} ≤ \binom{2m}{m}^{2}C^{m}(m!)^{2}(\frac{n}{m})^{m} ≤ \binom{2m}{m}^{2}C^{m}m!n^{m}. (2.13)

Notice that

\binom{2m}{m} ≤ 4^{m}. (2.14)

For the case m > n, notice that I_{n} ≤ n^{2}. Trivially,

𝔼 I_{n}^{m} ≤ n^{2m} ≤ m^{m}n^{m} ≤ C^{m}m!n^{m},

where the last step follows from Stirling’s formula.

For any random variable X we define

\overline{X} =: X − 𝔼 X.

We write (m, n]_{<}^{2} = {(j, k) ∈ (m, n]^{2}; j < k} (2.15)

For any A ⊂ {(j, k) ∈ (ℤ^{+})^{2}; j < k}, write

B(A) = ∑_{(j,k)∈A} δ(S_{j}, S_{k}). (2.16)

In our proofs we will use several decompositions of B_{n}. If J_{1}, …, J_{ℓ} are consecutive disjoint blocks of integers whose union is {1, …, n}, we have

B_{n} = ∑_{i} B((J_{i} × J_{i}) ∩ (0, n]_{<}^{2}) + ∑_{i<j} B(J_{i} × J_{j})

and also

B_{n} = ∑_{i} B((J_{i} × J_{i}) ∩ (0, n]_{<}^{2}) + ∑_{i} B(∪_{j=1}^{i−1}J_{j}) × J_{i}).

Lemma 2.3 There is a constant c > 0 such that

sup_{n} 𝔼 exp \{\frac{c}{n}| \overline{B}_{n} |\} < ∞. (2.17)

Proof. We first prove that there is c > 0 such that

M ≡ sup_{n} 𝔼 exp \{\frac{c}{2^{n}} | \overline{B}_{2^{n}} |\} < ∞. (2.18)

We have B_{2^{n}} (2.19)

= ∑_{j=1}^{n} ∑_{k=1}^{2^{j−1}} B(((2k − 2)2^{n−j}, (2k − 1)2^{n−j}] × ((2k − 1)2^{n−j}, (2k)2^{n−j}])

Write α_{j,k} = B(((2k − 2)2^{n−j}, (2k − 1)2^{n−j}] × ((2k − 1)2^{n−j}, (2k)2^{n−j}]) (2.20)

−𝔼 B(((2k − 2)2^{n−j}, (2k − 1)2^{n−j}] × ((2k − 1)2^{n−j}, (2k)2^{n−j}])

For each 1 ≤ j ≤ n, the random variables α_{j,k}, k = 1, ⋯, 2^{j−1} are i.i.d. with common distribution I_{2^{n−j}} − 𝔼 I_{2^{n−j}}. By the previous lemma there exists δ > 0 such that

(2.21)  sup_{n} sup_{j≤n} 𝔼 exp \{δ \frac{1}{2^{n−j}} |α_{j,1}|\} < ∞.

By [3, Lemma 1], there exists θ > 0 such that

(2.22)  C(θ) ≡ sup_{n} sup_{j≤n} 𝔼 exp \{θ2^{j/2} \frac{1}{2^{n}} | ∑_{k=1}^{2^{j−1}} α_{j,k}|\}
= sup_{n} sup_{j≤n} 𝔼 exp \{θ2^{−j/2} \frac{1}{2^{n−j}} | ∑_{k=1}^{2^{j−1}} α_{j,k}|\} < ∞.

Write

λ_{N} = ∏_{j=1}^{N} (1 − 2^{−j/2}) and λ_{∞} = ∏_{j=1}^{∞} (1 − 2^{−j/2}). (2.23)

Using Hölder’s inequality with 1/p = 1 − 2^{−n/2}, 1/q = 2^{−n/2} we have

𝔼 exp \{λ_{n} \frac{θ}{2^{n}} | ∑_{j=1}^{n} ∑_{k=1}^{2^{j−1}} α_{j,k}|\} (2.24)

≤ (𝔼 exp \{λ_{n−1} \frac{θ}{2^{n}} | ∑_{j=1}^{n−1} ∑_{k=1}^{2^{j−1}} α_{j,k}|\})^{1−2^{−n/2}}

×(𝔼 exp \{2^{n/2}λ_{n} \frac{θ}{2^{n}} | ∑_{k=1}^{2^{n−1}} α_{n,k}|\})^{2^{−n/2}}

≤ 𝔼 exp \{λ_{n−1} \frac{θ}{2^{n}} | ∑_{j=1}^{n−1} ∑_{k=1}^{2^{j−1}} α_{j,k}|\}C(θ)^{2^{−n/2}}

Repeating this procedure,

𝔼 exp \{λ_{n} \frac{θ}{2^{n}} | ∑_{j=1}^{n} ∑_{k=1}^{2^{j−1}} α_{j,k}|\} (2.25)

≤ C(θ)^{2^{−1/2}+···+2^{−n/2}} ≤ C(θ)^{2^{−1/2}(1−2^{−1/2})^{−1}}.

So we have

sup_{n} 𝔼 exp \{λ_{∞} \frac{θ}{2^{n}} | \overline{B}_{2^{n}} |\} < ∞. (2.26)

We now prove our lemma for general n. Given an integer n ≥ 2, we have the following unique representation: n = 2^{m_{1}} + 2^{m_{2}} + ⋯ + 2^{m_{l}} (2.27) where m_{1} > m_{2} > ⋯ m_{l} ≥ 0 are integers. Write n_{0} = 0 and n_{i} = 2^{m_{1}} + ⋯ + 2^{m_{i}}, i = 1, ⋯, l. (2.28) Then

∑_{1≤j<k≤n} δ(S_{j}, S_{k}) = ∑_{i=1}^{l} ∑_{n_{i−1}<j<k≤n_{i}} δ(S_{j}, S_{k}) + ∑_{i=1}^{l−1} B((n_{i−1}, n_{i}] × (n_{i}, n])

(2.29)  = : ∑_{i=1}^{l} B_{2^{m_{i}}}^{(i)} + ∑_{i=1}^{l−1} A_{i}.

By Hölder’s inequality, with M as in (2.18)

𝔼 exp \{\frac{c}{n}| ∑_{i=1}^{l}(B_{2^{m_{i}}}^{(i)} − 𝔼 B_{2^{m_{i}}}^{(i)})|\} (2.30)

≤ ∏_{i=1}^{l} (𝔼 exp \{\frac{c}{2^{m_{i}}} |B_{2^{m_{i}}}^{(i)} − 𝔼 B_{2^{m_{i}}}^{(i)}|\})^{\frac{2^{m_{i}}}{n}} ≤ ∏_{i=1}^{l} M^{2^{m_{i}}/n} = M.

Using Hölder’s inequality,

𝔼 exp \{\frac{c}{n} ∑_{i=1}^{l−1} A_{i}\} ≤ ∏_{i=1}^{l−1} (𝔼 exp \{\frac{c}{2^{m_{i}}} A_{i}\})^{\frac{2^{m_{i}}}{n}} . (2.31)

Notice that for each 1 ≤ i ≤ l − 1,

A_{i} =^{d} ∑_{j=1}^{2^{m_{i}}} ∑_{k=1}^{n−n_{i}} δ(S_{j}, S′_{k}) ≤ ∑_{j=1}^{2^{m_{i}}} ∑_{k=1}^{2^{m_{i}}} δ(S_{j}, S′_{k}), (2.32)

where the inequality follows from n − n_{i} = 2^{m_{i+1}} + ⋯ + 2^{m_{l}} ≤ 2^{m_{i}}. (2.33) Using (2.32) and Lemma 2.1, we can take c > 0 so that

(2.34) 𝔼 exp \{\frac{c}{2^{m_{i}}} A_{i}\} ≤ sup_{n} 𝔼 exp \{\frac{c}{n}I_{n}\} ≡ N < ∞.

Consequently,

𝔼 exp \{\frac{c}{n} ∑_{i=1}^{l−1} A_{i}\} ≤ ∏_{i=1}^{l−1} N^{2^{m_{i}}/n} ≤ N. (2.35)

In particular, this shows that

𝔼 \{\frac{c}{n} ∑_{i=1}^{l−1} A_{i}\} ≤ N. (2.36)

Combining (2.35) and (2.36) with (2.30) we have

sup_{n} 𝔼 exp \{\frac{c}{2n}|\overline{B}_{n}|\} < ∞. (2.37)

Lemma 2.4 (2.38) 𝔼 B_{n} = \frac{1}{2π\sqrt{det Γ}}n log n + o(n log n),

and if 𝔼 |S_{1}|^{2+2δ} < ∞ for some δ > 0 then

𝔼 B_{n} = \frac{1}{2π\sqrt{det Γ}}n log n + O(n). (2.39)

Proof.

𝔼 B_{n} = 𝔼 ∑_{1≤j<k≤n} δ(S_{j}, S_{k}) = ∑_{1≤j<k≤n} p_{k−j}(0) (2.40)

where p_{m}(x) = 𝔼 (S_{m} = x). If 𝔼 |S_{1}|^{2+2δ} < ∞, then by [12, Proposition 6.7],

p_{m}(0) = \frac{1}{2π\sqrt{det Γ}} \frac{1}{m} + o (\frac{1}{m^{1+δ}} ) . (2.41)

Since the last term is summable, it will contribute O(n) to (2.40). Also,

∑_{1≤j<k≤n} \frac{1}{k − j} = ∑_{m=1}^{n} ∑_{i=1}^{n−m} \frac{1}{m} = ∑_{m=1}^{n} \frac{n − m}{m} = n ∑_{m=1}^{n} \frac{1}{m} − n (2.42)

and our lemma follows from the well known fact that

∑_{m=1}^{n} \frac{1}{m} = log n + γ + O (\frac{1}{n}) (2.43)

where γ is Euler’s constant.

If we only assume finite second moments, instead of (2.41) we use (2.7) and proceed as above.

Lemma 2.5 For any θ > 0

sup_{n} 𝔼 exp \{\frac{θ}{n}(𝔼 B_{n} − B_{n})\} < ∞ (2.44)

and for any λ > 0

lim_{n→∞} \frac{1}{b_{n}} log ℙ\{𝔼 B_{n} − B_{n} ≥ λnb_{n}\} = −∞. (2.45)

Proof. By Lemma 2.3 this is true for some θ_{o} > 0. For any θ > θ_{o}, take an integer m ≥ 1 such that θm^{−1} < θ_{o}. We can write any n as n = rm + i with 1 ≤ i < m. Then

𝔼 B_{n} − B_{n} (2.46)

≤ ∑_{j=1}^{m} [𝔼 ∑_{(j−1)r<k,l≤jr} δ(S_{k}, S_{l}) − ∑_{(j−1)r<k,l≤jr} δ(S_{k}, S_{l})] + 𝔼 B_{n} − m𝔼 B_{r}.

We claim that 𝔼 B_{n} − m𝔼 B_{r} = O(n). (2.47)

To see this, write

𝔼 B_{n} − m𝔼 B_{r} = 𝔼 B_{n} − ∑_{l=1}^{m} 𝔼 B(((l − 1)r, lr]_{<}^{2}) (2.48)

Notice that

B_{n} − ∑_{l=1}^{m} B(((l − 1)r, lr]_{<}^{2}) (2.49)

= ∑_{l=1}^{m} B(((l − 1)r, lr] × (lr, mr]) + B((mr, n]_{<}^{2})

+B((0, mr] × (mr, n])

Since

B(((l − 1)r, lr] × (lr, mr]) =^{d} I_{r,(m−l)r} (2.50)

by (2.3) we have

𝔼 B(((l − 1)r, lr] × (lr, mr]) (2.51)

≤ C{(m − (l − 1))r) log(m − (l − 1))r)
−((m − l)r) log((m − l)r) − r log r}

Therefore

∑_{l=1}^{m} 𝔼 B(((l − 1)r, lr] × (lr, mr]) (2.52)

≤ C ∑_{l=1}^{m} {(m − (l − 1))r) log(m − (l − 1))r)

−((m − l)r) log((m − l)r) − r log r}

= C{mr log mr − mr log r} = Cmr log m.

Using (2.5) for 𝔼 B((0, mr] × (mr, n]) = 𝔼 I_{mr,i} and (2.38) for 𝔼 B((mr, n]_{<}^{2}) then completes the proof of (2.47).

Note that the summands in (2.46) are independent. Therefore, for some constant C > 0 depending only on θ and m,

𝔼 exp \{\frac{θ}{n}(𝔼 B_{n} − B_{n})\} ≤ C(𝔼 exp \{\frac{θ}{n}(𝔼 B_{r} − B_{r})\})^{m} (2.53)

which proves (2.44), since θ/n ≤ θ/mr < θ_{o}/r and r → ∞ as n → ∞. Then, by Chebychev’s inequality, for any fixed h > 0

ℙ\{𝔼 B_{n} − B_{n} ≥ λnb_{n}\} ≤ e^{−hλb_{n}}𝔼 exp \{\frac{h}{n}(𝔼 B_{n} − B_{n})\} (2.54)

so that by (2.44)

(2.55)  lim sup_{n→∞} \frac{1}{b_{n}} log ℙ\{𝔼 B_{n} − B_{n} ≥ λnb_{n}\} ≤ −hλ.

Since h > 0 is arbitrary, this proves (2.45).

3 Proof of Theorem 1.1

By the Gärtner-Ellis theorem ( [11, Theorem 2.3.6]), we need only prove

(3.1) lim_{n→∞} \frac{1}{b_{n}} log 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} |B_{n} − 𝔼 B_{n}|^{1/2}\} = \frac{1}{4}κ(2, 2)^{4}θ^{2} det(Γ)^{−1/2}.

Indeed, by the Gärtner-Ellis theorem the above implies that

(3.2) lim_{n→∞} \frac{1}{b_{n}} log ℙ\{|B_{n} − 𝔼 B_{n}| ≥ λnb_{n}\} = −λ\sqrt{det(Γ)}κ(2, 2)^{−4}.

Using (2.45) we will then have Theorem 1.1. It thus remains to prove (3.1).

Let f be a symmetric probability density function in the Schwartz space S(ℝ^{2}) of C^{∞} rapidly decreasing functions. Let ϵ > 0 be a small number and write f_{ϵ}(x) = ϵ^{−2}f(ϵ^{−1}x), x ∈ ℝ^{2} (3.3)

and

l(n, x) = ∑_{k=1}^{n} δ(S_{k}, x), l(n, x, ϵ) = ∑_{k=1}^{n} f_{ϵ(b_{n}^{−1}n)^{1/2}}(S_{k} − x). (3.4)

By [8, Theorem 3.1],

(3.5)  lim_{n→∞} \frac{1}{b_{n}} log 𝔼 exp \{\frac{θ}{\sqrt{2}}\sqrt{\frac{b_{n}}{n}} ( ∑_{x∈ℤ^{2}} l^{2}(n, x, ϵ))^{1/2}\}

= sup_{g∈F_{2}} \{\frac{θ}{\sqrt{2}}( ∫_{ℝ^{2}} |g^{2} ∗ f_{ϵ}(x)|^{2}dx)^{1/2} − \frac{1}{2} ∫_{ℝ^{2}}⟨∇g, Γ∇g⟩dx\}

where F_{2} = {g ∈ W^{1,2}(ℝ^{2}) | ∥g∥_{2} = 1} . (3.6)

As in the proof of [10, Theorem 1], (3.1) will follow from (3.5) and the next Theorem.

Theorem 3.1 For any θ > 0,

lim_{ϵ→0} lim_{n→∞} \frac{1}{b_{n}} log 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} |B_{n} − 𝔼 B_{n} − \frac{1}{2} ∑_{x∈ℤ^{2}} l^{2}(n, x, ϵ)|^{1/2}\} = 0.

Proof. Let l > 1 be a large but fixed integer. Divide [1, n] into l disjoint

subintervals D_{1}, ⋯, D_{l}, each of length [n/l] or [n/l] + 1. Write

D_{i}^{∗} = {(j, k) ∈ D_{i}^{2}; j < k} i = 1, ⋯, l. (3.7)

With the notation of (2.16) we have

B_{n} = ∑_{i=1}^{l} B(D_{i}^{∗}) + ∑_{1≤j<k≤l} B(D_{j} × D_{k}). (3.8)

Define a_{j}, b_{j} so that D_{j} = (a_{j}, b_{j}] (1 ≤ j ≤ l). Notice that

B(D_{j} × D_{k}) = ∑_{n_{1}∈D_{j},n_{2}∈D_{k}} δ(S_{n_{1}}, S_{n_{2}})
= ∑_{n_{1}∈D_{j},n_{2}∈D_{k}} δ((S_{n_{1}} − S_{b_{j}}) + S_{b_{j}}, S_{a_{k}} + (S_{n_{2}} − S_{a_{k}}))
(3.9)  = ∑_{n_{1}∈D_{j},n_{2}∈D_{k}} δ((S_{n_{1}} − S_{b_{j}}), Z + (S_{n_{2}} − S_{a_{k}}))

with Z =^{d} S_{a_{k}} − S_{b_{j}}, so that Z, S_{n_{1}} − S_{b_{j}}, S_{n_{2}} − S_{a_{k}} are independent. Then as in (2.6)

(3.10)  𝔼 B(D_{j} × D_{k}) = 𝔼 ∑_{n_{1}∈D_{j},n_{2}∈D_{k}} p_{b_{j}−n_{1}+n_{2}−a_{k}}(Z)
≤ ∑_{n_{1}∈D_{j},n_{2}∈D_{k}} p_{b_{j}−n_{1}+n_{2}−a_{k}}(0)

since sup_{x} p_{j}(x) = p_{j}(0) for a symmetric random walk. Then as in the proof of (2.4) we have that

𝔼 B(D_{j} × D_{k}) ≤ cn/l. (3.11)

Hence,

B_{n} − 𝔼 B_{n} (3.12)

= ∑_{i=1}^{l} [B(D_{i}^{∗}) − 𝔼 B(D_{i}^{∗})] + ∑_{1≤j<k≤l} B(D_{j} × D_{k}) − 𝔼 ∑_{1≤j<k≤l} B(D_{j} × D_{k})

= ∑_{i=1}^{l} [B(D_{i}^{∗}) − 𝔼 B(D_{i}^{∗})] + ∑_{1≤j<k≤l} B(D_{j} × D_{k}) + O(n)

where the last line follows from (3.11).

Write

ξ_{i}(n, x, ϵ) = ∑_{k∈D_{i}} f_{ϵ(b_{n}^{−1}n)^{1/2}}(S_{k} − x). (3.13)

Then

∑_{x∈ℤ^{2}} l^{2}(n, x, ϵ) = ∑_{i=1}^{l} ∑_{x∈ℤ^{2}} ξ_{i}^{2}(n, x, ϵ) + 2 ∑_{1≤j≤k≤l} ∑_{x∈ℤ^{2}} ξ_{j}(n, x, ϵ)ξ_{k}(n, x, ϵ).

(3.14)

Therefore, by (3.12)

|(B_{n} − 𝔼 B_{n}) − \frac{1}{2} ∑_{x∈ℤ^{2}} l^{2}(n, x, ϵ)| (3.15)

≤ ∑_{i=1}^{l} |B(D_{i}^{∗}) − 𝔼 B(D_{i}^{∗})| + \frac{1}{2} ∑_{i=1}^{l} ∑_{x∈ℤ^{2}} ξ_{i}^{2}(n, x, ϵ)

+ ∑_{1≤j<k≤l} |B(D_{j} × D_{k}) − ∑_{x∈ℤ^{2}} ξ_{j}(n, x, ϵ)ξ_{k}(n, x, ϵ)| + O(n).

The proof of Theorem 3.1 is completed in the next two lemmas. Lemma 3.2 For any θ > 0,

(3.16) lim sup_{n→∞} \frac{1}{b_{n}} log 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} ( ∑_{i=1}^{l} ∑_{x∈ℤ^{2}} ξ_{i}^{2}(n, x, ϵ))^{1/2}\}

≤ l^{−1} \frac{1}{2}κ(2, 2)^{4}θ^{2} det(Γ)^{−1/2}

and

(3.17) lim sup_{n→∞} \frac{1}{b_{n}} log 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} ( ∑_{i=1}^{l} |B(D_{i}^{∗})−𝔼 B(D_{i}^{∗})|)^{1/2}\} ≤ l^{−1}Hθ^{2},

where

(3.18) H = ( sup \{λ > 0; sup_{n} 𝔼 exp \{λ\frac{1}{n}|B_{n} − 𝔼 B_{n}|\} < ∞\})^{−1}.

Proof. Replacing θ by θ/\sqrt{l}, n by n/l, and b_{n} by b_{n}^{∗} = b_{ln} (notice that b_{n/l}^{∗} = b_{n})

(3.19)  lim sup_{n→∞} \frac{1}{b_{n}} log 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} ( ∑_{x∈ℤ^{2}} ξ_{i}^{2}(n, x, ϵ))^{1/2}\}

= lim sup_{n→∞} \frac{1}{b_{n/l}^{∗}} log 𝔼 exp \{\frac{θ}{\sqrt{l}}\sqrt{\frac{b_{n/l}^{∗}}{n/l}} ( ∑_{x∈ℤ^{2}} ξ_{i}^{2}(n, x, ϵ))^{1/2}\}.

Applying Jensen’s inequality on the right hand side of (3.5),

∫_{ℝ^{2}} |g^{2} ∗ f_{ε}(x)|^{2} = ∫_{ℝ^{2}} [ ∫_{ℝ^{2}} g^{2}(x − y)f_{ε}(y) dy]^{2} dx
≤ ∫ ∫ g^{4}(x − y)f_{ε}(y) dy dx = ∫ f_{ε}(y)[ ∫ g^{4}(x − y) dx] dy
= [ ∫ g^{4}(x) dx] ∫ f_{ε}(y) dy = ∫_{ℝ^{2}} g^{4}(y) dy.

Combining the last two displays with (3.5) we have that

(3.20) lim sup_{n→∞} \frac{1}{b_{n}} log 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} ( ∑_{x∈ℤ^{2}} ξ_{i}^{2}(n, x, ϵ))^{1/2}\}

≤ sup_{g∈F_{2}} \{\frac{θ}{\sqrt{l}}( ∫_{ℝ^{2}} |g(x)|^{4}dx)^{1/2} − \frac{1}{2} ∫_{ℝ^{2}}⟨∇g(x), Γ∇g(x)⟩dx\}

= l^{−1}θ^{2} det(Γ)^{−1/2} sup_{h∈F_{2}} \{( ∫_{ℝ^{2}} |h(x)|^{4}dx)^{1/2} − \frac{1}{2} ∫_{ℝ^{2}} |∇h(x)|^{2}dx\}

= \frac{1}{2}l^{−1} det(Γ)^{−1/2}κ(2, 2)^{4}θ^{2},

where the third line follows from the substitution g(x) = \sqrt{| det(A)|}f(Ax) with a 2 × 2 matrix A satisfying A^{τ}ΓA = l^{−1}θ^{2} det(Γ)^{−1/2}I_{2} (3.21) and the last line of [8, Lemma A.2 ]; here I_{2} is the 2 × 2 identity matrix.

Given δ > 0, there exist \overline{a}_{1} = (a_{1,1}, ⋯, a_{1,l}), ⋯, \overline{a}_{m} = (a_{m,1}, ⋯, a_{m,l}) in ℝ^{l} such that |\overline{a}_{1}| = ⋯ = |\overline{a}_{m}| = 1 and |z| ≤ (1 + δ) max\{\overline{a}_{1} · z, ⋯, \overline{a}_{m} · z\}, z ∈ ℝ^{l}. (3.22) In particular, with

z = (( ∑_{x∈ℤ^{2}} ξ_{1}^{2}(n, x, ϵ))^{1/2}, …, ( ∑_{x∈ℤ^{2}} ξ_{l}^{2}(n, x, ϵ))^{1/2}) (3.23)

we have

( ∑_{i=1}^{l} ∑_{x∈ℤ^{2}} ξ_{i}^{2}(n, x, ϵ))^{1/2} ≤ (1 + δ) max_{1≤j≤m} ∑_{i=1}^{l} a_{j,i}( ∑_{x∈ℤ^{2}} ξ_{i}^{2}(n, x, ϵ))^{1/2}. (3.24)

Hence

𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} ( ∑_{i=1}^{l} ∑_{x∈ℤ^{2}} ξ_{i}^{2}(n, x, ϵ))^{1/2}\} (3.25)

≤ ∑_{j=1}^{m} 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} (1 + δ) ∑_{i=1}^{l} a_{j,i}( ∑_{x∈ℤ^{2}} ξ_{i}^{2}(n, x, ϵ))^{1/2}\}

= ∑_{j=1}^{m} ∏_{i=1}^{l} 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} (1 + δ)a_{j,i}( ∑_{x∈ℤ^{2}} ξ_{i}^{2}(n, x, ϵ))^{1/2}\},

where the last line follows from independence of ∥ξ_{i}(n, x, ϵ)∥_{L^{2}(ℤ^{2})}, i = 1, …, l. Therefore

(3.26) lim sup_{n→∞} \frac{1}{b_{n}} log 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} ( ∑_{k=1}^{l} ∑_{x∈ℤ^{2}} ξ_{k}^{2}(n, x, ϵ))^{1/2}\}

≤ max_{1≤j≤m} \frac{1}{2}l^{−1}κ(2, 2)^{4}(1 + δ)^{2}θ^{2}( ∑_{i=1}^{l} a_{j,i}^{2})

= \frac{1}{2}l^{−1} det(Γ)^{−1/2}κ(2, 2)^{4}(1 + δ)^{2}θ^{2}.

Letting δ → 0^{+} proves (3.16).

By the inequality ab ≤ a^{2} + b^{2} we have that

𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} |B_{n} − 𝔼 B_{n}|^{1/2}\} (3.27)

≤ exp \{c^{2}θ^{2}b_{n}\}𝔼 exp \{c^{−2} \frac{1}{n}|B_{n} − 𝔼 B_{n}|\},

and taking c^{−2} ↑ H^{−1} we see that for any θ > 0,

(3.28) lim sup_{n→∞} \frac{1}{b_{n}} log 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} |B_{n} − 𝔼 B_{n}|^{1/2}\} ≤ Hθ^{2}.

Notice that for any 1 ≤ i ≤ l,

B(D_{i}^{∗}) − 𝔼 B(D_{i}^{∗}) =^{d} B_{#(D_{i})} − 𝔼 B_{#(D_{i})}. (3.29)

We have

𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} |B(D_{i}^{∗})−𝔼 B(D_{i}^{∗})|^{1/2}\} = 𝔼 exp\{\frac{θ}{\sqrt{l}}\sqrt{\frac{b_{n}}{n/l}}|B(D_{i}^{∗}−𝔼 B(D_{i}^{∗})|^{1/2}\}.

Replacing θ by θ/\sqrt{l}, n by n/l, and b_{n} by b_{n}^{∗} = b_{ln} (notice that b_{n/l}^{∗} = b_{n}) gives

(3.30) lim sup_{n→∞} \frac{1}{b_{n}} log 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} |B(D_{i}^{∗}) − 𝔼 B(D_{i}^{∗})|^{1/2}\} ≤ l^{−1}Hθ^{2}.

Thus (3.17) follows by the same argument we used to prove (3.16).

Lemma 3.3 For any θ > 0 and any 1 ≤ j < k ≤ l,

lim sup_{ϵ→0^{+}} lim sup_{n→∞} \frac{1}{b_{n}} log 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} |B(D_{j}×D_{k})−∑_{x∈ℤ^{2}} ξ_{j}(n, x, ϵ)ξ_{k}(n, x, ϵ)|^{1/2}\} = 0.

(3.31)

Proof. We now fix 1 ≤ j < k ≤ l and estimate

B(D_{j} × D_{k}) − ∑_{x∈ℤ^{2}} ξ_{j}(n, x, ϵ)ξ_{k}(n, x, ϵ). (3.32)

Without loss of generality we may assume that v =: [n/l] = #(D_{j}) = #(D_{k}). For y ∈ Z^{2} set

I_{n}(y) = ∑_{n_{1},n_{2}=1}^{n} δ(S_{n_{1}}, S′_{n_{2}} + y). (3.33)

Note that I_{n} = I_{n}(0). By (3.9) we have that

B(D_{j} × D_{k}) =^{d} I_{v}(Z) (3.34)

with Z independent of S, S′.

Similarly, we have

∑_{x∈ℤ^{2}} ξ_{j}(n, x, ϵ)ξ_{k}(n, x, ϵ)

= ∑_{x∈ℤ^{2}} ∑_{n_{1}∈D_{j},n_{2}∈D_{k}} f_{ϵ(b_{n}^{−1}n)^{1/2}}(S_{n_{1}} − x)f_{ϵ(b_{n}^{−1}n)^{1/2}}(S_{n_{2}} − x)

= ∑_{x∈ℤ^{2}} ∑_{n_{1}∈D_{j},n_{2}∈D_{k}} f_{ϵ(b_{n}^{−1}n)^{1/2}}(x)f_{ϵ(b_{n}^{−1}n)^{1/2}}(S_{n_{2}} − S_{n_{1}} − x)

= ∑_{n_{1}∈D_{j},n_{2}∈D_{k}} f_{ϵ(b_{n}^{−1}n)^{1/2}} ⊛ f_{ϵ(b_{n}^{−1}n)^{1/2}}(S_{n_{2}} − S_{n_{1}})

= ∑_{n_{1}∈D_{j},n_{2}∈D_{k}} f_{ϵ(b_{n}^{−1}n)^{1/2}} ⊛ f_{ϵ(b_{n}^{−1}n)^{1/2}}((S_{n_{2}} − S_{a_{k}}) − (S_{n_{1}} − S_{b_{j}}) + Z) (3.35)

where

f ⊛ f(y) = ∑_{x∈ℤ^{2}} f(x)f(y − x) (3.36)

denotes convolution in L^{1}(ℤ^{2}). It is clear that if f ∈ S(ℝ^{2}) so is f ⊛ f. For y ∈ Z^{2}, define the link

L_{n,ϵ}(y) = ∑_{n_{1},n_{2}=1}^{n} f_{ϵ} ⊛ f_{ϵ}(S′_{n_{2}} − S_{n_{1}} + y). (3.37)

By (3.35) we have that

∑_{x∈ℤ^{2}} ξ_{j}(n, x, ϵ)ξ_{k}(n, x, ϵ) =^{d} L_{v,(b_{n}^{−1}n)^{1/2}ϵ}(Z) (3.38)

with Z independent of S, S′.

Lemma 3.4 Let f ∈ S(ℝ^{2}) with Fourier transform f̂ supported on (−π, π)^{2}. Then for any r ≥ 1

∫ e^{−iλy}(f_{r} ⊛ f_{r})(y) dy = (f̂(rλ))^{2}, ∀λ ∈ ℝ^{2}. (3.39)

Proof. We have

∫ e^{−iλy}(f ⊛ f)(y) dy = ∑_{x∈ℤ^{2}} f(x) ∫ e^{−iλy}f(y − x) dy (3.40)
= f̂(λ) ∑_{x∈ℤ^{2}} f(x)e^{−iλx}

= f̂(λ) ∑_{x∈ℤ^{2}} (\frac{1}{(2π)^{2}} ∫ e^{ipx}f̂(p) dp) e^{−iλx}.

For x ∈ ℤ^{2}

∫ e^{ipx}f̂(p) dp = ∑_{u∈ℤ^{2}} ∫_{[−π,π]^{2}} e^{ipx}f̂(p + 2πu) dp (3.41)

and using Fourier inversion

∑_{x∈ℤ^{2}} (∫ e^{ipx}f̂(p) dp) e^{−iλx}

= ∑_{u∈ℤ^{2}} ∑_{x∈ℤ^{2}} (∫_{[−π,π]^{2}} e^{ipx}f̂(p + 2πu) dp) e^{−iλx} (3.42)

= (2π)^{2} ∑_{u∈ℤ^{2}} f̂(λ + 2πu).

Thus from (3.40) we find that

∫ e^{−iλy}f ⊛ f(y) dy = f̂(λ) ∑_{u∈ℤ^{2}} f̂(λ + 2πu). (3.43)

Since f̂_{r}(λ) = f̂(rλ) we see that for any r > 0

∫ e^{−iλy}(f_{r} ⊛ f_{r})(y) dy = f̂(rλ) ∑_{u∈ℤ^{2}} f̂(rλ + 2πru). (3.44)

Then if r ≥ 1, using the fact that f̂(λ) is supported in (−π, π)^{2}, we obtain (3.39).

Taking f ∈ S(ℝ^{2}) with f̂(λ) supported in (−π, π)^{2}, Lemma 3.3 will follow from Theorem 4.1 of the next section.

4 Intersections of Random Walks

Let S_{1}(n), S_{2}(n) be independent copies of the symmetric random walk S(n) in ℤ^{2} with a finite second moment.

Let f be a positive symmetric function in the Schwartz space S(ℝ^{2}) with ∫ f dx = 1 and f̂ supported in (−π, π)^{2}. Given ϵ > 0, and with the notation of the last section, let us define the link

I_{n,ϵ}(y) = ∑_{n_{1},n_{2}=1}^{n} f_{(b_{n}^{−1}n)^{1/2}ϵ} ⊛ f_{(b_{n}^{−1}n)^{1/2}ϵ}(S_{2}(n_{2}) − S_{1}(n_{1}) + y)) (4.1)

with I_{n,ϵ} = I_{n,ϵ}(0). Theorem 4.1 For any λ > 0

lim sup_{ϵ→0} lim sup_{n→∞} sup_{y}

(4.2)

\frac{1}{b_{n}} log 𝔼 (exp \{λ|\frac{I_{n}(y) − I_{n,ϵ}(y)}{b_{n}^{−1}n}|^{1/2}\}) = 0.

Proof of Theorem 4.1. We have

\frac{1}{b_{n}^{−1}n}I_{n}(y) (4.3)

= \frac{1}{b_{n}^{−1}n} ∑_{n_{1},n_{2}=1}^{n} δ(S_{1}(n_{1}), S_{2}(n_{2}) + y)

= \frac{1}{b_{n}^{−1}n(2π)^{2}} ∑_{n_{1},n_{2}=1}^{n} [ ∫_{[−π,π]^{2}} e^{ip·(S_{2}(n_{2})+y−S_{1}(n_{1}))} dp]

where from now on we work modulo ±π. Then by scaling we have

\frac{1}{b_{n}^{−1}n}I_{n}(y) (4.4)

= \frac{1}{(b_{n}^{−1}n)^{2}(2π)^{2}} ∑_{n_{1},n_{2}=1}^{n} [ ∫_{(b_{n}^{−1}n)^{1/2}[−π,π]^{2}} e^{ip·(S_{2}(n_{2})+y−S_{1}(n_{1}))/(b_{n}^{−1}n)^{1/2}} dp].

As in (4.3)-(4.4), using Lemma 3.4, the fact that ϵ(b_{n}^{−1}n)^{1/2} ≥ 1 for ϵ > 0 fixed and large enough n, and abbreviating ĥ = ( f̂)^{2}

\frac{1}{b_{n}^{−1}n}I_{n,ϵ}(y) (4.5)

= \frac{1}{b_{n}^{−1}n(2π)^{2}} ∑_{n_{1},n_{2}=1}^{n} [ ∫_{ℝ^{2}} e^{ip·(S_{2}(n_{2})+y−S_{1}(n_{1}))} ĥ(ϵ(b_{n}^{−1}n)^{1/2}p) dp]

= \frac{1}{(b_{n}^{−1}n)^{2}(2π)^{2}} ∑_{n_{1},n_{2}=1}^{n} [ ∫_{ℝ^{2}} e^{ip·(S_{2}(n_{2})+y−S_{1}(n_{1}))/(b_{n}^{−1}n)^{1/2}} ĥ(ϵp) dp].

Using our assumption that ĥ supported in [−π, π]^{2}, and that ϵ^{−1} ≤ (b_{n}^{−1}n)^{1/2} for ϵ > 0 fixed and large enough n, we have that

\frac{1}{b_{n}^{−1}n}I_{n,ϵ}(y) (4.6)

= \frac{1}{(b_{n}^{−1}n)^{2}(2π)^{2}} ∑_{n_{1},n_{2}=1}^{n}

[ ∫_{ϵ^{−1}[−π,π]^{2}} e^{ip·(S_{2}(n_{2})+y−S_{1}(n_{1}))/(b_{n}^{−1}n)^{1/2}} ĥ(ϵp) dp]

= \frac{1}{(b_{n}^{−1}n)^{2}(2π)^{2}} ∑_{n_{1},n_{2}=1}^{n}

[ ∫_{(b_{n}^{−1}n)^{1/2}[−π,π]^{2}} e^{ip·(S_{2}(n_{2})+y−S_{1}(n_{1}))/(b_{n}^{−1}n)^{1/2}} ĥ(ϵp) dp].

To prove (4.2) it suffices to show that for each λ > 0 we have

sup_{y} 𝔼 (exp \{λ|\frac{I_{n}(y) − I_{n,ϵ}(y)}{b_{n}^{−1}n}|^{1/2}\}) (4.7)

≤ Cb_{n}(1 − Cλϵ^{m/4})^{−1}(1 + Cλϵ^{1/4}b_{n}^{1/2})e^{Cλ^{2}ϵ^{1/2}b_{n}}.

for some C < ∞ and all ϵ > 0 sufficiently small.

We begin by expanding

𝔼 (exp \{λ|\frac{I_{n}(y) − I_{n,ϵ}(y)}{b_{n}^{−1}n}|^{1/2}\}) (4.8)

= ∑_{m=0}^{∞} \frac{λ^{m}}{m!} 𝔼 (|\frac{1}{b_{n}^{−1}n}(I_{n}(y) − I_{n,ϵ}(y))|^{m/2})

≤ ∑_{m=0}^{∞} \frac{λ^{m}}{m!} (𝔼 (\{\frac{1}{b_{n}^{−1}n}(I_{n}(y) − I_{n,ϵ}(y))\}^{2m}))^{1/4} .

By (4.4), (4.6) and the symmetry of S_{1} we have

𝔼 (\{\frac{1}{b_{n}^{−1}n}(I_{n}(y) − I_{n,ϵ}(y))\}^{m}) (4.9)

= \frac{1}{(b_{n}^{−1}n)^{2m}(2π)^{2m}} ∑_{j=1,...,m^{n_{1,j} ,n_{2,j} =1}}^{n} ∫_{(b_{n}^{−1}n)^{1/2}[−π,π]^{2m}}

𝔼 (e^{i ∑_{j=1}^{m} p_{j}·(S_{2}(n_{2,j})+y+S_{1}(n_{1,j}))/(b_{n}^{−1}n)^{1/2}}) ∏_{j=1}^{m}(1 − ĥ(ϵp_{j})) dp_{j}.

Then

|𝔼 (\{\frac{1}{b_{n}^{−1}n}(I_{n}(y) − I_{n,ϵ}(y))\}^{m}) | (4.10)

≤ \frac{1}{(b_{n}^{−1}n)^{2m}(2π)^{2m}} ∑_{j=1,...,m^{n_{1,j} =1}}^{n} ∑_{j=1,...,m^{n_{2,j} =1}}^{n} ∫_{(b_{n}^{−1}n)^{1/2}[−π,π]^{2m}}

|𝔼 (e^{i ∑_{j=1}^{m} p_{j}·S_{1}(n_{1,j})/(b_{n}^{−1}n)^{1/2}}) |

|𝔼 (e^{i ∑_{j=1}^{m} p_{j}·S_{2}(n_{2,j})/(b_{n}^{−1}n)^{1/2}}) | ∏_{j=1}^{m} |1 − ĥ(ϵp_{j})| dp_{j}.

By the Cauchy-Schwarz inequality ∫_{(b_{n}^{−1}n)^{1/2}[−π,π]^{2m}} |𝔼 (e^{i ∑_{j=1}^{m} p_{j}·S_{1}(n_{1,j})/(b_{n}^{−1}n)^{1/2}}) | (4.11)

|𝔼 (e^{i ∑_{j=1}^{m} p_{j}·S_{2}(n_{2,j})/(b_{n}^{−1}n)^{1/2}}) | ∏_{j=1}^{m} |1 − ĥ(ϵp_{j})| dp_{j}

≤ ∏_{i=1}^{2}_{{∫(b_{n}^{−1}n)^{1/2}[−π,π]^{2m}}

|𝔼 (e^{i ∑_{j=1}^{m} p_{j}·S(n_{i,j})/(b_{n}^{−1}n)^{1/2}}) |^{2} ∏_{j=1}^{m} |1 − ĥ(ϵp_{j})| dp_{j}}^{1/2} . (4.12)

Thus

|𝔼 (\{\frac{1}{b_{n}^{−1}n}(I_{n}(y) − I_{n,ϵ}(y))\}^{m}) |^{1/2} (4.13)

≤ ∑_{j=1,...,m^{n_{j} =1}}^{n} \frac{1}{(b_{n}^{−1}n)^{m}(2π)^{m}} _{\{∫(b_{n}^{−1}n)^{1/2}[−π,π]^{2m}}

|𝔼 (e^{i ∑_{j=1}^{m} p_{j}·S(n_{j})/(b_{n}^{−1}n)^{1/2}}) |^{2} ∏_{j=1}^{m} |1 − ĥ(ϵp_{j})| dp_{j}}^{1/2} .

For any permutation ψ of {1…. , m} let D_{m}(ψ) = {(n_{1}, …, n_{m})| 1 ≤ n_{ψ(1)} ≤ ⋯ ≤ n_{ψ(m)} ≤ n}. (4.14) Using the (non-disjoint) decomposition

{1, …, n}^{m} = ⋃_{π} D_{m}(ψ)

we have from (4.13) that

|𝔼 (\{\frac{1}{b_{n}^{−1}n}(I_{n}(y) − I_{n,ϵ}(y))\}^{m}) |^{1/2} (4.15)

≤ ∑_{ψ} ∑_{D_{m}(ψ)} \frac{1}{(b_{n}^{−1}n)^{m}(2π)^{m}} _{\{∫(b_{n}^{−1}n)^{1/2}[−π,π]^{2m}}

|𝔼 (e^{i ∑_{j=1}^{m} p_{j}·S(n_{j})/(b_{n}^{−1}n)^{1/2}}) |^{2} ∏_{j=1}^{m} |1 − ĥ(ϵp_{j})| dp_{j}}^{1/2} .

where the first sum is over all permutations ψ of {1…. , m}.

Set ϕ(u) = 𝔼 (e^{iu·S(1)}) . (4.16) It follows from our assumptions that ϕ(u) ∈ C^{2}, \frac{∂}{∂u_{i}} ϕ(0) = 0 and \frac{∂^{2}}{∂u_{i}∂u_{j}} ϕ(0) = −E (S_{(i)}(1)S_{(j)}(1)) where S(1) = (S_{(1)}(1), S_{(2)}(1)) so that for some δ > 0 ϕ(u) = 1 − E ((u · S(1))^{2}) /2 + o(|u|^{2}), |u| ≤ δ. (4.17) Then for some c_{1} > 0 ϕ(u) ≤ e^{−c_{1}|u|^{2}}, |u| ≤ δ. (4.18) Strong aperiodicity implies that |ϕ(u)| < 1 for u ≠ 0 and u ∈ [−π, π]^{2}. In particular, we can find b < 1 such that |ϕ(u)| ≤ b for δ ≤ |u| and u ∈ [−π, π]^{2}. But clearly we can choose c_{2} > 0 so that b ≤ e^{−c_{2}|u|^{2}} for u ∈ [−π, π]^{2}. Setting c = min(c_{1}, c_{2}) > 0 we then have ϕ(u) ≤ e^{−c|u|^{2}}, u ∈ [−π, π]^{2}. (4.19)

On D_{m}(ψ) we can write

∑_{j=1}^{m} p_{j} · S(n_{j}) = ∑_{j=1}^{m}(∑_{i=j}^{m} p_{ψ(i)})(S(n_{ψ(j)}) − S(n_{ψ(j−1)})). (4.20)

Hence on D_{m}(ψ)

𝔼 (e^{i ∑_{j=1}^{m} p_{j}·S(n_{j})/(b_{n}^{−1}n)^{1/2}}) = ∏_{j=1}^{m} ϕ((∑_{i=j}^{m} p_{ψ(i)})/(b_{n}^{−1}n)^{1/2})^{(n_{ψ(j)}−n_{ψ(j−1)})}. (4.21)

Now it is clear that

∑_{D_{m}(ψ)}_{{∫(b_{n}^{−1}n)^{1/2}[−π,π]^{2m}} (4.22)

| ∏_{j=1}^{m} ϕ((∑_{i=j}^{m} p_{ψ(i)})/(b_{n}^{−1}n)^{1/2})^{(n_{ψ(j)}−n_{ψ(j−1)})}|^{2} ∏_{j=1}^{m} |1 − ĥ(ϵp_{j})| dp_{j}}^{1/2}

= ∑_{1≤n_{ψ(1)}≤···≤n_{ψ(m)}≤n}_{{∫(b_{n}^{−1}n)^{1/2}[−π,π]^{2m}}

| ∏_{j=1}^{m} ϕ((∑_{i=j}^{m} p_{ψ(i)})/(b_{n}^{−1}n)^{1/2})^{(n_{ψ(j)}−n_{ψ(j−1)})}|^{2} ∏_{j=1}^{m} |1 − ĥ(ϵp_{j})| dp_{j}}^{1/2}

is independent of the permutation ψ. Hence writing

u_{j} = ∑_{i=j}^{m} p_{i} (4.23)

we have from (4.15) that

|𝔼 (\{\frac{1}{b_{n}^{−1}n}(I_{n}(y) − I_{n,ϵ}(y))\}^{m}) |^{1/2} (4.24)

≤ m! ∑_{1≤n_{1}≤···≤n_{m}≤n} \frac{1}{(b_{n}^{−1}n)^{m}(2π)^{m}} _{\{∫(b_{n}^{−1}n)^{1/2}[−π,π]^{2m}}

| ∏_{j=1}^{m} ϕ(u_{j}/(b_{n}^{−1}n)^{1/2})^{(n_{j}−n_{j−1})}|^{2} ∏_{j=1}^{m} |1 − ĥ(ϵp_{j})| dp_{j}}^{1/2} .

For each A ⊆ {2, 3, …, m} we use D_{m}(A) to denote the subset of {1 ≤ n_{1} ≤ ⋯ ≤ n_{m} ≤ n} for which n_{j} = n_{j−1} if and only if j ∈ A. Then we have

|𝔼 (\{\frac{1}{b_{n}^{−1}n}(I_{n}(y) − I_{n,ϵ}(y))\}^{m}) |^{1/2} (4.25)

≤ m! ∑_{A⊆\{2,3,...,m\}} ∑_{D_{m}(A)} \frac{1}{(b_{n}^{−1}n)^{m}(2π)^{m}} _{\{∫(b_{n}^{−1}n)^{1/2}[−π,π]^{2m}}

| ∏_{j=1}^{m} ϕ(u_{j}/(b_{n}^{−1}n)^{1/2})^{(n_{j}−n_{j−1})}|^{2} ∏_{j=1}^{m} |1 − ĥ(ϵp_{j})| dp_{j}}^{1/2} .

For any u ∈ ℝ^{d} let ũ denote the representative of u mod (b_{n}^{−1}n)^{1/2}2πℤ^{2} of smallest absolute value. We note that

| ˜−u| = |ũ|, and |{ \atop u + v}| = |ũ + ṽ| ≤ |ũ| + |ṽ|. (4.26)

Using the periodicity of ϕ we see that (4.19) implies that for all u

|ϕ(u/(b_{n}^{−1}n)^{1/2})| ≤ e^{−c|ũ|^{2}/(b_{n}^{−1}n)}. (4.27)

Then we have that on {1 ≤ n_{1} ≤ ⋯ ≤ n_{m} ≤ n}

| ∏_{j=1}^{m} ϕ(u_{j}/(b_{n}^{−1}n)^{1/2})^{(n_{j}−n_{j−1})}|^{2} ≤ ∏_{j=1}^{m} e^{−c|ũ_{j}|^{2}(n_{j}−n_{j−1})/(b_{n}^{−1}n)} (4.28)

Using |1 − ĥ(ϵp_{j})| ≤ cϵ^{1/2}|p_{j}|^{1/2} we bound the integral in (4.25) by

c^{m}ϵ^{m/2} ∫_{(b_{n}^{−1}n)^{1/2}[−π,π]^{2m}} ∏_{j=1}^{m} e^{−c|ũ_{j}|^{2}(n_{j}−n_{j−1})/(b_{n}^{−1}n)}|p_{j}|^{1/2} dp_{j}. (4.29)

Using (4.23) and (4.26) we have that

∏_{j=1}^{m} |p_{j}|^{1/2} ≤ ∏_{j=1}^{m}(|ũ_{j}|^{1/2} + |ũ_{j+1}|^{1/2}) (4.30)

and when we expand the right hand side as a sum of monomials we can be sure that no factor |ũ_{k}|^{1/2} appears more than twice. Thus we see that we can bound (4.29) by

C^{m}ϵ^{m/2} max_{h(j)} ∫_{(b_{n}^{−1}n)^{1/2}[−π,π]^{2m}} ∏_{j=1}^{m} e^{−c|ũ_{j}|^{2}(n_{j}−n_{j−1})/(b_{n}^{−1}n)}|ũ_{j}|^{h(j)/2} dp_{j} (4.31)

where the max runs over the the set of functions h(j) taking values 0, 1 or 2 and such that ∑_{j} h(j) = m. Here we used the fact that the number of ways to choose the {h(j)} is bounded by the number of ways of dividing m objects into 3 groups, which is 3^{m}. Changing variables, we thus need to bound

∫_{Λ_{n}} ∏_{j=1}^{m} e^{−c|ũ_{j}|^{2}(n_{j}−n_{j−1})/(b_{n}^{−1}n)}|ũ_{j}|^{h(j)/2} du_{j} (4.32)

where, see (4.23),

Λ_{n} = {(u_{1}, …, u_{m}) | u_{j} − u_{j+1} ∈ (b_{n}^{−1}n)^{1/2}[−π, π]^{2}, ∀j}. (4.33)

Let C_{n} denote the rectangle (b_{n}^{−1}n)^{1/2}[−π, π]^{2} and let us call any rectangle of the form 2πk + C_{n}, where k ∈ ℤ^{2}, an elementary rectangle. Note that any rectangle of the form v + C_{n}, where v ∈ ℝ^{2}, can be covered by 4 elementary rectangles. Hence for any v ∈ ℝ^{2} and 1 ≤ s ≤ n

∫_{v+C_{n}} e^{−c_{\frac{s}{b{−1 \atop n}n}} |ũ|^{2}}|ũ|^{h/2} du (4.34)

≤ 4 ∫_{R^{2}} e^{−c_{\frac{s}{(b{−1 \atop n}n)}} |u|^{2}}|u|^{h/2} du

≤ C (\frac{s}{b_{n}^{−1}n})^{−(1+h/4)} .

Similarly

∫_{v+C_{n}} |ũ|^{h/2} du ≤ C(b_{n}^{−1}n)^{(1+h/4)}. (4.35)

We now bound (4.32) by bounding successively the integration with respect to u_{1}, …, u_{m}. Consider first the du_{1} integral, fixing u_{2}, …, u_{m}. By (4.33) the du_{1} integral is over the rectangle u_{2}+C_{n}, hence the factors involving u_{1} can be bounded using (4.34). Proceeding inductively, using (4.33) when n_{j} − n_{j−1} > 0 and (4.35) when n_{j} = n_{j−1}, leads to the following bound of (4.32), and hence of (4.29) on D_{m}(A):

c^{m}ϵ^{m/2} ∫_{(b_{n}^{−1}n)^{1/2}[−π,π]^{2m}} ∏_{j=1}^{m} e^{−c|ũ_{j}|^{2}(n_{j}−n_{j−1})/(b_{n}^{−1}n)}|p_{j}|^{1/2} dp_{j} (4.36)

≤ C^{m}ϵ^{m/2} ∏_{j∈A}(b_{n}^{−1}n)^{(1+h(j)/4)} ∏_{j∈A^{c}} (\frac{(n_{j} − n_{j−1})}{b_{n}^{−1}n})^{−(1+h(j)/4)} .

Here A^{c} means the complement of A in {1, …, m}, so that A^{c} always contains 1. If A^{c} = {i_{1}, …, i_{k}} where i_{1} < ⋯ < i_{k} we then obtain for the sum in (4.25) over D_{m}(A), the bound

C^{m}ϵ^{m/4} max_{h(j)} \frac{1}{(b_{n}^{−1}n)^{m}} ∏_{j∈A}(b_{n}^{−1}n)^{(1+h(j)/4)} (4.37)

∑_{1≤n_{i_{1}}<···<n_{i_{k}} ≤n} ∏_{j∈A^{c}} (\frac{(n_{j} − n_{j−1})}{b_{n}^{−1}n})^{−(1+h(j)/4)/2}

Note that

(b_{n}^{−1}n)^{(1+h(j)/4)/2} \frac{1}{b_{n}^{−1}n} → 0 as n → ∞. (4.38)

Using this to bound the product over j ∈ A, and then bounding the sum by an integral, we can bound (4.37) by

C^{m}ϵ^{m/4} max_{h(j)} ∑_{1≤n_{i_{1}}<···<n_{i_{k}} ≤n} ∏_{j∈A^{c}} (\frac{(n_{j} − n_{j−1})}{b_{n}^{−1}n})^{−(1+h(j)/4)/2} \frac{1}{b_{n}^{−1}n} (4.39)

≤ C^{m}ϵ^{m/4} max_{h(j)} ∫_{0≤r_{i_{1}}<···<r_{i_{k}} ≤b_{n}} ∏_{j∈A^{c}}(r_{j} − r_{j−1})^{−(1/2+h(j)/8)} dr_{j}

≤ C^{m}ϵ^{m/4} max_{h(j)} \frac{b_{n}^{∑_{j∈A^{c}}(1/2−h(j)/8)}}{Γ(∑_{j∈A^{c}}(1/2 − h(j)/8))}

Using this together with (4.25), but with m replaced by 2m, and the fact that (2m!)^{1/2}/m! ≤ 2^{m}, we see that (4.7) is bounded by

∑_{m=0}^{∞} C^{m}λ^{m}ϵ^{m/4} ( ∑_{A⊆\{2,3,...,2m\}} max_{h(j)} \frac{b_{n}^{∑_{j∈A^{c}}(1/2−h(j)/8)}}{Γ(∑_{j∈A^{c}}(1/2 − h(j)/8))})^{1/2} . (4.40)

We have ∑_{A⊆{1,2,3,...,2m}} 1 = 2^{2m}. Then noting that ∑_{j∈A^{c}}(1/2 − h(j)/8) is an integer multiple of 1/8 which is always less than m, we can bound the last line by

∑_{l=0}^{∞} (∑_{m=l}^{∞} C^{m}λ^{m}ϵ^{m/4}) ∑_{j=0}^{7} (\frac{b_{n}^{l+j/8}}{Γ(l + j/8)})^{1/2} (4.41)

≤ Cb_{n} ∑_{l=0}^{∞} (∑_{m=l}^{∞} C^{m}λ^{m}ϵ^{m/4}) (\frac{b_{n}^{l}}{Γ(l)})^{1/2}

≤ Cb_{n}(1 − Cλϵ^{m/4})^{−1} ∑_{l=0}^{∞} C^{l}λ^{l}|ϵ|^{l/4}b_{n}^{l/2} (\frac{1}{Γ(l)})^{1/2}

for ϵ > 0 sufficiently small.

(4.7) then follows from the fact that for any a > 0

∑_{l=0}^{∞} a^{l} (\frac{1}{Γ(l)})^{1/2} (4.42)

= ∑_{m=0}^{∞} (a^{2m} (\frac{1}{Γ(2m)})^{1/2} + a^{2m+1} (\frac{1}{Γ(2m + 1)})^{1/2})

≤ C(1 + a) ∑_{m=0}^{∞} a^{2m} (\frac{1}{Γ(2m)})^{1/2}

≤ C(1 + a)e^{Ca^{2}}.

Remark 4.2 It follows from the proof that in fact for ρ > 0 sufficiently small, for any λ > 0

lim sup_{ϵ→0} lim sup_{n→∞} sup_{y}

(4.43)

\frac{1}{b_{n}} log 𝔼 (exp \{λ|\frac{I_{n}(y) − I_{n,ϵ}(y)}{ϵ^{ρ} b_{n}^{−1}n}|^{1/2}\}) = 0.

5 Theorem 1.2: Upper bound for 𝔼 B_{n} − B_{n}

Proof of Theorem 1.2.

We prove (1.5) for θ = 1:

−C_{1} ≤ lim inf_{n→∞} b_{n}^{−1} log ℙ{𝔼 B_{n} − B_{n} ≥ (2π)^{−1} det(Γ)^{−1/2}n log b_{n}}
(5.1) ≤ lim sup_{n→∞} b_{n}^{−1} log ℙ{𝔼 B_{n} − B_{n} ≥ (2π)^{−1} det(Γ)^{−1/2}n log b_{n}} ≤ −C_{2}

for any {b_{n}} satisfying (1.3). The case of general θ follows by replacing b_{n} by b_{n}^{θ} in (5.1).

In this section we prove the upper bound for (5.1). Let t > 0 and write K = [t^{−1}b_{n}]. Divide [1, n] into K > 1 disjoint subintervals (n_{0}, n_{1}], ⋯, (n_{K−1}, n_{K}], each of length [n/K] or [n/K] + 1. Notice that

𝔼 B_{n} − B_{n} ≤ ∑_{i=1}^{K} [𝔼 B((n_{i−1}, n_{i}]_{<}^{2}) − B((n_{i−1}, n_{i}]_{<}^{2})] (5.2)

+𝔼 B_{n} − ∑_{i=1}^{K} 𝔼 B((n_{i−1}, n_{i}]_{<}^{2})

By (2.39),

∑_{i=1}^{K} 𝔼 B((n_{i−1}, n_{i}]_{<}^{2}) = ∑_{i=1}^{K} 𝔼 B_{n_{i}−n_{i−1}} (5.3)

= ∑_{i=1}^{K} [\frac{1}{(2π)\sqrt{det Γ}}(n/K) log(n/K) + O(n/K)]

= \frac{1}{(2π)\sqrt{det Γ}}n log(n/K) + O(n)

With K > 1, the error term can be taken to be independent of t and {b_{n}}. Thus, by (2.39), there is constant log a > 0 independent of t and {b_{n}} such that

𝔼 B_{n} − ∑_{j=1}^{K} 𝔼 B((n_{i−1}, n_{i}]_{<}^{2}) (5.4)

≤ \frac{1}{(2π)\sqrt{det Γ}}n( log(t^{−1}b_{n}) + log a).

It is here that we use the condition that 𝔼 |S_{1}|^{2+δ} < ∞ for some δ > 0, needed for (2.39).

By first using Chebyshev’s inequality, then using (5.2), (5.4) and the independence of the B((n_{i−1}, n_{i}]_{<}^{2}), for any ϕ > 0,

ℙ{𝔼 B_{n} − B_{n} ≥ (2π)^{−1} det(Γ)^{−1/2}n log b_{n}} (5.5)

≤ exp \{ − ϕb_{n} log b_{n}\}𝔼 exp \{ − 2πϕ\sqrt{det Γ}\frac{b_{n}}{n} (B_{n} − 𝔼 B_{n})\}

≤ exp \{ϕb_{n}(log a − log t)\}(𝔼 exp \{ − 2πϕ\sqrt{det Γ}\frac{b_{n}}{n} (B_{[n/K]} − 𝔼 B_{[n/K]})\})^{K}

By [16, Theorem 1.2],

\sqrt{det Γ}\frac{b_{n}}{n} (B_{[n/K]} − 𝔼 B_{[n/K]}) ⟶^{d} γ_{t}, (n → ∞) (5.6)

where γ_{t} is the renormalized self-intersection local time of planar Brownian motion {W_{s}} up to time t. By Lemma 2.5 and the dominated convergence theorem,

𝔼 exp \{−2πϕ\sqrt{det Γ}\frac{b_{n}}{n} (B_{[n/K]}−𝔼 B_{[n/K]})\} ⟶ 𝔼 exp \{−2πϕtγ_{1}\}, (n → ∞)

(5.7) where we used the scaling γ_{t} =^{d} tγ_{1}.

Thus,

(5.8)  lim sup_{n→∞} b_{n}^{−1} log ℙ{𝔼 B_{n} − B_{n} ≥ (2π)^{−1} det(Γ)^{−1/2}n log b_{n}}

≤ ϕ(log a − log t) + \frac{1}{t} log 𝔼 exp \{ − 2πϕtγ_{1}\}

= ϕ log(aϕ) + \frac{1}{t} log 𝔼 exp \{ − (ϕt) log(θt) − 2π(ϕt)γ_{1}\}

By [2, p. 3233], the limit

(5.9) C ≡ lim_{t→∞} \frac{1}{t} log 𝔼 exp \{ − t log t − 2πtγ_{1}\}

exists. Hence

(5.10) lim sup_{n→∞} b_{n}^{−1} log ℙ{𝔼 B_{n} − B_{n} ≥ (2π)^{−1} det(Γ)^{−1/2}n log b_{n}}

≤ ϕ log(aϕ) + Cϕ.

Taking the minimizer ϕ = a^{−1}e^{−(1+C)} we have

(5.11)  lim sup_{n→∞} b_{n}^{−1} log ℙ{𝔼 B_{n} − B_{n} ≥ (2π)^{−1} det(Γ)^{−1/2}n log b_{n}}
≤ −a^{−1}e^{−(1+C)}.

This proves the upper bound for (5.1).

6 Theorem 1.2: Lower bound for 𝔼 B_{n} − B_{n}

In this section we complete the proof of Theorem 1.2 by proving the lower bound for (5.1).

Let B(x, r) be the ball of radius r centered at x. Let F_{k} = σ{X_{i} : i ≤ k}. Let us assume for simplicity that the covariance matrix for the random walk is the identity; routine modifications are all that are needed for the general case. We write Θ for (2π)^{−1} det (Γ)^{−1/2} = (2π)^{−1}. We write D(x, r) for the disc of radius r in ℤ^{2} centered at x.

Let K = [b_{n}] and L = n/K. Let us divide {1, 2, …, n} into K disjoint contiguous blocks, each of length strictly between L/2 and 3L/2. Denote the blocks J_{1}, …, J_{K}. Let v_{i} = #(J_{i}), w_{i} = ∑_{j=1}^{i} v_{j}. Let

B_{v_{i}}^{(i)} = ∑_{j,k∈J_{i},j<k} δ(S_{j}, S_{k}), A_{i} = ∑_{j∈J_{i−1},k∈J_{i}} δ(S_{j}, S_{k}). (6.1)

Define the following sets:

F_{i,1} = \{S_{w_{i}} ∈ D(i\sqrt{L}, \sqrt{L}/16)\},
F_{i,2} = \{S(J_{i}) ⊂ [(i − 1)\sqrt{L} − \sqrt{L}/8, i\sqrt{L} + \sqrt{L}/8] × [−\sqrt{L}/8, \sqrt{L}/8]\},
F_{i,3} = {B_{v_{i}}^{(i)} − 𝔼 B_{v_{i}}^{(i)} ≤ κ_{1}L},
F_{i,4} = \{∑_{j∈J_{i}} 1_{D(x,r\sqrt{L})}(S_{j}) ≤ κ_{2}rL for all x ∈ D(i\sqrt{L}, 3\sqrt{L}), 1/\sqrt{L} < r < 2\},
F_{i,5} = {A_{i} < κ_{3}L},

where κ_{1}, κ_{2}, κ_{3} are constants that will be chosen later and do not depend on K or L. Let C_{i} = F_{i,1} ∩ F_{i,2} ∩ F_{i,3} ∩ F_{i,4} ∩ F_{i,5} (6.2)

and E = ∩_{i=1}^{K}C_{i}. (6.3)

We want to show

ℙ(C_{i} | F_{w_{i−1}}) ≥ c_{1} > 0 (6.4)

on the event C_{1} ∩ ⋯ ∩ C_{i−1}. Once we have (6.4), then

ℙ(∩_{i=1}^{m}C_{i}) = 𝔼 (ℙ(C_{m} | F_{w_{m−1}}); ∩_{i=1}^{m−1}C_{i}) ≥ c_{1}ℙ(∩_{i=1}^{m−1}C_{i}), (6.5) and by induction

ℙ(E) = ℙ(∩_{i=1}^{K}C_{i}) ≥ c_{1}^{K} = e^{K log c_{1}} = e^{−c_{2}K}. (6.6)

On the set E, we see that S(B_{i}) ∩ S(B_{j}) = ∅ if |i − j| > 1. So we can write

B_{n} = ∑_{k=1}^{K}(B_{v_{k}}^{(k)} − 𝔼 B_{v_{k}}^{(k)}) + ∑_{k=1}^{K} 𝔼 B_{v_{k}}^{(k)} + ∑_{k=1}^{K} A_{k}. (6.7)

On the event E, each B_{v_{k}}^{(k)}−𝔼 B_{v_{k}}^{(k)} is bounded by κ_{1}L and each A_{k} is bounded by κ_{3}L. By (2.38), each 𝔼 B_{v_{k}}^{(k)} = Θv_{k} log v_{k} + O(L) = Θv_{k} log L + O(v_{k}). Therefore B_{n} ≤ κ_{1}KL + ΘKL log L + O(n) + κ_{3}KL, (6.8)

and using (2.38) again,

(6.9)  𝔼 B_{n} − B_{n} ≥ Θn log n − c_{3}n − Θn log(n/b_{n})
= Θn log b_{n} − c_{3}n

on the event E. We conclude that

ℙ(𝔼 B_{n} − B_{n} ≥ Θn log b_{n} − c_{3}n) ≥ e^{−c_{2}b_{n}}. (6.10)

We apply (6.10) with b_{n} replaced by b′_{n} = c_{4}b_{n}, where Θ log c_{4} = c_{3}. Then

(6.11) Θn log b′_{n} − c_{3}n = Θn log b_{n} + Θn log c_{4} − c_{3}n = Θn log b_{n}.

We then obtain

ℙ(𝔼 B_{n} − B_{n} ≥ Θn log b_{n}) = ℙ(𝔼 B_{n} − B_{n} ≥ Θn log b′_{n} − c_{3}n) ≥ e^{−c_{2}b′_{n}}, (6.12) which would complete the proof of the lower bound for (5.1), hence of Theorem 1.2.

So we need to prove (6.4). By scaling and the support theorem for Brownian motion (see [1, Theorem I.6.6]), if W_{t} is a planar Brownian motion and |x| ≤ \sqrt{L}/16, then

ℙ^{x} (W_{v_{i}} ∈ D(\sqrt{L}, \sqrt{L}/16) and (6.13)

\{W_{s}; 0 ≤ s ≤ v_{i}\} ⊂ [−\sqrt{L}/8, 9\sqrt{L}/8] × [−\sqrt{L}/8, \sqrt{L}/8]) > c_{5},

where c_{5} does not depend on L. Using Donsker’s invariance principle for random walks with finite second moments together with the Markov property, ℙ(F_{i,1} ∩ F_{i,2} | F_{w_{i−1}}) > c_{6}. (6.14) By Lemma 2.3, for L/2 ≤ ℓ ≤ 3L/2 ℙ(B_{ℓ} − 𝔼 B_{ℓ} > κ_{1}L) ≤ c_{6}/2 (6.15) if we choose κ_{1} large enough. Again using the Markov property, ℙ(F_{i,1} ∩ F_{i,2} ∩ F_{i,3} | F_{w_{i−1}}) > c_{6}/2. (6.16)

Now let us look at F_{i,4}. By [17, p. 75], ℙ(S_{j} = y) ≤ c_{7}/j with c_{7} independent of y ∈ ℤ^{2} so that

ℙ(S_{j} ∈ D(x, r\sqrt{L})) = ∑_{y∈D(x,r\sqrt{L})} ℙ(S_{j} = y) ≤ \frac{c_{8}r^{2}L}{j}. (6.17)

Therefore

(6.18) 𝔼 ∑_{j∈J_{1}} 1_{D(x,r\sqrt{L})}(S_{j}) ≤ ∑_{j=1}^{[2L]} ℙ(S_{j} ∈ D(x, r\sqrt{L}))
≤ r^{2}L + ∑_{j=r^{2}L}^{[2L]} \frac{c_{9}r^{2}L}{j}
≤ r^{2}L + c_{10}Lr^{2} log(1/r) ≤ c_{11}Lr^{2} log(1/r)

if 1/\sqrt{L} ≤ r ≤ 2. Let C_{m} = ∑_{j<m} 1_{D(x,r\sqrt{L})}(S_{j}) for m ≤ [2L] + 1 and let C_{m} = C_{[2L]+1} for m > L. By the Markov property and independence, 𝔼 [C_{∞} − C_{m} | F_{m}] ≤ 1 + 𝔼 [C_{∞} − C_{m+1} | F_{m}] (6.19)

≤ 1 + 𝔼^{S_{m}}C_{∞} ≤ c_{12}Lr^{2} log(1/r).

By [1, Theorem I.6.11], we have

𝔼 exp (c_{13} \frac{C_{[2L]+1}}{c_{12}Lr^{2} log(1/r)}) ≤ c_{14} (6.20)

with c_{13}, c_{14} independent of L or r. We conclude that for V > 0

ℙ( ∑_{j∈J_{1}} 1_{D(x,r\sqrt{L})}(S_{j}) > V Lr^{2} log(1/r)) ≤ c_{15}e^{−c_{16}V} . (6.21)

Suppose 2^{−s} ≤ r < 2^{−s+1} for some s ≥ 0. If x ∈ D(0, 3\sqrt{L}), then each point in the disc D(x, r\sqrt{L}) will be contained in D(x_{i}, 2^{−s+3}\sqrt{L}) for some x_{i}, where each coordinate of x_{i} is an integer multiple of 2^{−s−2}\sqrt{L}. There are at most c_{17}2^{2s} such balls, and Lr^{2} log(1/r) ≤ c_{18}2^{s/2}Lr, so for V > 0

ℙ( sup_{x∈D(0,3\sqrt{L}),2^{−s}≤r<2^{−s+1}} ∑_{j∈J_{1}} 1_{D(x,r\sqrt{L})}(S_{j}) > V rL) ≤ c_{19}2^{2s}e^{−c_{20}V 2^{s/2}}. (6.22)

If we now sum over positive integers s and take κ_{2} large enough, we see that ℙ(F_{1,4}^{c}) ≤ c_{6}/4. (6.23) By the Markov property, we then obtain ℙ(F_{i,1} ∩ F_{i,2} ∩ F_{i,3} ∩ F_{i,4} | F_{w_{i−1}}) > c_{6}/4. (6.24)

Finally, we examine F_{i,5}. We will show ℙ(F_{i,5}^{c} | F_{w_{i−1}}) ≤ c_{6}/8 (6.25) on the set ∩_{j=1}^{i−1}C_{j} if we take κ_{3} large enough. By the Markov property, it suffices to show

ℙ( ∑_{j=1}^{[2L]} 1_{(S_{j}∈G)} ≥ κ_{3}L) ≤ c_{6}/8 (6.26)

whenever G ∈ ℤ^{2} is a fixed nonrandom set consisting of [2L] points satisfying the property that #(G ∩ D(x, r\sqrt{L})) ≤ κ_{2}rL, x ∈ D(0, 3\sqrt{L}), 1/\sqrt{L} ≤ r ≤ 2. (6.27) We compute the expectation of

∑_{j=1}^{[2L]} 1_{(S_{j}∈G∩(D(0,2^{−k}\sqrt{L})\D(0,2^{−k+1}\sqrt{L})))}. (6.28)

When j ≤ 2^{−2k}L, then the fact that the random walk has finite second moments implies that the probability that |S_{j}| exceeds 2^{−k+1}\sqrt{L} is bounded by c_{21}j/(2^{−2k+2}L). When j > 2^{−2k}L, we use [17, p. 75], and obtain

ℙ(S_{j} ∈ G ∩ (D(0, 2^{−k}\sqrt{L}) ≤ c_{22} \frac{κ_{2}2^{−k}L}{j}. (6.29)

So

𝔼 ∑_{j=1}^{[2L]} 1_{G}(S_{j}) (6.30)

≤ ∑_{k} ∑_{[2L]≥j>2^{−2k}L} c_{22} \frac{κ_{2}2^{−k}L}{j} + ∑_{k} ∑_{j≤2^{−2k}L} c_{21} \frac{j}{2^{−2k+2}L}

≤ ∑_{k}(c_{23}κ_{2}k2^{−k}L + c_{24}2^{−2k}L) ≤ c_{25}L.

So if take κ_{3} large enough, we obtain (6.26).

This completes the proof of (6.4), hence of Theorem 1.2.

7 Laws of the iterated logarithm

7.1 Proof of the LIL for B_{n} − 𝔼 B_{n}

First, let S_{j}, S′_{j} be two independent copies of our random walk. Let

ℓ(n, x) = ∑_{i=1}^{n} δ(S_{i} , x), ℓ′(n, x) = ∑_{i=1}^{n} δ(S′_{i} , x) (7.1)

and note that

I_{k,n} = ∑_{i=1}^{k} ∑_{j=1}^{n} δ(S_{i}, S′_{j}) = ∑_{x∈ℤ^{2}} ℓ(k, x)ℓ′(n, x). (7.2)

Lemma 7.1 There exist constants c_{1}, c_{2} such that

ℙ(I_{k,n} > λ\sqrt{kn}) ≤ c_{1}e^{−c_{2}λ}. (7.3)

Proof. Clearly

(I_{k,n})^{m} = ∑_{x_{1}∈ℤ^{2}} ⋯ ∑_{x_{m}∈ℤ^{2}} ( ∏_{i=1}^{m} ℓ(k, x_{i}))( ∏_{i=1}^{m} ℓ′(n, x_{i})) (7.4)

Using the independence of S and S′,

𝔼 ((I_{k,n})^{m}) = ∑_{x_{1}∈ℤ^{2}} ⋯ ∑_{x_{m}∈ℤ^{2}} 𝔼 ( ∏_{i=1}^{m} ℓ(k, x_{i}))𝔼 ( ∏_{i=1}^{m} ℓ′(n, x_{i})). (7.5)

By Cauchy-Schwarz, this is less than

[ ∑_{x_{1}∈ℤ^{2}} ⋯ ∑_{x_{m}∈ℤ^{2}} (𝔼 ( ∏_{i=1}^{m} ℓ(k, x_{i})))^{2}]^{1/2} (7.6)

[ ∑_{x_{1}∈ℤ^{2}} ⋯ ∑_{x_{m}∈ℤ^{2}} (𝔼 ( ∏_{i=1}^{m} ℓ′(n, x_{i})))^{2}]^{1/2}

=: J_{1}^{1/2}J_{2}^{1/2}.

We can rewrite

J_{1} = ∑_{x_{1}∈ℤ^{2}} ⋯ ∑_{x_{m}∈ℤ^{2}} 𝔼 ( ∏_{i=1}^{m} ℓ(k, x_{i}))𝔼 ( ∏_{i=1}^{m} ℓ′(k, x_{i})) = 𝔼 ((I_{k})^{m}) , (7.7)

and similarly J_{2} = 𝔼 ((I_{n})^{m}).

Therefore, 𝔼 exp(aI_{k,n}/\sqrt{kn}) (7.8)

= ∑_{m=0}^{∞} \frac{a^{m}}{k^{m/2}n^{m/2}m!}𝔼 ((I_{k,n})^{m})
≤ ∑_{m} \frac{a^{m}}{k^{m/2}n^{m/2}m!}(𝔼 ((I_{k})^{m}))^{1/2}(𝔼 ((I_{n})^{m}))^{1/2}
≤ ( ∑ \frac{a^{m}}{m!}𝔼 (\frac{I_{k}}{k} )^{m})^{1/2}( ∑ \frac{a^{m}}{m!}𝔼 (\frac{I_{n}}{n} )^{m})^{1/2}
≤ (𝔼 e^{aI_{k}/k})^{1/2}(𝔼 e^{aI_{n}/n})^{1/2}.

By Lemma 2.2 this can be bounded independently of k and n if a is taken small, and our result follows.

We are now ready to prove the upper bound for the LIL for B_{n} − 𝔼 B_{n}. Write Ξ for \sqrt{det Γ} κ(2, 2)^{−4}. Recall that for any integrable random variable Z we let \overline{Z} denote Z − 𝔼 Z. Let ε > 0 and let q > 1 be chosen later. Our first goal is to get an upper bound on

ℙ( max_{n/2≤k≤n} \overline{B}_{k} > (1 + ε)Ξ^{−1}n log log n).

Let m_{0} = 2^{N}, where N will be chosen later to depend only on ε and n. Let A_{0} be the integers of the form n − km_{0} that are contained in {n/4, …, n}. For each i let A_{i} be the set of integers of the form n − km_{0}2^{−i} that are contained in {n/4, …, n}. Given an integer k, let k_{j} be the largest element of A_{j} that is less than or equal to k. For any k ∈ {n/2, …, n}, we can write

\overline{B}_{k} = \overline{B}_{k_{0}} + (\overline{B}_{k_{1}} − \overline{B}_{k_{0}}) + ⋯ + (\overline{B}_{k_{N}} − \overline{B}_{k_{N−1}}). (7.9)

If \overline{B}_{k} ≥ (1 + ε)Ξ^{−1}n log log n for some n/2 ≤ k ≤ n, then either

(a) \overline{B}_{k_{0}} ≥ (1 + \frac{ε}{2})Ξ^{−1}n log log n for some k_{0} ∈ A_{0}; or else

(b) for some i ≥ 1 and some pair of consecutive elements k_{i}, k′_{i} ∈ A_{i}, we have

\overline{B}_{k′_{i}} − \overline{B}_{k_{i}} ≥ \frac{ε}{40i^{2}} Ξ^{−1}n log log n. (7.10)

For each k_{0}, using Theorem 1.1 and the fact that k_{0} ≥ n/4, the probability in (a) is bounded by

(7.11) exp(−(1 + \frac{ε}{4}) log log k_{0}) ≤ c_{1}(log n)^{−(1+ \frac{ε}{4} )}.

There are at most n/m_{0} elements of A_{0}, so the probability in (a) is bounded by

\frac{n}{m_{0}} \frac{c_{1}}{(log n)^{1+ \frac{ε}{4}}} . (7.12)

Now let us examine the probability in (b). Fix i for the moment. Any two consecutive elements of A_{i} are 2^{−i}m_{0} apart. Recalling the notation (2.16) we can write

\overline{B}_{k} − \overline{B}_{j} = \overline{B}([j + 1, k]_{<}^{2}) + \overline{B}([1, j] × [j + 1, k]), (7.13)

So

ℙ(\overline{B}_{k} − \overline{B}_{j} ≥ \frac{ε}{40i^{2}} Ξ^{−1}n log log n) ≤ ℙ(\overline{B}([j + 1, k]_{<}^{2}) ≥ \frac{ε}{80i^{2}} Ξ^{−1}n log log n)
+ℙ (B([1, j] × [j + 1, k]) ≥ \frac{ε}{80i^{2}} Ξ^{−1}n log log n) . (7.14)

We bound the first term on the right by Lemma 2.3, and get the bound

(7.15) exp ( − \frac{cε}{80i^{2}} \frac{n log log n}{2^{−i}m_{0}}) ≤ exp ( − \frac{cε}{80i^{2}} 2^{i}(n/m_{0}) log log n)

if j and k are consecutive elements of A_{i}. Note that B([1, j] × [j + 1, k]) is equal in law to I_{j−1,k−j}. Using Lemma 7.1, we bound the second term on the right hand side of (7.14) by

c_{1} exp ( − c_{2} \frac{ε}{80i^{2}} \frac{n log log n}{\sqrt{2^{−i}m_{0}}\sqrt{j}} )

(7.16) ≤ c_{1} exp ( − c_{2} \frac{ε}{80i^{2}} 2^{i/2}(n/m_{0})^{1/2} log log n).

The number of pairs of consecutive elements of A_{i} is less than 2^{i+1}(n/m_{0}). So if we add (7.15) and (7.16) and multiply by the number of pairs, the probability of (b) occurring for a fixed i is bounded by

(7.17) c_{3} \frac{n}{m_{0}} 2^{i} exp ( − c_{4}2^{i/2}(n/m_{0})^{1/2} log log n/(80i^{2})).

If we now sum over i ≥ 1, we bound the probability in (b) by

(7.18) c_{5} \frac{n}{m_{0}} exp ( − c_{6}(n/m_{0})^{1/2} log log n).

We now choose m_{0} to be the largest power of 2 so that c_{6}(n/m_{0})^{1/2} > 2; recall n is big.

Let us use this value of m_{0} and combine (7.12) and (7.18). Let n_{ℓ} = q^{ℓ} and

(7.19) C_{ℓ} = \{ max_{n_{ℓ−1}≤k≤n_{ℓ}} \overline{B}_{k} ≥ (1 + ε)Ξ^{−1}n_{ℓ} log log n_{ℓ}\}.

By our estimates, ℙ(C_{ℓ}) is summable, so for ℓ large, by Borel-Cantelli we have

(7.20)  max_{n_{ℓ−1}≤k≤n_{ℓ}} \overline{B}_{k} ≤ (1 + ε)Ξ^{−1}n_{ℓ} log log n_{ℓ}.

By taking q sufficiently close to 1, this implies that for k large we have \overline{B}_{k} ≤ (1 + 2ε)Ξ^{−1}k log log k. Since ε is arbitrary, we have our upper bound.

The lower bound for the first LIL is easier. Let δ > 0 be small and let n_{ℓ} = [e^{ℓ^{1+δ}}]. Let

D_{ℓ} = \{\overline{B}([n_{ℓ−1} + 1, n_{ℓ}]_{<}^{2}) ≥ (1 − ε)Ξ^{−1}n_{ℓ} log log n_{ℓ}\}. (7.21)

Using Theorem 1.1, and the fact that n_{ℓ}/(n_{ℓ} −n_{ℓ−1}) is of order 1, we see that ∑_{ℓ} ℙ(D_{ℓ}) = ∞ if δ < ϵ/(1−ϵ). The D_{ℓ} are independent, so by Borel-Cantelli

\overline{B}([n_{ℓ−1} + 1, n_{ℓ}]_{<}^{2}) ≥ (1 − ε)Ξ^{−1}n_{ℓ} log log n_{ℓ} (7.22)

infinitely often with probability one. Note that as in (7.13) we can write

\overline{B}_{n_{ℓ}} = \overline{B}([n_{ℓ−1} + 1, n_{ℓ}]_{<}^{2}) + \overline{B}_{n_{ℓ−1}} + \overline{B}([1, n_{ℓ−1}] × [n_{ℓ−1} + 1, n_{ℓ}]). (7.23)

By the upper bound,

lim sup_{ℓ→∞} \frac{\overline{B}_{n_{ℓ−1}}}{n_{ℓ−1} log log n_{ℓ−1}} ≤ Ξ^{−1}

almost surely, which implies

(7.24)  lim sup_{ℓ→∞} \frac{\overline{B}_{n_{ℓ−1}}}{n_{ℓ} log log n_{ℓ}} = 0.

Since B([1, n_{ℓ−1}] × [n_{ℓ−1} + 1, n_{ℓ}]) ≥ 0 and by (2.5)

𝔼 B([1, n_{ℓ−1}] × [n_{ℓ−1} + 1, n_{ℓ}]) ≤ c_{1}\sqrt{n_{ℓ−1}}\sqrt{n_{ℓ} − n_{ℓ−1}} = o(n_{ℓ} log log n_{ℓ}), (7.25) using (7.22)-(7.25) yields the lower bound.

7.2 LIL for 𝔼 B_{n} − B_{n}

Let Δ = 2π\sqrt{det Γ}. Let us write J_{n} = 𝔼 B_{n} − B_{n}.

First we do the upper bound. Let m_{0}, A_{i}, and k_{j} be as in the previous subsection. We write, for n/2 ≤ k ≤ n,

J_{k} = J_{k_{0}} + (J_{k_{1}} − J_{k_{0}}) + ⋯ + (J_{k_{N}} − J_{k_{N−1}}). (7.26)

If max_{n/2≤k≤n} J_{k} ≥ (1 + ε)Δ^{−1}n log log log n, then either

(a) J_{k_{0}} ≥ (1 + \frac{ε}{2})Δ^{−1}n log log log n for some k_{0} ∈ A_{0}, or else

(b) for some i ≥ 1 and k_{i}, k′_{i} consecutive elements of A_{i} we have

(7.27) J_{k′_{i}} − J_{k_{i}} ≥ \frac{ε}{40i^{2}} Δ^{−1}n log log log n.

There are at most n/m_{0} elements of A_{0}. Using Theorem 1.2, the probability of (a) is bounded by

c_{1} \frac{n}{m_{0}} e^{−(1+ \frac{ε}{4} ) log log n}. (7.28)

To estimate the probability in (b), suppose j and k are consecutive elements of A_{i}. There are at most 2^{i+1}(n/m_{0}) such pairs. We have

(7.29) J_{k} − J_{j} = −\overline{B}([j + 1, k]_{<}^{2}) − \overline{B}([1, j] × [j + 1, k])
≤ −\overline{B}([j + 1, k]_{<}^{2}) + 𝔼 B([1, j] × [j + 1, k])
≤ −\overline{B}([j + 1, k]_{<}^{2}) + c_{2}\sqrt{j}\sqrt{k − j},

as in the previous subsection. Provided n is large enough, c_{2}\sqrt{j}\sqrt{k − j} = c_{2}\sqrt{j}\sqrt{2^{−i}m_{0}} will be less than \frac{ε}{80i^{2}} Δ^{−1}n log log log n for all i. So in order for J_{k} − J_{j} to be larger than \frac{ε}{40i^{2}} Δ^{−1}n log log log n, we must have −\overline{B}([j + 1, k]_{<}^{2}) larger than \frac{ε}{80i^{2}} Δ^{−1}n log log log n. We use Theorem 1.2 to bound this. Then multiplying by the number of pairs and summing over i, the probability is (b) is bounded by

(7.30)  ∑_{i=1}^{∞} 2^{i+1} \frac{n}{m_{0}} e^{− \frac{ε}{80i^{2}} _{\frac{n}{2^{−i}m_{0}}} log log n} ≤ c_{3} \frac{n}{m_{0}} e^{−c_{4}(n/m_{0}) log log n}.

We choose m_{0} to be the largest possible power of 2 such that c_{4}(n/m_{0}) > 2.

Combining (7.28) and (7.30), we see that if we set q > 1 close to 1, n_{ℓ} = [q^{ℓ}], and

(7.31) E_{ℓ} = { max_{n_{ℓ}/2≤k≤n_{ℓ}} J_{k} ≥ (1 + ε)Δ^{−1}n_{ℓ} log log log n_{ℓ}},

then ∑_{ℓ} ℙ(E_{ℓ}) is finite. So by Borel-Cantelli, the event E_{ℓ} happens for a last time, almost surely. Exactly as in the previous subsection, taking q close enough to 1 and using the fact that ε is arbitrary leads to the upper bound.

The proof of the lower bound is fairly similar to the previous subsection. Let n_{ℓ} = [e^{ℓ^{1+δ}}]. Theorem 1.2 and Borel-Cantelli tell us that F_{ℓ} will happen infinitely often, where

(7.32) F_{ℓ} = \{−\overline{B}([n_{ℓ−1} + 1, n_{ℓ}]_{<}^{2}) ≥ (1 − ε)Δ^{−1}n_{ℓ} log log log n_{ℓ}\}.

J_{n_{ℓ}} ≥ −\overline{B}([n_{ℓ−1} + 1, n_{ℓ}]_{<}^{2}) + J_{n_{ℓ−1}} − A(1, n_{ℓ−1}; n_{ℓ−1}, n_{ℓ}). (7.33)

By the upper bound,

(7.34) J_{n_{ℓ−1}} = O(n_{ℓ−1} log log log n_{ℓ−1}) = o(n_{ℓ} log log log n_{ℓ}).

By Lemma 7.1,

ℙ(B([1, n_{ℓ−1}]×[n_{ℓ−1}+1, n_{ℓ}]) ≥ εn_{ℓ} log log log n_{ℓ}) ≤ c_{1} exp (−c_{2} \frac{εn_{ℓ} log log log n_{ℓ}}{\sqrt{n_{ℓ−1}}\sqrt{n_{ℓ} − n_{ℓ−1}}} ).

(7.35) This is summable in ℓ, so

(7.36)  lim sup_{ℓ→∞} \frac{B([1, n_{ℓ−1}] × [n_{ℓ−1} + 1, n_{ℓ}])}{n_{ℓ} log log log n_{ℓ}} ≤ ε

almost surely. This is true for every ε, so the limsup is 0. Combining this with (7.34) and substituting in (7.33) completes the proof.

Acknowledgment. We thank Peter Mörters for pointing out an error in the proof of Lemma 2.2 in a previous version.

References

1. R.F. Bass, Probabilistic Techniques in Analysis, Springer, New York, 1995

2. R.F. Bass and X. Chen, Self-intersection local time: critical exponent and laws of the iterated logarithm. Ann. Probab.32 (2004) 3221-3247

3. R.F. Bass, X. Chen and J. Rosen, Large deviations for renormalized selfintersection local times of stable processes, Ann. Probab., 33 (2005), 984–1014.

4. R.F. Bass and J. Rosen, An almost sure invariance principle for the range of planar random walks, Ann. Probab., 33 (2005), 1856-1885.

5. R.F. Bass and J. Rosen, An almost sure invariance principle for renormalized intersection local times, Electron. J. Probab., 10, (2005), number 4, 124-164.

6. E. Bolthausen and U. Schmock. On self-attracting d-dimensional random walks. Ann. Probab. 25 (1997) 531–572.

7. D.C. Brydges and G. Slade, The diffusive phase of a model of selfinteracting walks. Probab. Theory Rel. Fields 103 (1995), 285–315.

8. X. Chen, Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks, Ann. Probab. 32 (2004) 3248-3300.

9. Chen, X. and Li, W. Large and moderate deviations for intersection local times. Probab. Theor. Rel. Fields. 128 (2004) 213-254.

10. X. Chen and J. Rosen, Exponential asymptotics for intersection local times of stable processes and random walks, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 901-928.

11. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. (2nd ed.) Springer, New York, 1998.

12. G. Lawler, Lecture notes on random walks, to appear.

13. J.-F. Le Gall, Propriétés d’intersection des marches aléatoires, I. Convergence vers le temps local d’intersection. Comm. Math. Phys., 104 (1986), 471–507.

14. J.-F. Le Gall, Wiener sausage and self intersection local times. J. Funct. Anal. 88 (1990) 299–341.

15. U. Mansmann, The free energy of the Dirac polaron, an explicit solution. Stochastics and Stochastic Reports 34 (1991) 93-125.

16. J. Rosen, Random walks and intersection local times, Ann. Probab. 18 (1990) 959-977.

17. F. Spitzer, Principles of Random Walk, Van Nostrand, Princeton, New Jersey (1964).

18. S. R. S. Varadhan. Appendix to Euclidean quantum field theory by K. Symanzyk. In R. Jost, editor, Local Quantum Theory. Academic Press, Reading, MA, 1969.

19. M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87 (1982/83) 567–576.