advertisement

J Stat Phys (2008) 133: 255–269 DOI 10.1007/s10955-008-9619-7 A Class of Weakly Self-Avoiding Walks Peter Mörters · Nadia Sidorova Received: 5 February 2008 / Accepted: 14 August 2008 / Published online: 5 September 2008 © Springer Science+Business Media, LLC 2008 Abstract We define a class of weakly self-avoiding walks on the integers by conditioning a simple random walk of length n to have a p-fold self-intersection local time smaller than nβ , where 1 < β < (p + 1)/2. We show that the conditioned paths grow of order nα , where α = (p − β)/(p − 1), and also prove a coarse large deviation principle for the order of growth. Keywords Random walk · Polymer measure · Self-intersection local time · Large deviation · Law of large numbers 1 Introduction and Main Results Weakly self-avoiding walks are defined by multiplying the distribution of a simple symmetric random walk path (Si : 1 ≤ i ≤ n) on Zd with a density which is decreasing in the p-fold intersection local time n (p) = 1{Si1 = · · · = Sip } 0≤i1 ,...,ip ≤n of the walk, where p ∈ N and p ≥ 2. In the classical Domb-Joyce model this density is given as 1 1 exp − n (2) , Zn T where T > 0 is a temperature parameter and Zn is a normalising factor. This model is wellunderstood in the one-dimensional case, where the resulting polymers grow like ∼c n and P. Mörters Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK e-mail: maspm@bath.ac.uk N. Sidorova () Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK e-mail: n.sidorova@ucl.ac.uk 256 P. Mörters, N. Sidorova laws of large numbers, central limit theorems and large deviation results are established, see [4] for a survey. A natural alternative to the Domb-Joyce model is to choose densities 1{n (p) < bn } P{n (p) < bn } where bn grows slower than En (p). In this paper we study this model in dimension d = 1. At a first glance, this may look harder to analyse than the Domb-Joyce model, but it turns out that interesting behaviour kicks in on a coarser scale and we are helped by the fact that on this scale we can understand, to a certain extent, what kind of behaviour of the walk realises the event {n (p) < bn } with maximal probability. As bn varies between the expectation and the minimum √ of n (p), we see weakly self-avoiding walks with typical growth of any order between n and n. To formulate our results precisely, we define the local time of the random walk at z ∈ Z by n (z) = n 1{Si = z}, i=0 which is exactly the number of visits to z until time n. Note that, for any integer p > 1, the p-fold self-intersection local time is n (p) = pn (z). (1) z∈Z p+1 Hence, n ≤ n (p) ≤ np and it is not hard to show (see Lemma 3) that En (p) n 2 (which means that the ratio of the two sides is bounded away from zero and infinity). We define S̄n = max |Si | 0≤i≤n and state a limit theorem for S̄n under the conditional probability. 2 Theorem 1 (Law of large numbers) Let εn ↓ 0 such that εnp−1 n → ∞. Then there exist constants 0 < c < C < ∞ such that √ − 1 √ − 1 P c nεn p−1 ≤ S̄n ≤ C nεn p−1 n (p) ≤ εn En (p) → 1. Remarks 2 • The condition εnp−1 n → ∞ ensures that εn En (p) grows faster than n, which is a strict lower bound for n (p). Hence the conditioning event has positive probability. • An important step in the proof of Theorem 1 is to analyse the asymptotic behaviour of the probability of the conditioning event. This result will be formulated as Theorem 3 in the next section. • It would√be interesting to see if there exists a constant c∗ > 0 such that, in probability, −1/(p−1) , but the methods of this paper do not allow to show this. S̄n ∼ c∗ nεn • In Theorem 1, we assume that p is an integer, but it is plausible that the same statement is true for any p > 1 if we use (1) as a definition of n (p). A Class of Weakly Self-Avoiding Walks 257 As a particular case of the law of large numbers, if we condition the random walk on the , we obtain a typical growth rate of event {n (p) ≤ nβ } for some 1 < β < p+1 2 p−β log S̄n ∼ log n p−1 for the weakly self-avoiding walk. Interestingly, it turns out that the probability of deviations from this behaviour decay with a speed dependent on the size of the deviation. The following coarse large deviation principle describes this behaviour. For its formulation we define log(2) (x) = log | log x| for all 0 < x < 1, as well as log(2) (1) = −∞ and log(2) (0) = ∞. Theorem 2 (Large deviation principle) Suppose 1 < β < p+1 . Then, for any 1 > a > 2 we have log S̄n 1 (2) β log P > a n (p) ≤ n = 2a − 1, lim n↑∞ log n log n and, for any 0 < a < p−β p−1 p−β , p−1 and c > 0 small enough P S̄n < cna n (p) ≤ nβ = 0. for all sufficiently large n. Remark In fact, we prove a stronger (but more technical) result, see Proposition 1 for the precise formulation. In particular, our result shows that the typical growth rate for any walk , while larger growth rates are possible but occur with subexponenwith n (p) ≤ nβ is p−β p−1 tially decaying probability, and smaller rates are not possible at all. Coming back to the beginning of this introduction, we see that, if p = 2, the Domb-Joyce model can be interpreted as a limiting case of our models when β ↓ 1, in which case both the self-intersection local time n (2), and the growth of the polymer are linear in n. Hence the self-avoidance in the Domb-Joyce model is significantly stronger than in our models. 2 Lower Deviations for Self-Intersection Local Times As an important step in the proof, we compute the asymptotics for the probability of lower deviations of n (p) from its expectation. This is also of independent interest. 2 Theorem 3 Let εn > 0 be such that εn → 0 and εnp−1 n → ∞. Then, 2 − p−1 − log P(n (p) ≤ εn En (p)) εn . Remark An analogous lower deviation regime also exists for self-intersections of planar random walks, see [1, Theorem 1.2]. Results for the lower deviations of self-intersection local times of one-dimensional Brownian motion are discussed in [3]. In the following, we fix p > 1 and abbreviate n = n (p). We assume that εn > 0 is such that εn → 0 2 and εnp−1 n → ∞. 258 P. Mörters, N. Sidorova We prepare the proof of Theorem 3 with two easy lemmas dealing with the curtailed Green function n P(Si = z) for z ∈ Z. Gn (z) = En (z) = i=0 The first lemma is elementary and is included for the sake of self-containment. Lemma 1 The curtailed Green function has the following properties: (1) Gn (z) = n; z∈Z 2 3 (2) n2 ; z∈Z Gn (z) √ √ (3) Gn (0) n and Gn (1) n. Proof Formula (1) is trivial, and (2) is proved in [2, (3.4)]. To show (3), we use the Stirling formula √ n! = exp n log n − n + log 2πn + δn , where δn → 0. We have Gn (0) = n n/2 P(Si = 0) = i=0 n/2 = n/2 2i (2i)! 2−2i 2−2i = 2 (i!) i i=0 i=0 √ exp{2i log(2i) − 2i + log 4πi i=0 √ + δ2i − 2i log i + 2i − 2 log 2πi − 2δi − 2i log 2} n/2 = n/2 1 √ √ √ exp{log 4πi + δ2i − 2 log 2πi − 2δi } i− 2 n i=0 i=0 and n/2 2i − 1 2i 2−2i+1 = 2−2i − 1. i i i=0 i=1 i=0 √ This implies Gn (0) − 1 ≤ Gn (1) ≤ Gn+1 (0) − 1, which gives Gn (1) n. Gn (1) = n n/2 P(Si = 1) = Lemma 2 For every q ∈ N, we have q−1 Gn/q (z)Gn/q (0) ≤ Eqn (z) ≤ q!Gn (z) Gnq−1 (0). Proof We first prove the lower bound. We have Eqn (z) = E n i1 =0 ≥ ··· q n 1{Sij = z} iq =0 j =1 0≤i1 ≤ qn i1 ≤i2 ≤i1 + qn ··· iq−1 ≤iq ≤iq−1 + qn P(Si1 = z) q j =2 P(Sij −ij −1 = 0) A Class of Weakly Self-Avoiding Walks = = ··· 0≤i1 ≤ qn n/q 259 P(Si1 = z) q P(Sij = 0) j =2 0≤iq ≤ qn n/q q−1 q−1 P(Si = z) P(Si = 0) = Gn/q (z)Gn/q (0). i=0 i=0 For the upper bound, we get Eqn (z) ≤ q! P(Si1 = · · · = Siq = z) 0≤i1 ≤···≤iq ≤n = q! P(Si1 = z) 0≤i1 ≤···≤iq ≤n ≤ q! q P(Sij −ij −1 = 0) j =2 P(Si1 = z) 0≤i1 ,...,iq ≤n q P(Sij = 0) = q!Gn (z)Gq−1 n (0), j =2 which completes the proof. In the next two lemmas, we study the asymptotic behaviour of the first two moments of n . Lemma 3 En n p+1 2 . Proof Applying Lemmas 1 and 2 with q = p, we obtain p+1 p−1 En = Epn (z) ≥ Gn/p (0) Gn/p (z) n 2 , z∈Z z∈Z and En ≤ p!Gnp−1 (0) Gn (z) n p+1 2 , z∈Z which completes the proof. Lemma 4 There exists c > 0 such that E2n ≤ cnp+1 for all n. Proof We have E2n =E 2 1{Si1 = · · · = Sip } 0≤i1 ,...,ip ≤n = P(Si1 = · · · = Sip = z, Sj1 = · · · = Sjp = w) z,w∈Z 0≤i1 ,...,ip ≤n 0≤j1 ,...,jp ≤n ≤ (2p)! 2p z,w∈Z 0≤l1 ≤···≤l2p ≤n A⊂{1,...,2p} i=1 |A|=p P(Sli −li−1 = aiz,w (A)), (2) 260 P. Mörters, N. Sidorova where l0 = 0 and, for all 2 ≤ i ≤ 2p, a1z,w (A) ⎧ z − w, ⎪ ⎪ ⎨ w − z, z,w ai (A) = ⎪ ⎪ ⎩0, if 1 ∈ A, if 1 ∈ / A, z, = w, if i ∈ A, i − 1 ∈ / A, if i ∈ / A, i − 1 ∈ A, if i, i − 1 ∈ A or if i, i − 1 ∈ / A. z,w = Observe that if 1 ∈ A then a1z,w = z and there is an index i(A) ∈ {2, . . . , 2p} such that ai(A) z,w w − z. On the other hand, if 1 ∈ / A then a1 = w and there is an index i(A) ∈ {2, . . . , 2p} z,w such that ai(A) = z − w. Further, denote ki = li − li−1 and notice that P(Ski = aiz,w (A)) ≤ P(Ski ∈ {0, 1}). We will use this estimate to bound all the factors in the product in (2) except those numbered 1 and i(A). This gives E2n ≤ (2p)! 2p P(Ski = aiz,w (A)) z,w∈Z 0≤k1 ,...,k2p ≤n A⊂{1,...,2p} i=1 ≤ (2p)! |A|=p P(Ski = aiz,w (A)) / z,w∈Z 0≤k1 ,...,k2p ≤n A⊂{1,...,2p} i ∈{1,i(A)} |A|=p × 1{1 ∈ A}P(Sk1 = z)P(Ski(A) = w − z) + 1{1 ∈ / A}P(Sk1 = w)P(Ski(A) = z − w) . Rearranging the terms, we obtain E2n ≤ (2p)! z,w∈Z 0≤k1 ,...,k2p 2p 1 2p P(Ski ∈ {0, 1}) 2 p i=3 ≤n × P(Sk1 = z)P(Sk2 = w − z) + P(Sk1 = w)P(Sk2 = z − w) 2p 2 ≤ (2p)! P(Ski ∈ {0, 1}) × 0≤k3 ,...,k2p ≤n i=3 P(Sk1 = z)P(Sk2 = w) 0≤k1 ,k2 ≤n z,w∈Z = (2p)! n 2 2 n 2p−2 P(Sk ∈ {0, 1}) k=0 = (2p)!2 n2 [Gn (0) + Gn (1)]2p−2 np+1 , where the last line follows from Lemma 1. A Class of Weakly Self-Avoiding Walks 261 We fix some small number 0 < η < 1. Define Si Sn Bnη = sup √ − η < 1, 1 < √ < 1 + η . n n 0≤i≤n Lemma 5 P(Bnη ) 1. Proof By Donsker’s invariance principle there is a standard one-dimensional Brownian motion (Bt )0≤t≤1 defined on the same probability space as (Si )i∈N0 such that Si P sup √ − Bi/n > δ → 0, n 0≤i≤n (3) for any δ > 0. Further, Si Si Sn P sup √ − η < 1, 1 < √ < 1 + η sup √ − Bi/n ≤ δ n n n 0≤i≤n 0≤i≤n Si ≥ P sup |Bt − η| < 1 − δ, 1 + δ < B1 < 1 + η − δ sup √ − Bi/n ≤ δ n 0≤t≤1 0≤i≤n → P sup |Bt − η| < 1 − δ, 1 + δ < B1 < 1 + η + δ > 0, 0≤t≤1 for δ < min{1 − η, η/2}, which, together with (3), implies the statement. Abbreviate an = En and let (mn ) be the sequence of natural numbers such that 2mn ≤ n < 2mn +1 . Lemma 6 (Upper bound) There is a constant C > 0 such that 2 − p−1 log P(n ≤ εn an ) ≤ −Cεn . Proof Let (kn ) be the sequence of natural numbers such that 2 2 cεnp−1 n ≤ 2kn < 2cεnp−1 n, where the constant c will be specified later. Note that, by choice of εn , we have 2kn → ∞. To obtain an upper bound for P(n ≤ εn an ), we only count self-intersections occurring within 2mn −kn disjoint intervals of length 2kn . We fix n and, for each 1 ≤ j ≤ 2mn −kn , denote (n,j ) Si = S(j −1)2kn +i − S(j −1)2kn , 0 ≤ i ≤ 2 kn . They are simple symmetric random walks starting at zero, which are independent. Denote, for z ∈ Z, kn k (n,j ) 2kn (z) = 2n i=0 (n,j ) 1{Si = z} = j2 i=(j −1)2kn 1{Si = z + S(j −1)2kn }. 262 P. Mörters, N. Sidorova For each 1 ≤ j ≤ 2mn −kn , denote by kn kn j2 Ynj = j2 ··· i1 =(j −1)2kn 1{Si1 = · · · = Sip } ip =(j −1)2kn the number of p-fold self-intersections in the j -th interval. We have Ynj = kn j2 kn ··· z∈Z i1 =(j −1)2kn = j2 1{Si1 = · · · = Sip = z + S(j −1)2kn } ip =(j −1)2kn (n,j ) p 2kn (z) , z∈Z and by Lemma 3 we get EYnj = E2kn 2 kn (p+1) 2 . Notice that n ≥ n −kn 2m Ynj − 1 j =1 and so 2mn −kn 2mn −kn P(n ≤ εn an ) ≤ P Ynj ≤ εn an + n ∼ P Ynj ≤ εn an j =1 (4) j =1 (it is easy to see that the equivalence follows from n = o(εn an )). Using Markov’s inequality and the independence of Ynj for fixed n and different j , we obtain, for each s > 0, 2mn −kn 2mn −kn P Ynj ≤ εn an ≤ exp{sεn an }E exp −s Ynj j =1 j =1 = exp sεn an + 2mn −kn log Ee−s2kn . (5) It is easy to check that ex ≤ 1 + x + x 2 for all x < 0 and so Ee−s2kn ≤ 1 − sE2kn + s 2 E22kn . Using log(1 + x) ≤ x for all x > −1, we obtain log Ee−s2kn ≤ −sE2kn + s 2 E22kn . (6) Combining (4), (5), and (6), we get P(n ≤ εn an ) ≤ min exp −s 2mn −kn E2kn − εn an + s 2 2mn −kn E22kn . s>0 The optimal value of s is given by −1 s = 2mn −kn E2kn − εn an 2kn −mn −1 E22kn . (7) A Class of Weakly Self-Avoiding Walks 263 By Lemma 3 there are constants c1 , c2 > 0 such that E2kn ≥ c1 2 kn (p+1) 2 an = En ≤ c2 n and p+1 2 . By choice of the sequences mn and kn , we obtain 2mn −kn E2kn − εn an ≥ c1 2mn 2 ≥ εn n p+1 2 kn (p−1) 2 c p−1 2 − c2 εn n p+1 2 c1 2−1 − c2 > 0, (8) where the last inequality holds if we choose c large enough. Computing the corresponding value in (7), we obtain [2mn −kn E2kn − εn an ]2 . (9) P(n ≤ εn an ) ≤ exp − 2mn −kn +2 E22kn By Lemma 3 we have εn an εn n we also have p+1 2 and, using the choice of kn and mn and again Lemma 3 2mn −kn E2kn n2 kn (p−1) 2 εn n p+1 2 , which, together with (8), implies 2mn −kn E2kn − εn an εn n p+1 2 . (10) Further, by Lemma 4 there is a constant c3 such that 2p 2mn −kn +2 E22kn ≤ c3 n2kn p εnp−1 np+1 . (11) Finally, combining (9), (10), and (11), we obtain log P(n ≤ εn an ) ≤ − − 2 [2mn −kn E2kn − εn an ]2 ≤ −Cεn p−1 , 2 m −k +2 n n 2 E2kn for some C > 0. Lemma 7 (Lower bound) There exist constants c, C > 0 such that, for each gn satisfying gn ≥ c, and 1 1 − p−1 gn n− 2 εn −→ 0, one has 1 − 2 1 − log P n ≤ εn an , S̄n > gn n 2 εn p−1 ≥ −Cεn p−1 gn2 . Proof Let (kn ) be a sequence of even natural numbers such that 2kn < n and kn → ∞, which will be specified later. To prove the lower bound, we describe a strategy of a random walk, the probability of which is large enough to provide the required bound, which implies that the p-fold self intersection local time is small and the maximal displacement is large. For this purpose, we divide the time interval into 2mn −kn time sub-intervals of length kn 2 and observe the path on the coarse time scale (that is, at times i2kn , 0 ≤ i ≤ 2mn −kn ). 264 P. Mörters, N. Sidorova As a strategy, we consider the event that on the coarse scale the path moves up in each step, whereas on the fine scale the path behaves typically. Hence, in one coarse time step, the path moves up by a distance of order 2kn /2 . Then we optimise over (kn ). This strategy guarantees that the path will have almost no self-intersections at times belonging to different sub-intervals. S̄n will be large, because the path is forced to go up (instead of fluctuating) on the coarse scale. Let 0 < η < 1/2 be fixed. Denote by Si − S(j −1)2kn Sj 2kn − S(j −1)2kn η sup − η < 1, 1 < <1+η Anj = 2kn /2 2kn /2 (j −1)2kn ≤i≤j 2kn the event that the random walk, considered on the i-th sub-interval, stays at distance of order 2kn /2 from its starting point and moves up by a distance of order 2kn /2 during the whole time. Further, denote by Aηn = −kn +1 2mn η Anj , j =1 the event that this happens on each sub-interval. η Using the independence of the events Anj , for j = 1, . . . , 2mn −kn +1 , we obtain by Lemma 5 log P(Aηn ) = log −kn +1 2mn j =1 P(Anj ) = 2mn −kn +1 log P(B2kn ) −n2−kn . η η (12) Consider the event Aηn . Notice that on this event we have kn kn −1 S̄n > 2mn −kn (1 − η)2 2 > 2mn − 2 1 1 − p−1 > gn n 2 εn , where the last inequality is satisfied if we choose kn in such a way that 2 2kn < nεnp−1 /(16gn2 ) −→ ∞. (13) Thus, the strategy leads to the desired growth of the walks. We now check that it also gives the right self-intersection local times. Note that for any 1 ≤ j1 , j2 ≤ 2mn −kn +1 such that |j2 − j1 | ≥ 2 the j1 -th and j2 -th pieces of length 2kn of (Si ) do not intersect. Indeed, let (j1 − 1)2kn ≤ i1 ≤ j1 2kn and (j2 − 1)2kn ≤ i1 ≤ j1 2kn . Then kn Si2 > S(j2 −1)2kn − (1 − η)2 2 > S(j2 −2)2kn + η2 ≥ Sj1 2kn + η2 kn 2 > S(j1 −1)2kn + (1 + η)2 kn 2 kn 2 > Si 1 . For each 1 ≤ j ≤ 2mn −kn +1 , define independent simple random walks starting at zero by (n,j ) Si = S(j −1)2kn +i − S(j −1)2kn , for 0 ≤ i < 2kn . (n,j ) Denote by 2kn −1 (z) the local time of S (n,j ) at z, and define independent random variables Ynj = (n,j ) p 2kn −1 (z) , for j ∈ {1, . . . , 2mn −kn +1 }. z∈Z A Class of Weakly Self-Avoiding Walks 265 As n < 2mn +1 we have, on the event An , that n ≤ n +1 −1 (j +1)2kn −1 2mn −k j =1 = (j +1)2kn −1 ··· i1 =(j −1)2kn (j +1)2kn −1 n +1 −1 2mn −k j =1 1{Si1 = · · · = Sip } ip =(j −1)2kn p 1{Si = z} . i=(j −1)2kn z∈Z Using the inequality (a + b)p ≤ 2p−1 (a p + bp ), which holds for all a, b ≥ 0 and p ∈ N, we obtain j 2kn −1 p (j +1)2kn −1 p n +1 −1 2mn −k p−1 n ≤ 2 1{Si = z} + 1{Si = z} j =1 = 2p−1 i=(j −1)2kn z∈Z i=j 2kn n +1 −1 2mn −k m −k +1 j =1 j =1 n 2 n Ynj . Ynj + Yn(j +1) ≤ 2p Let Znj , 1 ≤ j ≤ 2mn −kn +1 be a family of independent random variables such that Znj has η η the same distribution as Ynj conditioned on Anj . Since Anj are independent for all j , and η Ynj1 is independent of Anj2 for all j1 = j2 , we obtain (η) (η) 1 1 − P n ≤ εn an , S̄n > gn n 2 εn p−1 ≥ P n ≤ εn an Aηn P(Aηn ) 2mn −kn +1 2mn −kn +1 η p ≥P 2 Ynj ≤ εn an Anj P(Aηn ) j =1 j =1 2mn −kn +1 = P 2p (η) Znj ≤ εn an P(Aηn ). (14) j =1 We show that the first probability on the right hand side converges to one. By Lemma 5 we have η (η) EZnj = E[Ynj 1{Anj }] η P(Anj ) ≤ kn (p+1) E2kn −1 2 2 , η P(B2kn −1 ) (15) η E22kn −1 (η) E[Ynj2 1{Anj }] ≤ ≤ c1 2kn (p+1) , E (Znj )2 = η η P(Anj ) P(B2kn −1 ) for some c1 > 0. Observe that 2mn −kn +1 (η) p P 2 Znj > εn an j =1 =P 2 −mn +kn −1 −kn +1 2mn j =1 (η) Znj (η) − EZn1 >2 −p−mn +kn −1 (η) εn an − EZn1 266 P. Mörters, N. Sidorova 2mn −kn +1 −mn +kn −1 (η) (η) (η) ≤ P 2 Znj − EZn1 > 2−p−mn +kn −1 εn an − EZn1 . j =1 By Lemma 3 and (15), there are constants c2 , c3 > 0 such that an ≥ c2 n p+1 2 (η) EZn1 ≤ c3 2 and kn (p+1) 2 , which implies 2−p−mn +kn −1 εn an − EZn1 ≥ 2−p−1 n (η) p−1 2 2kn εn c2 − c3 2 kn (p+1) 2 > 0, where the last inequality holds if we choose 2 2kn < nεnp−1 (c2 2−p−1 /c3 ). (16) 1 Let c = (2p−3 c3 /c2 ) 2 , and note that, for all gn ≥ c, the inequality (13) implies (16). We now choose kn to be the even number satisfying 2 2 nεnp−1 /(64gn2 ) ≤ 2kn < nεnp−1 /(16gn2 ), so that (13) and (16) are satisfied, 2kn < n and 2kn → ∞, where the latter follows from the growth condition imposed on gn . Using the Chebyshev inequality, we obtain that P 2 p −kn +1 2mn (η) Znj (η) > ε n an ≤ j =1 VarZn1 (η) 2mn −kn +1 [2−p−mn +kn −1 εn an − EZn1 ]2 . It follows from the choice of (kn ) that (η) 2 2−p−mn +kn −1 εn an − EZn1 2(p+1) ≥ c4 εn p−1 np+1 gn−4 , for some c4 > 0 independent of gn . From (15) we obtain, for some c5 , c6 > 0 independent of gn , 2(p+1) VarZn1 ≤ c5 2kn (p+1) ≤ c6 εn p−1 np+1 gn−2(p+1) . (η) Using these two formulas and the asymptotics for kn , we obtain P 2 p −kn +1 2mn (η) Znj 2 2 > εn an ≤ c7 gn−2p εnp−1 ≤ c8 εnp−1 −→ 0, j =1 where c7 , c8 > 0 are independent of gn . Hence P 2 p −kn +1 2mn j =1 (η) Znj 2 ≤ εn an ≥ 1 − c8 εnp−1 → 1. A Class of Weakly Self-Avoiding Walks 267 It follows now from (14) and (12) that 1 1 − log P n ≤ εn an , S̄n > gn n 2 εn p−1 2mn −kn +1 (η) p Znj ≤ εn an + log P(Aηn ) ≥ log P 2 j =1 2 − 2 ≥ log 1 − c8 εnp−1 − c9 n2−kn ≥ −Cgn2 εn p−1 , with some c9 , C > 0 independent of gn . Proof of Theorem 3 Let gn = c be a constant sequence, where c is taken from Lemma 7. Then the assumptions on the growth of gn hold by choice of εn . Hence we can use Lemma 7 to obtain a lower bound, 1 − 2 1 − log P(n ≤ εn an ) ≥ log P n ≤ εn an , S̄n > cn 2 εn p−1 ≥ −Cc2 εn p−1 . The upper bound from Lemma 6 completes the proof. 3 Growth of the Weakly Self-Avoiding Walk We now state a more general version of Theorem 2, which also includes Theorem 1. The result of Theorem 2 follows immediately by specialising to the case εn = nβ−(p+1)/2 . 2 Proposition 1 Let εn > 0 be such that εn → 0 and εnp−1 n → ∞. There exists c1 > 0 such that eventually, 1 1 − P S̄n ≤ c1 n 2 εn p−1 n (p) ≤ εn En (p) = 0, and there exists c2 > 0 such that 1 − 2 1 − − log P S̄n ≥ c2 n 2 εn p−1 n (p) ≤ εn En (p) εn p−1 . 1 1 − p−1 In particular, S̄n n 2 εn in probability. 1 1 − p−1 Moreover, for any gn ≥ c2 such that gn n− 2 εn → 0, one has 1 − 2 1 − − log P S̄n > gn n 2 εn p−1 n (p) ≤ εn En (p) gn2 εn p−1 . 1 − 1 Proof of Proposition 1 Denote fn = c1 n 2 εn p−1 , where c1 > 0 will be specified later. On the event {S̄n < fn }, we have n (z) = 0 for |z| ≥ fn , and |z|<fn n (z) = n. (17) 268 P. Mörters, N. Sidorova Consider the function ϕ defined on the simplex S = {x ∈ Rm : xi ≥ 0 ∀i, x1 + · · · + xm = a} p p by ϕ(x) = x1 + · · · + xm . As ϕ has the global minimum at the point (a/m, . . . , a/m) we p p have x1 + · · · + xm ≥ a p m1−p . Applying this together with the condition (17), we obtain eventually n = pn (z) ≥ np (2fn − 1)1−p ≥ np 31−p fn1−p = (3c1 )1−p εn n p+1 2 . |z|<fn Since an = En n p+1 2 by Lemma 3, one can pick c1 small enough in order to ensure that {n ≤ εn an } ⊂ {S̄n > fn }, which finishes the proof of the first statement. −1/(p−1) Now fix gn such that gn ≥ c2 , and gn n−1/2 εn → 0 (where c2 > 0 will be specified later). Define 1 1 − p−1 fn = gn n 2 εn . By the reflection principle we have P(S̄n > fn ) ≥ P max Si > fn = 2P(Sn > fn ), 0≤i≤n and P(S̄n > fn ) ≤ P max Si > fn + P min Si < −fn 0≤i≤n 0≤i≤n = 2P max Si > fn = 4P(Sn > fn ). 0≤i≤n Hence P(S̄n > fn ) P(Sn > fn ). Further, the Azuma-Hoeffding inequality gives log P(Sn > fn ) ≤ − fn2 . 2n In particular, this implies 1 1 − log P S̄n > gn n 2 εn p−1 , n ≤ εn an g2 − 2 f2 ≤ log P S̄n > fn ≤ − n + log 4 = − n εn p−1 + log 4. 2n 2 (18) The corresponding lower bound is given by Lemma 7. For c2 > c, we have 1 − 2 1 − log P S̄n > gn n 2 εn p−1 , n ≤ εn an ≥ −Cεn p−1 gn2 , where C is independent of gn . Recall that, by Theorem 3, − 2 log P n ≤ εn an −εn p−1 , which, together with (18) and (19), implies (19) A Class of Weakly Self-Avoiding Walks 269 1 1 − log P S̄n > gn n 2 εn p−1 n ≤ εn an 1 − 2 1 − = log S̄n > gn n 2 εn p−1 , n ≤ εn an − log P n ≤ εn an −gn2 εn p−1 , if c2 is chosen large enough so that the first probability dominates even in the case when gn is constant. Acknowledgements We gratefully acknowledge support by the research council EPSRC through grant EP/C500229/1 and an ARF awarded to the first author. References 1. Bass, R.F., Chen, X., Rosen, J.: Moderate deviations and law of the iterated logarithm for the renormalized self-intersection local times of planar random walks. Electron. J. Probab. 11, 993–1030 (2006) 2. Lawler, G.F.: Intersections of Random Walks. Birkhäuser, Boston (1991) 3. Mörters, P., Ortgiese, M.: Small value probabilities via the branching tree heuristic. Bernoulli 14, 277–299 (2008) 4. van der Hofstad, R., König, W.: A survey of one-dimensional polymers. J. Stat. Phys. 103, 915–944 (2001)