The continuum limit getdoc9b2f. 332KB Jun 04 2011 12:04:44 AM

Proposition 14. For ℓ ≤ n we have E ” ∆ 1 2 ϕ ℓ,λ ϕ ℓ,λ = x — = 1 n b 1 λ t + 1 n osc 1 + O k −32 = O k −1 E ” ∆ 1 2 ϕ ℓ,λ ∆ 1 2 ϕ ℓ,λ ′ ϕ ℓ,λ = x, ϕ ℓ,λ ′ = y — = 1 n a 1 t, x, y + 1 n osc 2 + O k −32 E ” ∆ 1 2 ϕ ℓ+ 1 2 , λ ϕ ℓ+ 1 2 , λ = x — = 1 n b 2 λ t + 1 n osc 3 + O k −32 = O k −1 E ” ∆ 1 2 ϕ ℓ+ 1 2 , λ ∆ 1 2 ϕ ℓ+ 1 2 , λ ′ ϕ ℓ+ 1 2 , λ = x, ϕ ℓ+ 1 2 , λ ′ = y — = 1 n a 2 t, x, y + 1 n osc 4 + O k −32 , E ” |∆ 1 2 ϕ ℓ,λ | d ϕ ℓ,λ = x — , E ” |∆ 1 2 ϕ ℓ+ 1 2 , λ | d ϕ ℓ,λ = x — = O k −d2 , d = 2, 3 where t = ℓn , b 1 λ = λ 4ˆ s + ℜ ˆ ρ 2pˆ s + ℑ ˆ ρ 2 2 βˆ s 2 , b 2 λ = λ 4ˆ s − ℜ d d t ρ ℑρ + ℑρ 2 2 βˆ s 2 , a 1 = 1 βˆ s 2 ℜ ” e ix − y — + 1+ ℜ ˆ ρ 2 βˆ s 2 , a 2 = 1 βˆ s 2 ℜ ” e ix − y — + 1 + ℜρ 2 βˆ s 2 . The oscillatory terms are osc 1 = ℜ−v ℓ − iq 1 2e −i x ˆ ρ −2 ℓ η ℓ + ℜie −2i x ˆ ρ −4 ℓ η 2 ℓ q 1 4, osc 2 = q 3 ℜe −i x ˆ ρ −2 ℓ η ℓ + e −i y ˆ ρ −2 ℓ η ℓ 2 + ℜq 1 e −i x ˆ ρ −2 ℓ η ℓ + e −i y ˆ ρ −2 ℓ η ℓ + e −ix+ y ˆ ρ −4 ℓ η 2 ℓ 2, osc 3 = ℜ−ˆv ℓ − iq 2 2e −i x η ℓ + ℜie −2i x η 2 ℓ q 2 4, osc 4 = q 3 ℜe −i x η ℓ + e −i y η ℓ 2 + ℜq 2 e −i x η ℓ + e −i y η ℓ + e −ix+ y η 2 ℓ 2. Proof. We start with the identity ϕ ℓ+ 1 2 , λ − ϕ ℓ,λ = ϕ ℓ+1,λ∗ ˆ Q −1 ℓ − ϕ ℓ,λ∗ ˆ Q −1 ℓ = ϕ ℓ,λ∗ ˆ Q −1 ℓ S ℓ,λ − ϕ ℓ,λ∗ ˆ Q −1 ℓ = ashS ℓ,λ , e i ϕ ℓ,λ ¯ η ℓ ˆ ρ −2 ℓ , −1. Here we used the definition of the angular shift with the fact that S ℓ,λ and any affine transfor- mation will preserve ∞ ∈ H which corresponds to −1 in U. A similar identity can be proved for ∆ 1 2 ϕ ℓ+ 1 2 , λ . The proof now follows exactly the same as in [13], it is a straightforward application of Lemma 12 using the estimates on v ℓ,λ , ˆ v ℓ,λ , V ℓ , ˆ V ℓ .

5.2 The continuum limit

In this section we will prove that ϕ n t, λ = ϕ ⌊tn ⌋,λ converges to the solution of a one-parameter family of stochastic differential equations on t ∈ [0, 1. The main tool is the following proposition, proved in [13] based on [11] and [4]. Proposition 15. Fix T 0, and for each n ≥ 1 consider a Markov chain X n ℓ ∈ R d with ℓ = 1, . . . , ⌊nT ⌋. Let Y n ℓ x be distributed as the increment X n ℓ+1 − x given X n ℓ = x. We define b n t, x = nE[Y n ⌊nt⌋ x], a n t, x = nE[Y n ⌊nt⌋ xY n ⌊nt⌋ x T ]. 330 Suppose that as n → ∞ we have |a n t, x − a n t, y| + |b n t, x − b n t, y| ≤ c|x − y| + o1 48 sup x, ℓ E[ |Y n ℓ x| 3 ] ≤ cn −3 2 , 49 and that there are functions a, b from R × [0, T ] to R d 2 , R d respectively with bounded first and second derivatives so that sup x ∈R d2 ,t Z t a n s, x ds − Z t as, x ds + sup x ∈R d ,t Z t b n s, x ds − Z t bs, x ds → 0. 50 Assume also that the initial conditions converge weakly, X n d =⇒ X . Then X n ⌊nt⌋ , 0 ≤ t ≤ T converges in law to the unique solution of the SDE d X = b d t + σ d B, X 0 = X , t ∈ [0, T ], where B is a d-dimensional standard Brownian motion and σ : R d × [0, T ] is a square root of the matrix valued function a, i.e. at, x = σt, x σt, x T . We will apply this proposition to ϕ ℓ,λ with ℓ ≤ n 1 − ǫ and ℓ ∈ Z2, so the single steps of the proposition correspond to half steps in our setup. The following lemma shows that the oscillatory terms in the estimates of Proposition 14 average out in the ‘long run’. Its proof relies on Proposition 14 and Lemma 26 of the Appendix. Lemma 16. Let |λ|, |λ ′ | ≤ ¯λ and ǫ 0. Then for any ℓ 1 ≤ n 1 − ǫ, ℓ 1 ∈ Z 1 n ∼ X ≤ℓℓ 1 E ” ∆ 1 2 ϕ ℓ,λ | ϕ ℓ,λ = x — = 1 n ℓ 1 −1 X ℓ=0 b λ t + O n − 1 2 + n 1 2 1 n −32 51 1 n ∼ X ≤ℓℓ 1 E ” ∆ 1 2 ϕ ℓ,λ ∆ 1 2 ϕ ℓ,λ ′ | ϕ ℓ,λ = x, ϕ ℓ,λ ′ = y — = 1 n ℓ 1 −1 X ℓ=0 at, x, y + O n − 1 2 + n 1 2 1 n −32 where t = ℓn , the functions b λ , a are defined as b λ = λ 2ˆ s + ℜ ˆ ρ 2pˆ s + ℑ ˆ ρ 2 + ρ 2 2 βˆ s 2 − ℜ d d t ρ ℑρ , a = 2 βˆ s 2 ℜ ” e ix − y — + 2 + ℜ ˆ ρ 2 + ρ 2 βˆ s 2 , 52 and the implicit constants in O depend only on ǫ, β, ¯λ. The indices in the summation ∼ P run through half integers. Proof of Lemma 16. We will only prove the first statement, the second one being similar. Note that b λ t = b 1 λ t + b 2 λ t. Summing the first and third estimates in Proposition 14 we get 51 with an error term 1 n ℓ 1 −1 X ℓ=0 ℜe 1, ℓ η ℓ + 1 n ℓ 1 −1 X ℓ=0 ℜe 2, ℓ η 2 ℓ + O n −1 2 , 53 331 where the first two terms will be denoted ζ 1 , ζ 2 . Here e 1, ℓ = € −v λ − iq 1 2 ˆ ρ 2 ℓ + −ˆv λ − iq 2 2 Š e −i x , e 2, ℓ = i ˆ ρ −4 ℓ q 1 + q 2 e −2i x 4 where for this proof c denotes varying constants depending on ǫ. Using the fact that v λ , ˆ v λ , q 1 , q 2 and their first derivatives are continuous on [0, 1 − ǫ] we get |e i, ℓ | c, |e i, ℓ − e i, ℓ+1 | cn −1 . 54 Applying Lemma 26 of the Appendix to the first sum in 53: |ζ 1 | ≤ 1 n |e 1, ℓ 1 ||F 1 1, ℓ 1 | + 1 n ℓ 1 −1 X ℓ=1 |e 1, ℓ − e 1, ℓ+1 ||F 1 1, ℓ |. Since ℓ 1 ≤ n 1 − ǫ we have |F 1 1, ℓ | ≤ c1 + n 1 2 1 k −12 ≤ cn 1 2 1 n −12 + 1 and |ζ 1 | ≤ cn −32 n 1 2 1 + n −1 . Recall that k = n − ℓ. For the estimate of ζ 2 we first note that |e 2, ℓ | = 1 2 β n k | ˆ ρ −2 ℓ + ρ 2 ℓ | = 1 2 β n k | ˆ ρ 2 ℓ ρ 2 ℓ + 1|. 55 We will use Lemma 26 if | ˆ ρ 2 ℓ ρ 2 ℓ + 1| is ‘big’, and a direct bound with 55 if it is small. To be more precise: we divide the sum into three pieces, we cut it at indices ℓ ∗ 1 and ℓ ∗ 2 so that | ˆ ρ 2 ℓ ρ 2 ℓ + 1| ≤ n −12 if k ∈ [k ∗ 2 , k ∗ 1 ] and | ˆ ρ 2 ℓ ρ 2 ℓ + 1| ≥ n −12 otherwise. 56 Note that one or two of the resulting partial sums may be empty. We can always find such indices because arg ˆ ρ 2 ℓ ρ 2 ℓ is monotone if µ n ≥ p m − n and if µ n p m − n then arg ˆ ρ 2 ℓ ρ 2 ℓ decreases if k pm 1 n 1 then it increases. See the proof of Lemma 26. We denote the three pieces by ζ 2,i , i = 1, 2, 3 and bound them separately. Since k ≥ ǫn , Lemma 26 gives |ζ 2,1 | ≤ cn 1 2 1 n −32 + n −12 . The term |ζ 2,3 | can be bounded exactly the same way, so we only need to deal with ζ 2,2 . Here we use 55 to get a direct estimate: |ζ 2,2 | ≤ 1 2 β X k ∈[k ∗ 2 ,k ∗ 1 ]∩[ǫn ,n ] 1 k | ˆ ρ −2 ℓ + ρ 2 ℓ | ≤ cn −12 . Collecting all our estimates the statement follows. Now we have the ingredients to prove the continuum limit. 332 Proposition 17. Assume that m n → κ ∈ [1, ∞], nn → ν ∈ [1, ∞] and that eventually µ n p m − n or µ n ≤ p m − n. Then the continuous functions p n t −1 , ρ n t, ˆ ρ n t converge to following limits on [0, 1: p −1 t = κ − t −12 , ρt = ± r ν − 1 ν − t + i r 1 − t ν − t , ˆ ρt = r κ − 1 κ − t + i r 1 − t κ − t , where the sign in ℜρ depends on the eventual sign of µ n − p m − n. If κ = ∞ then p −1 t = 0, ˆ ρt = 1 and if ν = ∞ then ρt = ±1. Let B and W be independent real and complex standard Brownian motions, and for each λ ∈ R consider the strong solution of d ϕ λ = – λ 2ˆ s − ℜρ ′ ℑρ + ℑρ 2 + ˆ ρ 2 2 βˆ s 2 + ℜ ˆ ρ 2pˆ s ™ d t + p 2 ℜe −iϕ λ dW p β ˆ s + p 2 + ℜρ 2 + ˆ ρ 2 p β ˆ s d B, ϕ λ 0 = π. 57 Then we have ϕ λ,⌊n t ⌋ d =⇒ ϕ λ t, as n → ∞, where the convergence is in the sense of finite dimensional distributions for λ and in path-space D[0, 1 for t. Proof. The proof is very similar to the proof of Theorem 25 in [13]. One needs to check that for any fixed vector λ 1 , . . . , λ d the Markov chain ϕ ℓ,λ i , 1 ≤ i ≤ d, ℓ ≤ ⌊1 − ǫn ⌋, ℓ ∈ Z2 satisfies the conditions of Proposition 15 and to identify the variance matrix of the limiting diffusion. Note that because our Markov chain lives on the half integers one needs to slightly rephrase the proposition, but this is straightforward. The Lipshitz condition 48 and the moment condition 49 are easy to check from Proposition 14. The averaging condition 50 is satisfied because of Lemma 16, using the fact that because of the conditions of the proposition, the functions b λ t, at, x, y converge. This proves that the rescaled version of ϕ ℓ,λ j , 1 ≤ j ≤ d converges in distribution to an SDE in R d where the drift term is given by the limit of b λ j , j = 1 . . . d and the diffusion matrix is given by at, x j,k = 2 βˆ s 2 ℜ ” e ix k −x j — + 2+ ℜ ˆ ρ 2 +ρ 2 βˆ s 2 . The only step left is to verify that the limiting SDE can be rewritten in the form 57. This follows easily using the fact that if Z is a complex Gaussian with i.i.d. standard real and imaginary parts and ω 1 , ω 2 ∈ C then Eℜω 1 Z ℜω 2 Z = ℜ ¯ ω 1 ω 2 . The following corollary describes the scaling limit of the relative phase function α ℓ,λ . Corollary 18. Let Z be a complex Brownian motion with i.i.d. standard real and imaginary parts and consider the strong solution α λ t of the SDE system 17. Then α ⌊n t ⌋,λ d =⇒ α λ t as n → ∞ where the convergence is in the sense of finite dimensional distributions for λ and in path-space D[0, 1 for t. 333 Proof. We just need to show that for any subsequence of n we can choose a further subsequence so that the convergence holds. By choosing an appropriate subsequence we can assume that m n , n n both converge and that µ n − p m − n is always positive or nonnegative. Then the con- ditions of Proposition 17 are satisfied and α λ = ϕ λ − ϕ will satisfy the SDE 17 with a complex Brownian motion Z t := R t e i ϕ t dW t . From this the statement of the corollary follows. 6 Middle stretch In this section we will study the behavior of α ℓ,λ and ϕ ℓ,λ in the interval [ ⌊1 − ǫn ⌋, n 2 ] with n 2 = j n − K n 1 3 1 ∨1 k . The constant K will eventually go to ∞, so we can assume that K C with C large enough.

6.1 The relative phase function