decompose getdoc0885. 605KB Jun 04 2011 12:04:03 AM

7 Cutoff Estimates In this section, we cutoff the interaction with the far away particles. We fix a good index L ∈ G and a good external point configuration y ∈ Y L . Consider the measure µ y = e −H y Z y with H y x = N    n X i=1 x 2 i 2 − 2N −1 X 1 ≤i j≤n log |x j − x i | − 2N −1 X k,i log |x i − y k |    7.1 The measure µ y is supported on the interval I y = y −1 , y 1 . For any fixed

y, decompose

H y = H 1 + H 2 , H 2 x = n X i=1 V 2 x i , 7.2 where V 2 x = N 2 x 2 − 2 X |k|≥n B log |x − y k | 7.3 and H 1 x = −2 X 1 ≤i j≤n log |x j − x i | − n X i=1 V 1 x i 7.4 with V 1 x = −2 X |k|n B log |x − y k | where B is a large positive number with B ǫ 12. We define the measure µ 1 y dx := e −H 1 x d x Z 1 . 7.5 Lemma 7.1. Let L ∈ G and y ∈ Y L . For B ≥ 20, we have sup x ∈I n y d µ 1 y d µ y x − 1 ≤ Cn −B9+2 7.6 This lemma will imply that one can cutoff all y k ’s in the potential with |k| ≥ n B . Proof. Let δV 2 := max x ∈I y V 2 − min x ∈I y V 2 , then, by 6.15 and y ∈ Y L , we have δV 2 ≤ |I y |kV ′ 2 k ∞ ≤ C nN −1 kV ′ 2 k ∞ In Lemma 7.2 we will give an upper bound on kV ′ 2 k ∞ , and then we have, for B ≥ 20, that δV 2 ≤ C n −B9+1 . 553 Since d µ 1 y d µ y x − 1 = e − P n i=1 V 2 x i −min V 2 − 1 ≤ CnδV 2 ≤ C n −B9+2 , we obtain 7.6. ƒ Lemma 7.2. For B ≥ 20 and for any L ∈ G 1 , y ∈ Y L we have sup x ∈I y |V ′ 2 x| = sup x ∈I y −2 X |k|≥n B 1 x − y k + N x ≤ C N n γ12−B8 . 7.7 Proof. Recall that y ∈ Y L ⊂ Ω 1 implies that the density of the y’s is close the semicircle law in the sense of 6.9. Let d := n B N ̺ sc y −1 . 7.8 Since y ∈ Ω 1 , we know that | y −1 |, | y 1 | ≤ 2 − κ2 see 6.14, thus ̺ sc y −1 ≥ c 0. Taking the imaginary part of 4.3 for |z| ≤ 2 and renaming the variables, we have the identity x = 2 Z R ̺ sc y x − y d y. Furthermore, with ¯ y = 1 2 y −1 + y 1 we have Z | y− ¯y|≤d ̺ sc y x − y d y ≤ Cd since ¯ y is away from the spectral edge thus ̺ sc is continuously differentiable on the interval of integration [ ¯ y − d, ¯y + d]. Thus N x − 2N Z | y− ¯y|≥d ̺ sc y x − y d y ≤ CN d ≤ Cn B therefore to prove 7.7 it is sufficient to show that sup x ∈I y 1 N X |k|≥n B 1 x − y k − Z | y− ¯y|≥d ̺ sc y x − y d y ≤ Cn γ12−B8 7.9 We will consider only k ≥ n B and compare the sum with the integral on the regime y ≥ ¯y + d, the sum for k ≤ −n B is similar. Define dyadic intervals I m = [ ¯ y + 2 m d, ¯ y + 2 m+1 d], m = 0, 1, 2, . . . , log N Since y ∈ Y L ⊂ Ω 1 , i.e. max | y k | ≤ K, there will be no y k above the last interval I log N . We subdivide each I m into n B 2 equal disjoint subintervals of length 2 m d n −B2 I m = n B 2 [ ℓ=1 I m, ℓ , I m, ℓ = [ y ∗ m, ℓ−1 , y ∗ m, ℓ ] with y ∗ m, ℓ := y 1 + 2 m d1 + ℓn −B2 . 554 For y ∈ Y L ⊂ Ω 1 , the estimate 4.22 holds for y 1 and y n B , i.e. N̺ sc y 1 y n B − y 1 − n B − 1 ≤ Cn γ+3B4 ≤ C n 4B 5 if B ≥ 20, which means that | y n B − y 1 + d| ≤ C n 4B 5 N + n B N 1 ̺ sc y −1 − 1 ̺ sc y 1 ≤ C n 4B 5 N + C n B+1 N 2 ≤ C n 4B 5 N 7.10 using B ǫ 12, n B ≤ N 1 2 , i.e. | y n B − ¯y + d| ≤ C n 4B 5 N 7.11 by using the definition of d from 7.8, the fact that ̺ sc y ±1 is separated away from zero and that |I y | ≤ C nN −1 from 6.14. Therefore we can estimate 1 N X k ≥n B 1 x − y k − 1 N log N X m=0 n B 2 X ℓ=1 X j ∈J L 1 y j ∈ I m, ℓ x − y j ≤ 1 N X j ∈J L 1 j n B , y j ≥ ¯y + d |x − y j | + 1 N X j ∈J L 1 j ≥ n B , y j ¯y + d |x − y j | ≤ C n 1 −B5 . 7.12 To see the last estimate, we notice that in the first summand we have ¯ y + d ≤ y j ≤ y n B ≤ ¯y + d + C n 4B 5 N −1 by 7.11, i.e. all these y j ’s lie in an interval of length C n 4B 5 N −1 , so their number is bounded by C n 4B 5 by 6.10. Thus the first term in the right hand side of 7.12 is bounded by C n 4B 5 N −1 d −1 ≤ C n 1 −B5 ; the estimate of the second term is similar. Using that max y ∈I m, ℓ 1 x − y − min y ∈I m, ℓ 1 x − y ≤ |I m, ℓ | max x ∈I y max y ∈I m, ℓ 1 x − y 2 ≤ C 2 m d n −B2 2 m d 2 ≤ C 2 m d n B 2 we have 1 N log N X m=0 n B 2 X ℓ=1 X j ∈J L 1 y j ∈ I m, ℓ x − y j − 1 N log N X m=0 n B 2 X ℓ=1 N I m, ℓ x − y ∗ m, ℓ ≤ C N log N X m=0 n B 2 X ℓ=1 N I m, ℓ 2 m d n B 2 ≤ C log N X m=0 n B 2 X ℓ=1 1 n B ≤ C n −B2 log N ≤ C n −B4 . 7.13 In the second line we used that N I m, ℓ ≤ KN|I m, ℓ | by 6.10 since y ∈ Ω 1 and I m, ℓ ∩ I y = ;. 555 We use that for y ∈ Ω 1 we can apply 6.9 for I = I m, ℓ and we get 1 N X m ≥0 n B 2 X ℓ=1 N I m, ℓ x − y ∗ m, ℓ − 1 N log N X m=0 n B 2 X ℓ=1 N |I m, ℓ |̺ sc y ∗ m, ℓ x − y ∗ m, ℓ ≤ C n γ12 N log N X m=0 n B 2 X ℓ=1 N |I m, ℓ | 3 4 |x − y ∗ m, ℓ | ≤ C n γ12 N log N X m=0 n B 2 X ℓ=1 2 m n B 2 3 4 2 m n B N −1 ≤ C n γ12−B8 , 7.14 where we used that |I m, ℓ | = 2 m d n −B2 ≤ C · 2 m n B 2 N −1 see 7.8 and that |x − y ∗ m, ℓ | ≥ 2 m −1 d ≥ c · 2 m n B N −1 . Finally, the second term on the left hand side of 7.14 is a Riemann sum of the integral in 7.9 with an error log N X m=0 n B 2 X ℓ=1 |I m, ℓ |̺ sc y ∗ m, ℓ x − y ∗ m, ℓ − Z | y− ¯y|≥d ̺ sc y x − y d y ≤ log N X m=0 n B 2 X ℓ=1 C N 2 m n B 2 |I m, ℓ | 2 ≤ C n −B2 log N , 7.15 since on each interval I m, ℓ we could estimate the derivative of the integrand as sup y ∈I m, ℓ d d y ̺ sc y x − y ≤ C N 2 m n B 2 . Combining 7.12, 7.13, 7.14 and 7.15, we have proved 7.9 which completes the proof of Lemma 7.2. ƒ 8 Derivative Estimate of Orthogonal Polynomials In the next few sections, we will prove the boundedness and small distance regularity of the density. Our proof follows the approach of [26] cf: Lemma 3.3 and 3.4 in [26], but the estimates are done in a different way due to the singularity of the potential. For the rest of this paper, it is convenient to rescale the local equilibrium measure to the interval [ −1, 1] as we now explain. Suppose L ∈ G and y ∈ Y L . We change variables by introducing the transformation T : I y → [−1, 1], e w = T w := 2w − ¯y |I y | , with ¯ y := y −1 + y 1 2 and its inverse w = T −1 e w = ¯ y + e w |I y | 2 , then T I y = [−1, 1]. Let e µ ey be the measure µ 1 y see 7.5 rescaled to the interval [ −1, 1], i.e., e µ ey d ex := 1 e Z n, ey exp h − n n X i=1 U ey ex i + 2 X 1 ≤i j≤n log |ex i − ex j | i d ex 8.1 556 on [ −1, 1] n with U ey ex := − 2 n X |k|n B log |ex − ey k |. 8.2 The ℓ-point correlation functions of µ y and e µ ey are related by p ℓ n x 1 , x 2 , . . . x n = p ℓ n ¯ y + ex 1 |I y | 2 , . . . ¯ y + ex n |I y | 2 = 2 |I y | ℓ ep ℓ n ex 1 , ex 2 , . . . ex n . 8.3 Let p j λ, j = 0, 1, . . . denote the real orthonormal polynomials on [−1, 1] corresponding to the weight function e −nU ey λ , i.e. deg p j = j and Z 1 −1 p j λp k λe −nU ey λ d λ = δ jk and define ψ j λ := p j λe −nU ey λ2 8.4 to be orthonormal functions with respect to the Lebesgue measure on [ −1, 1]. Everything depends on

y, but y is fixed in this section and we will omit this dependence from the notation.