Local Taylor approximation Uniform control in large dimensions for capacities

with C a constant independent of k. Therefore, for N N , ln 1 − u N k 1 − v k = ln ‚ 1 + v k − u N k 1 − v k Œ ≤ C ′ N 2 . 3.20 Hence ln U N ,N V N ≤ C ′ N −→ N →+∞ 0. 3.21 As P |v k | +∞, we get lim N →+∞ V N = V µ 0, and thus 3.13 is proved. 4 Estimates on capacities To prove Theorem 3.1, we prove uniform estimates of the denominator and numerator of 2.6, namely the capacity and the mass of the equilibrium potential.

4.1 Uniform control in large dimensions for capacities

A crucial step is the control of the capacity. This will be done with the help of the Dirichlet principle 2.7. We will obtain the asymptotics by using a Laplace-like method. The exponential factor in the integral 2.9 is largely predominant at the points where h is likely to vary the most, that is around the saddle point O. Therefore we need some good estimates of the potential near O.

4.1.1 Local Taylor approximation

This subsection is devoted to the quadratic approximations of the potential which are quite subtle. We will make a change of basis in the neighborhood of the saddle point O that will diagonalize the quadratic part. Recall that the potential G γ,N is of the form G γ,N x = − 1 2N x, [Id − D]x + 1 4N kxk 4 4 . 4.1 where the operator D is given by D = γ ” Id − 1 2 Σ + Σ ∗ — and Σx j = x j+1 . The linear operator Id−D = −∇ 2 F γ,N O has eigenvalues −λ k,N and eigenvectors v k,N with components v k,N j = ω jk , with ω = e i2 πN . Let us change coordinates by setting ˆ x j = N −1 X k=0 ω − jk x k . 4.2 Then the inverse transformation is given by x k = 1 N N −1 X j=0 ω jk ˆ x j = x k ˆ x. 4.3 332 Note that the map x → ˆx maps R N to the set b R N = ¦ ˆ x ∈ C N : ˆ x k = ˆ x N −k © 4.4 endowed with the standard inner product on C N . Notice that, expressed in terms of the variables ˆ x, the potential 1.2 takes the form G γ,N xˆ x = 1 2N 2 N −1 X k=0 λ k,N |ˆx k | 2 + 1 4N kxˆxk 4 4 . 4.5 Our main concern will be the control of the non-quadratic term in the new coordinates. To that end, we introduce the following norms on Fourier space: kˆxk p, F = 1 N N −1 X i=0 |ˆx| p 1 p = 1 N 1 p kˆxk p . 4.6 The factor 1 N is the suitable choice to make the map x → ˆx a bounded map between L p spaces. This implies that the following estimates hold see [16], Vol. 1, Theorem IX.8: Lemma 4.1. With the norms defined above, we have i the Parseval identity, kxk 2 = kˆxk 2, F , 4.7 and ii the Hausdorff-Young inequalities: for 1 ≤ q ≤ 2 and p −1 + q −1 = 1, there exists a finite, N - independent constant C q such that kxk p ≤ C q kˆxk q, F . 4.8 In particular kxk 4 ≤ C 4 3 kˆxk 4 3,F . 4.9 Let us introduce the change of variables, defined by the complex vector z, as z = ˆ x N . 4.10 Let us remark that z = 1 N P N −1 k=1 x k ∈ R. In the variable z, the potential takes the form e G γ,N z = G γ,N xN z = 1 2 N −1 X k=0 λ k,N |z k | 2 + 1 4N kxNzk 4 4 . 4.11 Moreover, by 4.7 and 4.10 kxNzk 2 2 = kNzk 2 2, F = 1 N kNzk 2 2 . 4.12 333 In the new coordinates the minima are now given by I ± = ±1, 0, . . . , 0. 4.13 In addition, zB N − = zB ρ p N I − = B ρ I − where the last ball is in the new coordinates. Lemma 4.1 will allow us to prove the following important estimates. For δ 0, we set C δ =    z ∈ b R N : |z k | ≤ δ r k,N p |λ k,N | , 0 ≤ k ≤ N − 1    , 4.14 where λ k,N are the eigenvalues of the Hessian at O as given in 2.14 and r k,N are constants that will be specified below. Using 3.15, we have, for 3 ≤ k ≤ N2, λ k,N ≥ k 2 ‚ 1 − π 2 12 Œ µ − 1. 4.15 Thus λ k,N verifies λ k,N ≥ ak 2 , for 1 ≤ k ≤ N2, with some a, independent of N. The sequence r k,N is constructed as follows. Choose an increasing sequence, ρ k k ≥1 , and set r 0,N = 1 r k,N = r N −k,N = ρ k , 1 ≤ k ≤ š N 2  . 4.16 Let, for p ≥ 1, K p = X k ≥1 ρ p k k p 1 p . 4.17 Note that if K p is finite then, for all p 1 p , K p 1 is finite. With this notation we have the following key estimate. Lemma 4.2. For all p ≥ 2, there exist finite constants B p , such that, for z ∈ C δ , kxNzk p p ≤ δ p N B p 4.18 if K q is finite, with 1 p + 1 q = 1. Proof. The Hausdorff-Young inequality Lemma 4.1 gives us: kxNzk p ≤ C q kNzk q, F . 4.19 Since z ∈ C δ , we get kNzk q q, F ≤ δ q N q −1 N −1 X k=0 r q k,N λ q 2 k . 4.20 Then N −1 X k=0 r q k,N λ q 2 k = 1 λ q 2 + 2 ⌊N2⌋ X k=1 r q k,N λ q 2 k ≤ 1 λ q 2 + 2 a q 2 ⌊N2⌋ X k=1 ρ q k k q ≤ 1 λ q 2 + 2 a q 2 K q q = D q q 4.21 334 which is finite if K q is finite. Therefore, kxNzk p p ≤ δ p N q−1 p q C p q D p q , 4.22 which gives us the result since q − 1 p q = 1. We have all what we need to estimate the capacity.

4.1.2 Capacity Estimates