Bound on the level repulsion and potential for good configurations

Proof. The probability of Ω m was estimated in 4.17. The probability of the second event in 4.18 can be estimated by 4.13 from Proposition 4.2 and from N sc 0 = 12. The third event is treated by the large deviation estimate on N I for any interval I with length |I| ≥ log N 2 N see Theorem 4.6 from [15]; note that there is a small error in the statement of this theorem, since the conditions y ≥ log NN and |I| ≥ log NN should actually be replaced by the stronger assumptions y ≥ log N 2 N and |I| ≥ log N 2 N which are used in its proof: P {N I ≥ KN|I|} ≤ e −c p K N |I| . 4.20 The fourth event is a large deviation of the largest eigenvalue, see, e.g. Lemma 7.4. in [13]. ƒ In case of good configurations, the location of the eigenvalues are close to their equilibrium localition given by the semicircle law. The following lemma contains the precise statement and it will be proven in Appendix C. Lemma 4.3. Let λ 1 λ 2 . . . λ N denote the eigenvalues in increasing order and let κ 0. Then on the set Ω and if N ≥ N κ, it holds that |λ a − N −1 sc aN −1 | ≤ Cκ −12 n −γ6 4.21 for any N κ 3 2 ≤ a ≤ N1 − κ 3 2 recall the definition of N sc from 4.12, and N̺ sc λ a λ b − λ a − b − a ≤ Cκ −12 n γ |b − a| 3 4 + N −1 |b − a| 2 4.22 for any N κ 3 2 ≤ a b ≤ N1 − κ 3 2 and |b − a| ≤ C N n −γ6 .

4.1 Bound on the level repulsion and potential for good configurations

Lemma 4.4. On the set Ω and with the choice n given in 4.15, we have 1 N E 1−κ 3 2 N X ℓ=N κ 3 2 X j 6=ℓ 1 Ω [N λ j − λ ℓ ] 2 ≤ C n 2 γ . 4.23 and 1 N E 1−κ 3 2 N X ℓ=N κ 3 2 X j 6=ℓ 1 Ω N λ ℓ − λ j ≤ C n 2 γ 4.24 with respect to any Wigner ensemble satisfying the conditions 2.4 and 2.5 Proof. First we partition the interval [ −2 + κ, 2 − κ] into subintervals I r = n γ N −1 r − 1 2 , n γ N −1 r + 1 2 , r ∈ Z, |r| ≤ r 1 := 2 − κN n −γ , 4.25 that have already been used in the proof of Lemma 4.3. On the set Ω we have the bound N I r ≤ KN|I r | ≤ C n γ 4.26 540 on the number of eigenvalues in each interval I r . Moreover, the constraint N κ 3 2 ≤ ℓ ≤ N1 − κ 3 2 implies, by 4.21, that |λ ℓ | ≤ 2 − κ for sufficiently small κ, thus λ ℓ ∈ I r with |r| ≤ r 1 . We estimate 4.23 as follows: A := 1 N E 1 Ω ∗ X j ℓ 1 [N λ j − λ ℓ ] 2 = 1 N E 1 Ω X j ℓ X k ∈Z X |r|≤r 1 1 λ ℓ ∈ I r 12 k ≤ N|λ j − λ ℓ | ≤ 2 k+1 [N λ j − λ ℓ ] 2 ≤ 1 N E 1 Ω X |r|≤r 1 X j ℓ X k ∈Z 2 −2k 1 n λ ℓ ∈ I r , 2 k ≤ N|λ j − λ ℓ | ≤ 2 k+1 o 4.27 where the star in the first summation indicates a restriction to N κ 3 2 ≤ j ℓ ≤ 1 − κ 3 2 N . By 4.26, for any fixed r, the summation over ℓ with λ ℓ ∈ I r contains at most C n γ elements. The summation over j contains at most C n γ elements if k 0, since λ ℓ ∈ I r and |λ j − λ ℓ | ≤ 2 k+1 N −1 ≤ N −1 imply that λ j ∈ I r ∪ I r+1 . If k ≥ 0, then the j-summation has at most C2 k + n γ elements since in this case λ j ∈ S {I s : |s − r| ≤ C · 2 k n −γ + 1}. Thus we can continue the above estimate as A ≤ C n 2 γ N X k X |r|≤r 1 2 −2k P n ∃I ⊂ I r −1 ∪ I r ∪ I r+1 : |I| ≤ 2 k+1 N −1 , N I ≥ 2 o + C n γ N X k ≥0 X |r|≤r 1 2 −2k n γ + 2 k . 4.28 The second sum is bounded by C n 3 γ . In the first sum, we use the level repulsion estimate by decomposing I r −1 ∪ I r ∪ I r+1 = S m J m into intervals of length 2 k+2 N −1 that overlap at least by 2 k+1 N −1 , more precisely J m = h n γ N −1 r − 1 − 1 2 + 2 k+1 N −1 m − 1, n γ N −1 r − 1 − 1 2 + 2 k+1 N −1 m + 1 i , where m = 1, 2, . . . , 3n γ · 2 −k−1 . Then P n ∃I ⊂ I r −1 ∪ I r ∪ I r+1 : |I| ≤ 2 k+1 N −1 , N I ≥ 2 o ≤ 3n γ ·2 −k−1 X m=1 P N J m ≥ 2 Using the level repulsion estimate given in Theorem 3.4 of [15] here the condition 2.5 is used and the fact that J m ⊂ I r −1 ∪ I r ∪ I r+1 ⊂ [−2 + κ, 2 − κ] since |r| ≤ r 1 , we have P N J m ≥ 2 ≤ CN|J m | 4 and thus A ≤ C n 3 γ N −1 X k= −∞ X |r|≤r 1 2 −2k 2 −k−1 2 k+2 4 ≤ C n 2 γ . and this completes the proof of 4.23. 541 For the proof of 4.24, we note that it is sufficient to bound the event when N |λ j − λ ℓ | ≥ 1 after using 4.23. Inserting the partition 4.25, we get 1 N E 1 Ω ∗ X j ℓ 1N |λ ℓ − λ j | ≥ 1 N λ ℓ − λ j = 1 N X |r|,|s|≤r E 1 Ω X j ℓ 1 λ j ∈ I r , λ ℓ ∈ I s 1N |λ ℓ − λ j | ≥ 1 N λ ℓ − λ j ≤ C N X |r|,|s|≤r E 1 Ω N I r N I s n γ [|s − r| − 1] + + 1 ≤ C n 2 γ N X |r|,|s|≤r 1 n γ [|s − r| − 1] + + 1 ≤ C n γ log N . Recalling the choice of n completes the proof of Lemma 4.4. ƒ 5 Global entropy

5.1 Evolution of the entropy