assuming that the boundary term Z
∂ Ξ
∂
i
h ∂
2 i j
h e
−H
= 0 5.9
in the integration by parts vanishes. To see 5.9, consider a segment
λ
i
= λ
i+1
of the boundary ∂ Ξ. From the explicit representation
5.11, 5.12 in the next section, we will see that f
t
≥ 0 is a meromorphic function in each variable in the domain Ξ for any t
0. It can be represented as by λ
i+1
− λ
i β
F λ with some β ∈ Z,
where F is analytic and 0 F ∞ near λ
i
= λ
i+1
. Since f
t
≥ 0, we obtain that the exponent β is non-negative and even. Therefore f
1 2
t
behaves as |λ
i+1
− λ
i
|
β2
with a non-negative integer exponent
β2 near λ
i
= λ
i+1
. It then follows that ∂
i
p f
∂
2 i j
p f e
−H
vanishes at the boundary due to the factor
λ
i+1
− λ
i 2
in e
−H
, i.e. the integral 5.9 indeed vanishes.
5.2 Bound on the entropy
Lemma 5.1. Let s = N
−1
. For any α
1 4
we have S
µ
f
s
:= S f
s
µ|µ ≤ C N
1+ α
5.10 with C depending on
α.
Proof. In the proof we consider the probability density uλ and the equilibrium measure µ extended
to R
N
see 3.12, i.e. the eigenvalues are not ordered. Clearly S f
s
µ|µ = S f
s
e µ|e
µ and we estimate the relative entropy of the extended measures.
Given the density f
λ
e
µdλ of the eigenvalues of the Wigner matrix as an initial distribution, the
eigenvalue density f
s
λ for the matrix evolved under the Dyson’s Brownian motion is given by
f
s
λ euλ =
Z
R
N
g
s
λ, ν f ν
euνdν
5.11 with a kernel
g
s
λ, ν =
N
N 2
2π
N 2
c
N N −12
1 − c
2 N
2
∆
N
λ
∆
N
ν
det
exp
−Ncλ
j
− ν
k 2
21 − c
2
j,k
, 5.12
where c = cs = e
−s2
for brevity. The derivation of 5.12 follows very similarly to Johansson’s presentation of the Harish-ChandraItzykson-Zuber formula see Proposition 1.1 of [21] with the
difference that in our case the matrix elements move by the Ornstein-Uhlenbeck process 3.1 in- stead of the Brownian motion.
In particular, formula 5.12 implies that f
s
is an analytic function for any s 0 since
f
s
λ =
h
s
λ
∆
N
λ
Z
R
N
det
exp
−Ncλ
j
− ν
k 2
21 − c
2
j,k
f
ν euν
∆
N
ν dν
with an explicit analytic function h
s
λ. Since the determinant is analytic in λ, we see that f
s
λ is
meromorphic in each variables and the only possible poles of f
s
λ come from the factors λ
i
−λ
j −1
544
in ∆
N
λ near the coalescence points. But f
s
λ is a non-negative function, so it cannot have a
singularity of order −1, thus these singular factors cancel out from a factor λ
i
− λ
j
from the integral. Alternatively, using the Laplace expansion the determinant, one can explicitly see that
each 2 by 2 subdeterminant from the i-th and j-th columns carry a factor ±λ
i
− λ
j
. Then, by Jensen inequality from 5.11 and from the fact that f
ν euν is a probability density, we
have S
e µ
f
s
= Z
R
N
f
s
log f
s
d e
µ ≤ ZZ
R
N
×R
N
log g
s
λ, ν euλ
g
s
λ, ν f ν
euνdλ dν.
Expanding this last expression we find, after an exact cancellation of the term N 2 log2π,
S
e µ
f
s
≤ ZZ
R
N
×R
N
N 2
log N −
N N − 1
2 log c
− N
2 log1
− c
2
+ log ∆
N
λ − log ∆
N
ν
+ log det
exp
−Ncλ
j
− ν
k 2
21 − c
2
j,k
− N
2
2 log N
+ N
2
N
X
i=1
λ
2 i
− 2 log ∆
N
λ +
N
X
j=1
log j
g
s
λ, ν f ν
euνdλdν.
Since s = N
−1
, we have log c = −12N and log1 − c
2
= − log N + ON
−1
. Hence S
e µ
f
s
≤ ZZ
R
N
×R
N
C N log N + log ∆
N
λ − log ∆
N
ν + log det
exp
−Ncλ
j
− ν
k 2
21 − c
2
j,k
− N
2
2 log N +
N 2
N
X
i=1
λ
2 i
− 2 log ∆
N
λ +
N
X
i=1
log j g
s
λ, ν f ν
euνdλdν.
5.13 For the determinant term, we use that each entry is at most one, thus
log det
exp
−Ncλ
j
− ν
k 2
21 − c
2
j,k
≤ log N. The last term in 5.13 can be estimated using Stirling’s formula and Riemann integration
N
X
j=1
log j ≤
N
X
j=1
log
j e
j
+ C log2π j
≤ Z
N +1 1
dx x log x −
N
X
j=1
j + C N log N ≤
N
2
log N 2
− 3
4 N
2
+ C N log N 5.14
thus the
1 2
N
2
log N terms cancel. For the N
2
terms we need the following approximation
545
Lemma 5.2. With respect to any Wigner ensemble whose single-site distribution satisfies 2.4–2.6 and for any
α 14 we have E
h N 2
N
X
i=1
λ
2 i
− 2 log ∆
N
λ
i =
3 4
N
2
+ ON
1+ α
, 5.15
where the constant in the error term depends on α and on the constants in 2.4–2.6.
Note that 2.6, 2.5 hold for both the initial Wigner ensemble with density f and for the evolved
one with density f
t
. These conditions ensure that Theorem 3.5 of [15] is applicable. Proof of Lemma 5.2. The quadratic term can be computed explicitly using 3.4:
N 2
E
N
X
i=1
λ
2 i
= N
2 E
Tr H
2
= 1
2 N
2
= N
2
2 Z
x
2
̺
sc
x dx, 5.16
The second determinant term will be approximated in the following lemma whose proof is post- poned to Appendix D.
Lemma 5.3. With respect to any Wigner ensemble whose single-site distribution satisfies 2.4–2.6 and for any
α 14 we have E
log ∆
N
λ =
N
2
2 Z Z
log |x − y| ̺
sc
x̺
sc
y dx d y + ON
1+ α
. 5.17
Finally, explicit calculation then shows that 1
2 Z
x
2
̺
sc
x dx − Z Z
log |x − y| ̺
sc
x̺
sc
y dx d y = 3
4 ,
and this proves Lemma 5.2. Hence, continuing the estimate 5.13, we have the bound
S
e µ
f
s
≤ C N
1+ α
+ ZZ
R
N
×R
N
log ∆
N
λ − log ∆
N
ν g
s
λ, ν f ν
euνdνdλ
≤ C N
1+ α
+ N
4 E
N
X
j=1
[λ
2 j
s − λ
2 j
0] = C N
1+ α
, 5.18
where we used Lemma 5.2 both for the initial Wigner measure and for the evolved one and finally we used that the E Tr H
2
is preserved, see 3.4. This completes the proof of 5.10.
6 Local equilibrium
6.1 External and internal points
