TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
F. BEKERMAN⇤ , A. FIGALLI† , A. GUIONNET‡
Abstract. We construct approximate transport maps for non-critical -matrix models,
that is, maps so that the push forward of a non-critical -matrix model with a given potential
is a non-critical -matrix model with another potential, up to a small error in the total
variation distance. One of the main features of our construction is that these maps enjoy
regularity estimates which are uniform in the dimension. In addition, we find a very useful
asymptotic expansion for such maps which allow us to deduce that local statistics have the
same asymptotic behavior for both models.
1. Introduction.
Given a potential V : RN ! R and
(1.1)
PN
V (d 1 , . . . , d
´Q
N)
:=
1
ZVN
> 0, we consider the -matrix model
Y
P
N N
i=1 V ( i ) d
| i
j| e
1 . . . d N,
1i<jN
PN
N i=1 V ( i )
where ZVN =
d 1 . . . d N . It is well known (see e.g [Vil])
j| e
1i<jN | i
N
N
N
that for any V, W : R ! R such that ZV , ZV +W < 1 there exists a map T N : RN !RN
N
N
that transports PN
V into PV +W , that is, for any bounded measurable function f : R !R we
have
ˆ
ˆ
N
N
f T ( 1 , . . . , N ) PV (d 1 , . . . , d N ) = f ( 1 , . . . , N ) PN
V +W (d 1 , . . . , d N ) .
However, the dependency in the dimension N of this transport map is in general unclear
unless one makes very strong on the densities [Caf00] that unfortunately are never satisfied
in our situation. Hence, it seems extremely difficult to use these maps T N to understand the
relation between the asymptotic of the two models.
The main contribution of this paper is to show that a variant of this approach is indeed
possible and provides a very robust and flexible method to compare the asymptotics of local
statistics. In the more general context of several-matrices models, it was shown in [GS] that
the maps T N are asymptotically well approximated by a function of matrices independent of
N , but it was left open the question of studying corrections to this limit. In this article we
consider one-matrix models, and more precisely their generalization given by -models, and
we construct approximate transport maps with a very precise dependence on the dimension.
This allows us to compare local fluctuations of the eigenvalues and show universality.
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge,
MA 02139-4307 USA. email: bekerman@mit.edu.
†
The University of Texas at Austin, Mathematics Dept. RLM 8.100, 2515 Speedway Stop C1200, Austin,
Texas 78712-1202, USA. email: figalli@math.utexas.edu.
‡
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge,
MA 02139-4307 USA. email: guionnet@math.mit.edu.
⇤
1
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
2
We mention that universality was first proved in the = 2 case, where orthogonal polynomial techniques can be used (see e.g. [Meh04, CK06, LL08]), and then extended to the
case = 1, 4 [DG09]. The local fluctuations of more general -ensembles were only derived
recently [VV09, RRV11] in the Gaussian case. Universality in the -ensembles was addressed
in [BEYa, BEYb, KRV, Shca].
The main idea of this paper is that we can build a smooth approximate transport map
which at the first order is simply a tensor product, and then it has first order corrections
of order N 1 (see Theorem 1.4). This continues the long standing idea developed in loop
equations theory to study the correlation functions of -matrix models by clever change of
variables, see e.g. [AM90, BEMPF12]. On the contrary to loop equations, the change of
variable has to be of order one rather than infinitesimal.
The first order term of our map is simply the monotone transport map between the
asymptotic equilibrium measures; then, corrections to this first order are constructed so that
densities match up to a priori smaller fluctuating terms. As we shall see, our transport map
is constructed as the flow of a vector field obtained by approximately solving a linearized
Jacobian equation and then making a suitable ansatz on the structure of the vector field.
The errors are controlled by deriving bounds on covariances and correlations functions thanks
to loop equations, allowing us to obtain a self-contained proof of universality (see Theorem
1.5). Although optimal transportation will never be really used, it will provide us the correct
intuition in order to solve this problem (see Sections 2.2 and 2.3).
We notice that this last step could also be performed either by using directly central
limit theorems, see e.g. [Shca, BG13, Shcb], or local limit laws [BEYa, BEYb]. However,
our approach has the advantage of being pretty robust and should generalize to many other
mean field models. In particular, it should be possible to generalize it to the several-matrices
models, at least in the perturbative regime considered in [GS], but this would not solve yet
the question of universality as the transport map would be a non-commutative function of
several matrices. Even in the case of GUE matrices, there are not yet any results about the
local statistics of the eigenvalues of such non-commutative functions, except for a few very
specific cases.
We now describe our results in detail. Given a potential V : R ! R, we consider the
probability measure (1.1). We assume that V goes to infinity faster than log |x| (that is
V (x)/ log |x| ! +1 as |x| ! +1) so that in particular ZVN is finite.
We will use µV to denote the equilibrium measure, which is obtained as limit of the spectral
measure and is characterized as the unique minimizer (among probability measures) of
ˆ
ˆ
(1.2)
IV (µ) := V (x)dµ(x)
log |x y|dµ(x)dµ(y) .
2
We assume hereafter that another smooth potential W is given so that V +W goes to infinity
faster than log |x|. We denote Vt := V + tW , and we shall make the following assumption:
Hypothesis 1.1. We assume that µV0 and µV1 have a connected support and are non-critical,
that is, there exists a constant c̄ > 0 such that, for t = 0, 1,
p
dµVt
= St (x) (x at )(bt x)
with St c̄ a.e. on [at , bt ].
dx
Remark 1.2. The assumption of a connected support could be removed here, following the
lines of [Shca, BG]. Indeed, only a generalization of Lemma 3.2 is required, which is not
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
3
difficult. However, the non-criticality assumption cannot be removed easily, as criticality
would result in singularities in the transport map.
Finally, we assume that the eigenvalues stay in a neighborhood of the support [at ✏, bt +✏]
with large enough PN
C N p for some
V -probability, that is with probability greater than 1
p large enough. By [BG, Lemma 3.1], the latter is fulfilled as soon as:
Hypothesis 1.3. For t = 0, 1,
(1.3)
UVt (x) := Vt (x)
ˆ
dµVt (y) log |x
y|
achieves its minimal value on [a, b]c at its boundary {a, b}
All these assumptions are verified for instance if Vt is strictly convex for t 2 {0, 1}.
The main goal of this article is to build an approximate transport map between PN
V and
N
PV +W : more precisely, we construct a map T N : RN !RN such that, for any bounded
measurable function ,
ˆ
ˆ
C(log N )3
N
N
(1.4)
T dPV
dPN
k k1
V +W 
N
for some constant C independent of N , and which has a very precise expansion in the
dimension (in the following result, T0 : R ! R is a smooth transport map of µV onto µV +W ,
see Section 4):
Theorem 1.4. Assume that V 0 , W are of class C 30 and satisfy Hypotheses 1.1 and 1.3.
Then there exists a map T N = (T N,1 , . . . , T N,N ) : RN !RN which satisfies (1.4) and has the
form
1
ˆ := ( 1 , . . . , N ),
T N,i ( ˆ ) = T0 ( i ) + T1N,i ( ˆ ) 8 i = 1, . . . , N,
N
where T0 : R ! R and T1N,i : RN ! R are smooth and satisfy uniform (in N ) regularity
estimates. More precisely, T N is of class C 23 and we have the decomposition T1N,i = X1N,i +
1
X N,i where
N 2
(1.5)
sup kX1N,k kL4 (PVN )  C log N,
1kN
kX2N kL2 (PVN )  CN 1/2 (log N )2 ,
for some constant C > 0 independent of N . In addition, with probability greater than
1 N N/C ,
p
N,k
N,k0
(1.6)
max
|X
(
)
X
(
)|

C
log
N
N| k
k0 |.
1
1
0
1k,k N
As we shall see in Section 5, this result implies universality as follows (compare with
[BEYb, Theorem 2.4]):
Theorem 1.5. Assume V 0 , W 2 C 30 , and let T0 be as in Theorem 1.4 above. Denote P̃VN
the distribution of the increasingly ordered eigenvalues i under PN
V . There exists a constant
Ĉ > 0, independent of N , such that the following two facts hold true:
(1) Given m 2 N, assume that, under P̃VN ,pthe spacings N ( i
[N ✏, N (1 ✏)] are bounded by MN,m ⌧ N with probability 1
around i 2
pN,m . Then, for any
i+k )1km
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
4
Lipschitz function f on Rm ,
ˆ
f N(
i+1
i ), . . . , N ( i+m
i)
dP̃NV +W
ˆ
0
V
f T00 ( i )N ( i+1
i ), . . . , T0 ( i )N ( i+m
i ) dP̃N
✓
◆
✓
◆
2
p (log N )3 MN,m
(log N )3
MN,m log N
p
 Ĉ
+ pN,m kf k1 + Ĉ
m
+
+
krf k1 .
N
N
N
N
(2) Let aV (resp. aV +W ) denote the smallest point in the support of µV (resp. µV +W ), so
that supp(µV ) ⇢ [aV , 1) (resp. supp(µV +W ) ⇢ [aV +W , 1)). Given m 2 N, assume
that, under P̃VN , the numbers N 2/3 ( i aV )1im are bounded by MN,m ⌧ N 1/3 with
probability 1 pN,m . Then, for any Lipschitz function f on Rm ,
ˆ
f N 2/3 ( 1 aV +W ), . . . , N 2/3 ( m aV +W ) dP̃NV +W
ˆ ⇣
⌘
f N 2/3 T00 (aV ) 1 aV , . . . , N 2/3 T00 (aV ) m aV dP̃NV
✓
◆
✓
◆
2
p (log N )3 MN,m
(log N )3
log N
 Ĉ
+ pN,m kf k1 + Ĉ
m
+ 2/3 + 1/3 krf k1 .
N
N 4/3
N
N
The same bound holds around the largest point in the support of µV .
Remark 1.6. The condition that V 0 , W 2 C 30 in the theorem above is clearly non-optimal
(compare with [BEYa]). For instance, by using Stieltjes transform instead of Fourier transform in some of our estimates, we could reduce the regularity assumptions on V 0 , W to C 21
by a slightly more cumbersome proof. In addition, by using [BEYb, Theorem 2.4] we could
also weaken our regularity assumptions in Theorem 1.4, as we could use that result to estimate the error terms in Section 3.4. However, the main point of this hypothesis for us is to
stress that we do not need to have analytic potentials, as often required in matrix models
theory. Moreover, under this assumption we can provide self-contained and short proofs of
Theorems 1.4 and 1.5.
Note that Theorem 1.4 is well suited to prove universality of the spacings distribution in
the bulk as stated in Theorem 1.5, but it is not clear how to directly deduce the universality
of the rescaled density, see e.g [BEYa, Theorem 2.5(i)]. Indeed, this corresponds to choosing
test functions whose uniform norm blows up like some power of the dimension, so to apply
Theorem 1.4 we should have an a priori control on the numbers of eigenvalues inside sets of
N
size of order N 1 under both PN
V and PV +W . Notice however that [BEYa, Theorem 2.5(ii)]
requires
1, while our results hold for any > 0. In particular, the edge universality
proved in Theorem 1.5(2) is completely new for 2 (0, 1). In addition our strategy is very
robust and flexible. For instance, although we shall not pursue this direction here, it looks
likely to us that one could use it prove the universality of the asymptotics of the law of
{N ( i x)}1im under P̃NV for given i and x.
The paper is structured as follows: In Section 2 we describe the general strategy to construct our transport map as the flow of vector fields obtained by approximately solving a
linearization of the Monge-Ampère equation (see (2.2)). As we shall explain there, this idea
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
5
comes from optimal transport theory. In Section 3 we make an ansatz on the structure of
an approximate solution to (2.2) and we show that our ansatz actually provided a smooth
solution which enjoys very nice regularity estimates that are uniform as N ! 1. In Section
N
4 we reconstruct the approximate transport map from PN
V to PV +W via a flow argument.
The estimates proved in this section will be crucial in Section 5 to show universality.
Acknowledgments: AF was partially supported by NSF Grant DMS-1262411. AG was
partially supported by the Simons Foundation and by NSF Grant DMS-1307704.
2. Approximate Monge-Ampère equation
2.1. Propagating the hypotheses. The central idea of the paper is to build transport
maps as flows, and in fact to build transport maps between PVN and PVNt where t 7! Vt
is a smooth function so that V0 = V , V1 = V + W . In order to have a good interpolation
between V and V +W , it will be convenient to assume that the support of the two equilibrium
measures µV and µV +W (see (1.2)) are the same. This can always be done up to an affine
transformation. Indeed, if L is the affine transformation which maps [a1 , b1 ] (the support
of µV1 ) onto [a0 , b0 ] (the support of µV0 ), we first construct a transport map from PVN to
V +W
L⌦N
= PNV +W̃ where
] PN
(2.1)
W̃ = V
L
1
+W
L
1
V,
and then we simply compose our transport map with (L 1 )⌦N to get the desired map from
PVN to PVN+W . Hence, without loss of generality we will hereafter assume that µV and µV +W
have the same support. We then consider the interpolation µVt with Vt = V + tW , t 2 [0, 1].
We have:
Lemma 2.1. If Hypotheses 1.1 and 1.3 are fulfilled for t = 0, 1, Hypothesis 1.1 is also fulfilled
for all t 2 [0, 1]. Moreover, we may assume without loss of generality that V goes to infinity
as fast as we want up to modify PVN and PVN+W by a negligible error (in total variation).
Proof. Let ⌃ denote the support of µV and µV +W . Following [BG13, Lemma 5.1], the measure
µVt is simply given by
µVt = (1
t)µV + tµV +W .
Indeed, µV is uniquely determined by the fact that there exists a constant c such that
ˆ
log |x y|dµV (x) V  c
with equality on the support of µV , and this property is verified by linear combinations. As
a consequence the support of µVt is ⌃, and its density is bounded away from zero on ⌃. This
shows that Hypothesis 1.1 is fulfilled for all t 2 [0, 1].
Furthermore, we can modify PVN and PVN+W outside an open neighborhood of ⌃ without
changing the final result, as eigenvalues will quit this neighborhood only with very small
probability under our assumption of non-criticality according to the large deviation estimate
[BG]:
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
1
ln PVN [9 i :
N !1 N
1
lim inf ln PVN [9 i :
N !1 N
lim sup
with Ũ V := U V
6
inf Ũ V (x),
i
2 F] 
2
x2F
i
2 ⌦]
2
x2⌦
inf Ũ V (x).
⇤
inf U V .
Thanks to the above lemma and the discussion immediately before it, we can assume that
µV and µV +W have the same support, that W is bounded, and that V goes to infinity faster
than xp for some p > 0 large enough.
N
2.2. Monge-Ampère equation. Given the two probability densities PN
Vt to PVs as in (1.1)
with 0  t  s  1, by optimal transport theory it is well-known that there exists a
N
N
N
(convex) function N
t,s such that r t,s pushes forward PVt onto PVs and which satisfies the
Monge-Ampère equation
det(D2
N
t,s )
=
⇢t
⇢s (r
N
t,s )
,
⇢⌧ :=
dPN
V⌧
d 1... d
N
(see for instance [Vil, Chapters 3 and 4] or the recent survey paper [DF] for an account on
optimal transport theory and its link to the Monge-Ampère equation).
Because t,t (x) = |x|2 /2 (since r t,t is the identity map), we can differentiate the above
equation with respect to s and set s = t to get
(2.2)
N
t
X @i
= cN
t
i<j
where
N
t
:= @s
N
t,s |s=t
N
t
@j
i
j
N
t
+N
X
i
W ( i) + N
X
Vt0 ( i )@i
N
t ,
i
and
cN
t
:=
N
ˆ X
N
W ( i ) dPN
Vt = @t log ZVt .
i
N
Although this is a formal argument, it suggests to us a way to construct maps T0,t
: RN ! R N
N
N
N
N
N
N
N
sending PN
T0,t
sends PN
V onto PVt : indeed, if T0,t sends PV onto PVt then r t,s
V onto PVs .
N
N
N
Hence, we may try to find T0,s
of the form T0,s
=r N
T0,t
+ o(s t). By differentiating
t,s
N
N
this relation with respect to s and setting s = t we obtain @t T0,t
= r tN (T0,t
).
N
N
N
Thus, to construct a transport map T from PV onto PV +W we could first try to find
N
N
solving the ODE ẊtN = r tN (XtN ) and setting
t by solving (2.2), and then construct T
N
N
N
T := X1 . We notice that, in general, T is not an optimal transport map.
Unfortunately, finding an exact solution of (2.2) enjoying “nice” regularity estimates that
are uniform in N seems extremely difficult. So, instead, we make an ansatz on the structure
of tN (see (3.2) below): the idea is that at first order eigenvalues do not interact, then at
order 1/N eigenvalues interact at most by pairs, and so on. As we shall see, in order to
construct a function which enjoys nice regularity estimates and satisfies (2.2) up to a error
that goes to zero as N ! 1, it will be enough to stop the expansion at 1/N . Actually,
while the argument before provides us the right intuition, we notice that there is no need to
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
7
assume that the vector field generating the flow XtN is a gradient, so we will consider general
N
N
N
N
vector fields YN
that approximately solve
t = (Y1,t , . . . , YN,t ) : R ! R
(2.3)
X YN
i,t
N
divYN
t = ct
YN
j,t
i
i<j
+N
j
X
W ( i) + N
i
X
Vt0 ( i )YN
i,t ,
i
We begin by checking that the flow of an approximate solution of (2.3) gives an approximate transport map.
2.3. Approximate Jacobian equation. Here we show that if a C 1 vector field YN
t approximately satisfies (2.3), then its flow
N
ẊtN = YN
t (Xt ),
X0N = Id,
produces almost a transport map.
N
N
More precisely, let YN
be a smooth vector field and denote
t : R ! R
N
RN
t (Y )
:=
cN
t
X YN
i,t
i
i<j
Lemma 2.2. Let
YN
t . Then
ˆ
YN
j,t
+N
j
X
W ( i) + N
i
X
Vt0 ( i )YN
i,t
divYN
t .
i
: RN ! R be a bounded measurable function, and let XtN be the flow of
(XtN ) dPN
V
ˆ
dPN
Vt
 k k1
ˆ
0
t
N
kRN
ds.
s (Y )kL1 (PN
V )
s
1
Proof. Since YN
t 2 C , by Cauchy-Lipschitz Theorem its flow is a bi-Lipchitz homeomorphism.
If JXtN denotes the Jacobian of XtN and ⇢t the density of PN
Vt , by the change of variable
formula it follows that
ˆ
ˆ
dPN
Vt =
(XtN )JXtN ⇢t (XtN ) dx
thus
(2.4)
ˆ
(XtN ) dPN
V
ˆ
dPN
Vt
 k k1
ˆ
|⇢0
JXtN ⇢t (XtN )| dx =: At
N
Using that @t (JXtN ) = divYN
t JXt and that the derivative of the norm is smaller than the
norm of the derivative, we get
ˆ
|@t At | 
@t JXtN ⇢t (XtN ) dx
ˆ
N
N
N
N
N
N
N
= |divYN
t JXt ⇢t (Xt ) + JXt (@t ⇢t )(Xt ) + JXt r⇢t (Xt ) · @t Xt | dx
ˆ
N
N
N
= |RN
t (Y)|(Xt ) JXt ⇢t (Xt ) dx
ˆ
N
= |RN
t (Y)| dPVt .
Integrating the above estimate in time completes the proof.
By taking the supremum over all functions
with k k1  1, the lemma above gives:
⇤
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
8
N
N
N
N
Corollary 2.3. Let XtN be the flow of YN
t , and set P̂t := (Xt )# PV the image of PV by
XtN . Then
ˆ t
N
N
N
kP̂t
P V t kT V 
kRN
ds.
s (Y )kL1 (PN
V )
s
0
3. Constructing an approximate solution to (2.2)
Fix t 2 [0, 1] and define the random measures
X
1 X
(3.1)
LN :=
and
M
:=
N
i
N i
i
i
N µVt .
As we explained in the previous section, a natural ansatz to find an approximate solution of
(2.2) is given by
(3.2)
ˆ h
¨
i
1
1
N
0,t (x) +
1,t (x) dMN (x) +
2,t (x, y) dMN (x) dMN (y),
t ( 1 , . . . , N ) :=
N
2N
where (without loss of generality) we assume that 2,t (x, y) = 2,t (y, x).
Since we do not want to use gradient of functions but general vector fields (as this gives us
more flexibility), in order to find an ansatz for an approximate solution of (2.3) we compute
first the gradient of :
ˆ
1 0
1 N
N
0
N
@i t = 0,t ( i ) +
( i ) + ⇠1,t ( i , MN ),
⇠1,t (x, MN ) := @1 2,t (x, y) dMN (y).
N 1,t
N
This suggests us the following ansatz for the components of YN
t :
(3.3)
ˆ
1
1
N
Yi,t ( 1 , . . . , N ) := y0,t ( i ) + y1,t ( i ) + ⇠ t ( i , MN ), ⇠ t (x, MN ) := zt (x, y) dMN (y),
N
N
for some functions y0,t , y1,t : R ! R, zt : R2 ! R.
Here and in the following, given a function of two variables , we write 2 C s,v to denote
that it is s times continuously differentiable with respect to the first variable and v times
with respect to the second.
The aim of this section is to prove the following result:
Proposition 3.1. Assume V 0 , W 2 C r with r
30. Then, there exist y0,t 2 C r 2 , y1,t 2
C r 8 , and zt 2 C s,v for s + v  r 5, such that
✓
◆
X YN
X
X
YN
i,t
j,t
N
N
N
0
Rt := ct
+N
W ( i) + N
Vt ( i )Yi,t
divYN
t
i<j
i
j
i
i
satisfies
(log N )3
t
N
for some positive constant C independent of t 2 [0, 1].
kRN
C
t kL1 (PN
V )
The proof of this proposition is pretty involved, and will take the rest of the section.
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
9
3.1. Finding an equation for y0,t , y1,t , zt . Using (3.3) we compute
divYN
t
=N
ˆ
y00,t (x) dLN (x)
+
ˆ
y01,t (
) dLN (x) +
ˆ
@1 ⇠ t (x, MN ) dLN (x) + ⌘(LN ),
where, given a measure ⌫, we set
⌘(⌫) :=
Therefore, recalling that MN = N (LN
RN
t
ˆ
@2 zt (y, y) d⌫(y).
µVt ) we get
ˆ
ˆ
y0,t (x) y0,t (y)
2
0
2
=
dLN (x) dLN (y) + N
Vt y0,t dLN + N
W dLN
x y
ˆ
y1,t (x) y1,t (y)
N
dLN (x) dLN (y) + N Vt0 y1,t dLN
2
x y
¨
ˆ
N
⇠ t (x, MN ) ⇠ t (y, MN )
dLN (x) dLN (y) + N Vt0 (x) ⇠ t (x, MN ) dLN (x)
2
x y
⇣
⌘ˆ
1
⌘(MN ) N 1
y00,t dLN
N
2
⇣
⌘ˆ
⇣
⌘ˆ
0
1
y1,t dLN
1
@1 ⇠ t (x, MN ) dLN (x) ⌘(µVt ) + c̃N
t ,
2
2
N2
2
¨
¨
where c̃N
t is a constant and we use the convention that, when we integrate a function of the
f (x) f (y)
form x y with respect to LN ⌦ LN , the diagonal terms give f 0 (x).
We now observe that LN converges towards µVt as N ! 1 [AG97], and the latter minimizes IVt (see (1.2)). Hence, considering µ" := (x + "f )# µVt and writing that IVt (µ" )
IVt (µVt ), by taking the derivative with respect to " at " = 0 we get
ˆ
(3.4)
Vt0 (x)f (x) dµVt (x)
=
2
¨
f (x)
x
f (y)
dµVt (x) dµVt (y)
y
for all smooth bounded functions f : R ! R. Therefore we can recenter LN by µVt in the
formula above: more precisely, if we set
(3.5)
⌅f (x) :=
ˆ
f (x)
x
f (y)
dµVt (y) + Vt0 (x)f (x),
y
then
N
2
ˆ
Vt0
f dLN
N2
2
¨
f (x)
x
f (y)
dLN (x) dLN (y)
y
ˆ
¨
f (x)
= N ⌅f dMN
2
x
f (y)
dMN (x) dMN (y)
y
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
10
Applying this identity to f = y0,t , y1,t , ⇠ t (·, MN ) and recalling the definition of ⇠ t (·, MN ) (see
(3.3)), we find
ˆ
N
Rt = N [⌅y0,t + W ] dMN
◆
ˆ
ˆ ✓
⇣
⌘
0
+
⌅y1,t +
1 y0,t + @1 zt (z, ·)dµVt (z)
dMN
2
✓
◆
¨
y0,t (x) y0,t (y)
+
dMN (x) dMN (y) ⌅zt (·, y)[x]
+ CtN + EN ,
2
x y
where
zt (x, y) zt (x̃, y)
dµVt (x̃) + Vt0 (x)zt (x, y),
x x̃
N
Ct is a deterministic term, and EN is a reminder that we will prove to be negligible:
ˆ
⌘ˆ
1
1⇣
EN :=
@2 zt (x, x) dMN (x)
1
y01,t dMN
N
N
2
⌘¨
1⇣
1
@1 zt (x, y) dMN (x) dMN (y)
N
2
¨
(3.6)
y1,t (x) y1,t (y)
dMN (x) dMN (y)
2N
x y
˚
zt (x, y) zt (x̃, y)
dMN (x) dMN (y) dMN (x̃).
2N
x x̃
⌅zt (·, y)[x] =
ˆ
Hence, for RN
t to be small we want to impose
⌅y0,t =
(3.7)
⌅zt (·, y)[x] =
⌅y1,t =
W + c,
y0,t (x) y0,t (y)
+ c(y),
2
x y
ˆ
⇣
⌘
0
1 y0,t + @1 zt (z, ·) dµVt (z) + c0 ,
2
where c, c0 are some constant to be fixed later, and c(y) does not depend on x.
3.2. Inverting the operator ⌅. We now prove a key lemma, that will allow us to find the
desired functions y0,t , y1,t , zt .
Lemma 3.2. Given V : R ! R, assume that µV has support given by [a, b] and that
p
dµV
(x) = S(x) (x a)(b x)
dx
with S(x) c̄ > 0 a.e. on [a, b].
Let g : R!R be a C k function and assume that V is of class C p . Set
ˆ
f (x) f (y)
⌅f (x) :=
dµV (x) + V 0 (x)f (x)
x y
Then there exists a unique constant cg such that the equation
⌅f (x) = g(x) + cg
TRANSPORT MAPS FOR
has a solution of class C (k
constant Cj such that
2)^(p 3)
-MATRIX MODELS AND UNIVERSALITY
. More precisely, for j  (k
2) ^ (p
11
3) there is a finite
(3.8)
kf kC j (R)  Cj kgkC j+2 (R) ,
P
where, for a function h, khkC j (R) := jr=0 kh(r) kL1 (R) .
Moreover f (and its derivatives) behaves like (g(x) + cg )/V 0 (x) (and its corresponding
derivatives) when |x| ! +1.
This solution will be denoted by ⌅ 1 g.
Note that Lf (x) = ⌅f 0 (x) can be seen as the asymptotics of the infinitesimal generator of
the Dyson Brownian motion taken in the set where the spectral measure approximates µV .
This operator is central in our approach, as much as the Dyson Brownian motion is central
to prove universality [ESYY12, BEYa, BEYb].
Proof. As a consequence of (3.4), we have
ˆ
1
(3.9)
PV
dµV (y) = V 0 (x)
x y
on the support of µV .
Therefore solving the equation ⌅f (x) = g(x) + cg on the support of µV amounts to
ˆ
f (y)
(3.10)
PV
dµV (y) = g(x) + cg
8 x 2 [a, b].
x y
Let us write
p
d(x) := dµV /dx = S(x) (x a)(b x)
with S positive inside the support [a, b]. We claim that S 2 C p 3 ([a, b]).
1
c
Indeed,
´ by (3.4)1 with f (x) = (z x) for z 2 [a, b] , we find that the Stieltjes transform
G(z) = (z y) dµV (y) satisfies, for z outside [a, b],
ˆ 0
V (y) V 0 (<(z))
2
0
G(z) = G(z)V (<(z)) + F (z),
with F (z) =
dµV (y) .
2
z y
Solving this quadratic equation so that G ! 0 as |z| ! 1 yields
⌘
p
1⇣ 0
(3.11)
G(z) =
V (<(z))
[V 0 (<(z))]2 + 2 F (z) .
Notice that V 0 (<(z))2 + 2 F (z) becomes real as z goes to the real axis. Hence, since
⇡ 1 =G(z) converges to the density of µV as z goes to the real axis (see e.g [AGZ10,
Theorem 2.4.3]), we get
⇥
⇤
(3.12)
S(x)2 (x a)(b x) = ( ⇡) 2 V 0 (x)2 + 2 F (x) .
This implies in particular that ´{a,
the two points of the real line where V 0 (x)2 +2 F (x)
´ 1b} are
vanishes. Moreover F (x) =
V 00 (↵y + (1 ↵)x) d↵ dµV (y) is of class C p 2 on R (recall
0
that V 2 C p by assumption), therefore (V 0 )2 + 2 F 2 C p 2 (R). Since we assumed that S
does not vanish in [a, b], from (3.12) we deduce that S is of class C p 3 on [a, b].
To solve (3.10) we apply Tricomi’s formula [Tri57, formula 12, p.181] and we find that, for
x 2 [a, b],
ˆ bp
p
(y a)(b y)
f (x) (x a)(b x)d(x) = P V
(g(y) + cg )dy + c2 := h(x)
y x
a
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
12
for some constant c2 , hence
h(x) =
f (x)(xp a)(b x)S(x)
´ b (y a)(b y)
= PV a
(g(y) + cg )dy + c2
y x
´bp
´ b p(y a)(b
= P V a (y a)(b y) g(y)y g(x)
dy
+
(g(x)
+
c
)P
V
g
x
y x
a
´bp
g(y) g(x)
a+b
= a (y a)(b y) y x dy ⇡ x
(g(x) + cg ) + c2 ,
2
where we used that, for x 2 [a, b],
ˆ bp
(y
PV
a
a)(b
y x
y)
dy =
Set
ˆ bp
h0 (x) =
(y
a)(b
a
y)
⇣
⇡ x
g(y)
y
y)
dy + c2
a + b⌘
.
2
g(x)
dy.
x
Then h0 is of class C
(recall that g is of class C k ). We next choose cg and c2 such that h
vanishes at a and b (notice that this choice uniquely identifies cg ).
We note that f 2 C (k 2)^(p 3) ([a, b]). Moreover, we can bound its derivatives in terms of
the derivatives of h0 , g and S: if we assume j  p 3, we find that there exists a constant
Cj , which depends only on the derivatives of S, such that
⇣
⌘
(p+1)
(j)
(p+1)
kf kL1 ([a,b])  Cj max kh0
kL1 ([a,b]) + kg
kL1 ([a,b])  Cj max kg (p) kL1 ([a,b]) .
k 1
pj
pj+2
Let us define
k(x) :=
PV
ˆ
f (y)
dµV (y)
x y
g(x)
cg
8 x 2 R.
By (3.10) we see that k ⌘ 0 on [a, b]. To ensure that ⌅f = g + cg also outside the support
of µV we want
✓
◆
ˆ
1
0
f (x)
PV
dµV (y) V (x) = k(x)
8 x 2 [a, b]c .
x y
Let us consider the function ` : R ! R defined as
ˆ
1
(3.13)
`(x) := P V
dµV (y)
x y
V 0 (x).
p
Notice that, thanks to (3.11), `(x) = G(x) V 0 (x) =
[V 0 (x)]2 + 2 F (x). Hence,
comparing this expression with (3.12), and recalling that S c̄ > 0 in [a, b], we deduce that
[V 0 (x)]2 + 2 F (x) is smooth and has simple zeroes both at a and b, therefore [V 0 (x)]2 +
2 F (x) > 0 in [a ✏, b + ✏]\[a, b] for some ✏ > 0.
This shows that ` does not vanish in [a ✏, b + ✏]\[a, b]. Recalling that can freely modify
V outside [a ✏, b + ✏] (see proof of Lemma 2.1), we can actually assume that ` vanishes at
{a, b} and does not vanish in the whole [a, b]c .
We claim that ` is Hölder 1/2 at the boundary points, and in fact is equivalent to a square
root there. Indeed, it is immediate to check that ` is of class C p 1 except possibly at the
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
boundary points {a, b}. Moreover
ˆ
ˆ b
1
1 p
PV
dµV (y) = S(a)
(y a)(b y) dy
x y
y
a x
✓ˆ 1
◆
ˆ b
p
y a
0
+
S (↵a + (1 ↵)y)d↵
(y
y 0
a x
The first term can be computed exactly and we have, for some c 6= 0,
s
✓
ˆ b
a+b
⇣x
p
x
1
2
(3.14)
(y a)(b y) dy = c(b a)
y
b a
b
a x
13
a)(b
a+b
2
a
⌘2
y) dy.
1
4
◆
which is Hölder 1/2, and in fact behaves as a square root at the boundary points. On the
other hand, since S is of class C p 3 on [a, b] with p 4, the second function is differentiable,
with derivative at a given by
✓ˆ 1
◆
ˆ b
p
1
0
S (↵a + (1 ↵)y)d↵
(y a)(b y) dy,
y 0
a a
which is a convergent integral. The claim follows.
Thus, for x outside the support of µV we can set
f (x) := `(x) 1 k(x).
With this choice ⌅f = g + cg and f is of class C (k 2)^(p 3) on R \ {a, b}.
We now want to show that f is of class C (k 2)^(p 3) on the whole R. For this we need to
check the continuity of f and its derivatives at the boundary points, say at a (the case of b
being similar). We take hereafter r  (k 2) ^ (p 3), so that f has r derivatives inside
[a, b] according to the above considerations.
Let us first deduce the continuity of f at a. We write, with f (a+ ) = limx#a f (x),
k(x) = f (a+ )`(x) + k1 (x)
with
k1 (x) :=
✓
PV
ˆ
f (y)
dµV (y)
x y
PV
ˆ
◆
f (a+ )
dµV (y) + g(x) + cg + f (a+ )Vt0 (x).
x y
Notice that since f = ` 1 k outside [a, b], if we can show that ` 1 (x)k1 (x) ! 0 as x " a then
we would get f (a ) = f (a+ ), proving the desired continuity.
To prove it we first notice that k1 vanishes at a (since both k and ` vanish inside [a, b]),
hence
✓
◆
ˆ
ˆ
f (y) f (a+ )
f (y) f (a+ )
k1 (x) =
PV
dµV (y) P V
dµV (y) + g̃(x) g̃(a)
x y
a y
ˆ
f (y) f (a+ )
= (a x) P V
dµV (y) + g̃(x) g̃(a),
(x y)(a y)
with g̃ := g + f (a+ )V 0 2 C 1 . Assume 1  (k 2) ^ (p 3). Since f is of class C 1 inside [a, b]
we have |f (y) f (a+ )|  C|y a|, from which we deduce that |k1 (x)|  C|x a| for x  a.
Hence ` 1 (x)k1 (x) ! 0 as x " a (recall that ` behaves as a square root near a), which
proves that
lim f (x) = lim f (x)
x"a
x#a
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
and shows the continuity of f at a.
We now consider the next derivative: we write
⇥
k(x) = f (a) + f 0 (a+ )(x
with
k2 (x) := (a
+ g̃(x)
14
⇤
a) `(x) + k2 (x)
f (a+ ) (y a)f 0 (a+ )
dµV (y)
(x y)(a y)
g̃(a) + f 0 (a+ )(x a)Vt0 (x).
x) P V
ˆ
f (y)
Since k = ` ⌘ 0 on [a, b] we have k2 (a) = k20 (a+ ) = k20 (a ) = 0. Hence, since f is of class C 2
on [a, b], we see that |k2 (x)|  C|x a|2 for x  a, therefore k2 (x)/`(x) is of order |x a|3/2 ,
thus
f (x) = f (a) + f 0 (a+ )(x a) + O(|x a|3/2 )
for x  a,
which shows that f has also a continuous derivative.
We obtain the continuity of the next derivatives similarly. Moreover, away from the
boundary point the j-th derivative of f outside [a, b] is of the same order than that of g/V 0 ,
while near the boundary points it is governed by the derivatives of g nearby, therefore
(3.15)
kf (j) kL1 ([a,b]c )  Cj0 max kg (r) kL1 (R) .
rj+2
Finally, it is clear that f behaves like (g(x) + cg )/V 0 (x) when x goes to infinity.
⇤
3.3. Defining the functions y0,t , y1,t , zt . To define the functions y0,t , y1,t , zt according to
(3.7), notice that Lemma 2.1 shows that the hypothesis of Lemma 3.2 are fulfilled. Hence,
as a consequence of Lemma 3.2 we find the following result (recall that 2 C s,v means that
is s times continuously differentiable with respect to the first variable and v times with
respect to the second).
Lemma 3.3. Let r
7. If W, V 0 2 C r , we can choose y0,t of class C r 2 , zt 2 C s,v for
s + v  r 5, and y1,t 2 C r 8 . Moreover, these functions (and their derivatives) go to zero
at infinity like 1/V 0 (and its corresponding derivatives).
Proof. By Lemma 3.2 we have y0,t = ⌅ 1 W 2 C r 2 . For zt , we can rewrite
ˆ 1
⌅zt (·, y)[x] =
y0 (↵x + (1 ↵)y) d↵ + c(y)
2 0 0,t
ˆ 1
=
[y00,t (↵x + (1 ↵)y) + c↵ (y)] d↵
2 0
where we choose c↵ (y) to be the unique constant provided by Lemma
´ 13.2 which ensures that
1 0
⌅ [y0,t (↵x + (1 ↵)y) + c↵ (y)] is smooth. This gives that c(y) = 0 c↵ (y)d↵. Since ⌅ 1 is
a linear integral operator, we have
ˆ 1
zt (x, y) =
⌅ 1 [y00,t (↵ · +(1 ↵)y)](x) d↵ .
2 0
As the variable y is only a translation, it is not difficult to check that zt 2 C s,v for any
s + v  r 5. It follows that
ˆ
⇣
⌘
0
1 y0,t + @1 zt (z, ·) dµVt (z) + c0
2
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
is of class C r 6 and therefore by Lemma 3.2 we can choose y1,t 2 C r 8 , as desired.
The decay at infinity is finally again a consequence of Lemma 3.2.
15
⇤
3.4. Getting rid of the random error term EN . We show that the L1PN -norm of the error
Vt
term EN defined in (3.6) goes to zero. To this end, we first make some general consideration
on the growth of variances.
Following [MMS, Theorem 1.6], up to assume that Vt goes sufficiently fast at infinity
(which we did, see Lemma 2.1), we have that there exists a constant ⌧0 > 0 so that for all
⌧ ⌧0 ,
r
✓
◆
log N
2
N
PVt D(LN , µVt ) ⌧
 e c⌧ N log N ,
N
where D is the 1-Wasserstein distance
ˆ
D(µ, ⌫) := sup
f (dµ d⌫) .
kf 0 k1 1
Since MN = N (LN
µVt ) we get
1
D(LN , µVt ) =
N
(3.16)
hence for ⌧
(3.17)
⌧0
PN
Vt
✓
sup
kf kLip 1
ˆ
f dMN
sup
kf 0 k1 1
ˆ
f dMN ,
p
⌧ N log N
◆
e
c⌧ 2 N log N
.
´
This already shows that, if f is sufficiently smooth, f (x, y)dMN (x) dMN (y) is of order at
most N log N . More precisely
✓ˆ
◆
ˆ
ˆ
ˆ
i⇣x
i⇠x
ˆ
f (x, y) dMN (x) dMN (y) = f (⇣, ⇠)
e dMN (x) e dMN (x) d⇠ d⇣,
2
so that with probability greater than 1 e c⌧0 N log N we have
ˆ
ˆ
2
(3.18)
f (x, y) dMN (x) dMN (y)  ⌧0 N log N
|fˆ(⇣, ⇠)| |⇣| |⇠| d⇣ d⇠ .
To improve this estimate, we shall use loop equations´ as well as Lemma 3.2. Given a
function g and a measure ⌫, we use the notation ⌫(g) := g d⌫.
Lemma 3.4. Let g be a smooth function. Then, if M̃N = N LN N EVt [LN ], there exists a
finite constant C such that
ˆ
(1)
1
MN (g) dPN
Vt  C m(g) =: BN (g)
N (g) :=
ˆ ⇣
⌘2
⇣
⌘
(2)
N
2
1
0
2
M̃N (g) dPVt  C m(g) + m(g)kgk1 + k⌅ gk1 kg k1 =: BN
(g)
N (g) :=
ˆ ⇣
⌘4
(4)
(g)
:=
M̃
(g)
dPN
N
Vt
N
⇣
⌘
(2)
1
0
3
2 (2)
4
4
 C k⌅ gk1 kg k1 N (g) + kgk1 m(g) + m(g) N (g) + m(g) =: BN
(g),
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
16
where
m(g) := 1
1
0
k(⌅ g) k1 +
2
log N
2
ˆ
ˆ 1 g|(⇠) |⇠|3 d⇠.
|⌅
Proof. First observe that, by integration by parts, for any C 1 function f
ˆ ✓ X
ˆ X
X f ( i) f ( j ) ◆
N
0
(3.19)
N
V ( i )f ( i )
dPVt =
f 0 ( i ) dPN
Vt
i
i
i<j
j
i
which we can rewrite as the first loop equation
ˆ
ˆ ⇣
ˆ
⌘ˆ
f (x)
N
0
(3.20)
MN (⌅f ) dPVt =
1
f dLN +
2
2N
x
f (y)
dMN (x)dMN (y) dPN
Vt .
y
We denote
⇣
FN (g) := 1
2
⌘ˆ
0
1
(⌅ g) dLN +
2N
⌅ 1 g(x)
x
ˆ
⌅ 1 g(y)
dMN (x) dMN (y)
y
so that taking f := ⌅ 1 g in (3.20) we deduce
ˆ
ˆ
N
MN (g) dPVt = FN (g) dPN
Vt .
To bound the right hand side above, we notice that ⌅ 1 g goes to zero at infinity like 1/V 0
(see Lemma 3.2). Hence we can write its Fourier transform and get
ˆ
⌅ 1 g(x)
x
⌅ 1 g(y)
dMN (x) dMN (y)
y
ˆ
ˆ
1
ˆ
= i d⇠ ⇠ ⌅ g(⇠)
0
1
d↵
ˆ
e
i↵⇠x
dMN (x)
ˆ
ei(1
↵)⇠y
dMN (y),
so that we deduce (recall (3.16))
supp
D(LN ,µVt )⌧0
log N/N
FN (g)  (1 + ⌧02 ) m(g).
On the other hand, as the mass of MN is always bounded by 2N , we
p deduce that FN (g)
is bounded everywhere by N m(g). Since the set {D(LN , µVt )
⌧0 N log N } has small
probability (see (3.17)), we conclude that
ˆ
c⌧02 N log N
(3.21)
MN (g) dPN
m(g) + (1 + ⌧02 )m(g)  C m(g),
Vt  N e
which proves our first bound.
Before proving the next estimates, let us make a simple remark: using the definition of
MN and M̃N it is easy to check that, for any function g,
ˆ
(3.22)
MN (g) M̃N (g) =
MN (g) dPN
Vt .
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
17
To get estimates on the covariance we obtain the second loop equation by changing V (x)
into V (x) + g(x) in (3.19) and differentiating with respect to at = 0. This gives
ˆ
ˆ
N
MN (⌅f )M̃N (g) dPVt = LN (f g 0 ) dPN
Vt

ˆ
ˆ
ˆ
(3.23)
⇣
⌘
f (x) f (y)
+
1
f 0 dLN +
dMN (x)dMN (y) M̃N (g) dPN
Vt .
2
2N
x y
´
We now notice that MN (⌅f ) M̃N (⌅f ) is deterministic and M̃N (g) dPN
Vt = 0, hence the
left hand side is equal to
ˆ
M̃N (⌅f )M̃N (g) dPN
Vt .
We take f := ⌅ 1 g and we
pargue similarly to above (that is, splitting the estimate depending
whether D(LN , µVt ) ⌧0 N log N or not, and use that |M̃N (g)|  N kgk1 ) to deduce that
´
(2)
|M̃N (f )|2 dPN
Vt satisfies
N (g) :=
ˆ
(2)
0
1
|FN (g)||M̃N (g)|dPN
Vt
N (g)  kg ⌅ gk1 +
ˆ
1
0
2
c⌧02 N log N
(3.24)
 k⌅ gk1 kg k1 + N e
kgk1 m(g) + C m(g) |M̃N (g)|dPN
Vt
= k⌅ 1 gk1 kg 0 k1 + N 2 e
c⌧02 N log N
m(g)kgk1 + C m(g)
(2)
1/2
.
N (g)
Solving this quadratic inequality yields
h
i
(2)
2
1
0
(g)

C
m(g)
+
m(g)kgk
+
k⌅
gk
kg
k
1
1
1
N
for some finite constant C.
We finally turn to the fourth moment. If we make an infinitesimal change of potential
V (x) into V (x) + 1 g2 (x) + 2 g3 (x) and differentiate at 1 = 2 = 0 into (3.23) we get,
denoting g = g3 ,
(3.25)
ˆ
ˆ X
N
MN (⌅f )M̃N (g1 )M̃N (g2 )M̃N (g3 ) dPVt =
LN (f g 0 (1) )M̃N (g
ˆ ⇣
1
2
⌘ˆ
0
f dLN +
2N
ˆ
f (x)
x
(2) )M̃N (g (3) )
dPN
Vt +
f (y)
dMN (x) dMN (y) MN (g1 )M̃N (g2 )M̃N (g3 ) dPN
Vt ,
y
where we sum over the permutation of {1, 2, 3}. Taking ⌅f = g1 = g2 = g3 = g, by (3.22),
(3.21), and Cauchy-Schwarz inequality we get
h
i
(4)
(2)
(4)
0
1
3
3/4
2 (2)
+ m(g) N (g) ,
N (g)  C kg ⌅ gk1 N (g) + kgk1 m(g) + m(g) N (g)
which implies
(4)
N (g)
h
0
 C kg ⌅
1
(2)
gk1 N (g)
+
kgk31 m(g)
+
(2)
m(g)2 N (g)
Applying the above result with g = ei · we get the following:
4
i
+ m(g) .
⇤
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
18
Corollary 3.5. Assume that V 0 , W 2 C r with r 8. Then there exists a finite constant C
such that, for all 2 R,
ˆ
7 2
(3.26)
|MN (ei · )|2 dPN
Vt  C[log N (1 + | | )] ,
ˆ
7 4
(3.27)
|MN (ei · )|4 dPN
Vt  C[log N (1 + | | )] .
Proof. In the case g(x) = ei x we estimate the norms of ⌅ 1 g by using Lemma 3.2, and we
get a finite constant C such that
k⌅ 1 g 0 k1  C| |3 ,
k⌅ 1 gk1  C| |2 ,
whereas, since ⌅ 1 g goes fast to zero at infinity (as 1/V 0 ), for j  r 3 we have (see Lemma
3.2)
1
j+2
ˆ 1 g|(⇠)  C k⌅ gkC j  C 0 kgkC j+2  C 0 1 + | | .
|⌅
1 + |⇠|j
1 + |⇠|j
1 + |⇠|j
Hence, we deduce that there exists a finite constant C 0 such that
✓
◆
ˆ
1 + | |7 3
3
m(g)  C log N | | + 1 + d⇠
|⇠| = C 0 log N 1 + | |7 ,
1 + |⇠|5
1
(g)  C 0 log N 1 + | |7 ,
BN
2
BN
(g)  C 0 (log N )2 1 + | |7
4
BN
(g)  C 0 (log N )4 1 + | |7
2
4
,
.
Finally, for k = 2, 4, using (3.22) and (3.21) we have
✓ˆ
◆
ˆ
k
i · k
N
k 1
i · k
N
1
|MN (e )| dPVt  2
|M̃N (e )| dPVt + BN (g)
from which the result follows.
⇤
We can now estimate EN .
The linear term can be handled in the same way as we shall do now for the quadratic and
cubic terms (which are actually more delicate), so we just focus on them.
We have two quadratic terms in MN which sum up into
¨
⌘¨
y1,t (x) y1,t (y)
1⇣
1
EN =
1
@1 zt (x, y) dMN (x) dMN (y)
dMN (x) dMN (y).
N
2
2N
x y
Writing
y1,t (x)
x
y1,t (y)
=
y
we see that
¨
y1,t (x)
x
ˆ
0
1
y01,t (↵x
+ (1
↵)y) d↵ =
y1,t (y)
dMN (x) dMN (y) =
y
ˆ 1 ✓ˆ
0
ˆ
0
d⇠ yc
1,t (⇠)
ˆ
0
i(↵x+(1
0
yc
1,t (⇠)e
↵)y)⇠
◆
d⇠ d↵
1
d↵ MN (ei↵⇠· )MN (ei(1
↵)⇠·
),
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
so using (3.26) we get
✓ˆ
ˆ
(log N )2
2
1
N
|EN | dPVt  C
d⇠ |ŷ1,t |(⇠) |⇠| 1 + |⇠|7
N
¨
+
d⇠ d⇣ |ẑt |(⇠, ⇣) |⇠| 1 + |⇠|7
19
1 + |⇣|
7
◆
.
It is easy to see that the right hand side is finite if y1,t and zt are smooth enough (recall that
these functions and their derivatives decay fast at infinity). More precisely, to ensure that
C
2
|ŷ1,t |(⇠) |⇠| 1 + |⇠|7 
2 L1 (R)
2
1 + |⇠|
and
C
|ẑt |(⇠, ⇣) |⇠| 1 + |⇠|7 1 + |⇣|7 
2 L1 (R2 ),
1 + |⇠|3 + |⇣|3
we need y1,t 2 C 17 and zt 2 C 11,7 \ C 8,10 , so (recalling Lemma 3.3) V 0 , W 2 C 25 is enough
to guarantee that the right hand side is finite.
Using (3.26), (3.27), and Hölder inequality, we can similarly bound the expectation of the
cubic term
˚
zt (x, y) zt (x̃, y)
2
EN =
dMN (x) dMN (y) dMN (x̃)
2N
x x̃
¨
ˆ 1
d
d⇠ d⇣ @1 zt (⇠, ⇣)
d↵ MN (ei↵⇠· )MN (ei(1 ↵)⇠· )MN (ei⇣· )
=i
2N
0
to get
ˆ
¨
(log N )3
2
2
N
|EN | dPVt  C
d⇠ d⇣ |ẑt (⇠, ⇣)| |⇠| 1 + |⇠|7
1 + |⇣|7 .
N
Again the right hand side is finite if zt 2 C 18,7 \ C 15,10 , which is ensured by Lemma 3.3 if
V 0 , W are of class C 30 .
3.5. Control on the deterministic term CtN . By what we proved above we have
ˆ
(log N )3
N
N
|RN
C
|
dP

C
,
t
t
Vt
N
thus, in particular,
(log N )3
N
N
Ct
E[Rt ]  C
.
N
Notice now that, by construction,
X
N
RN
=
LY
+
N
W ( i ) + cN
t
t
t
with cN
t =
E[N
P
i
i
W ( i )] and
LY := divY +
X Yi
i<j
i
Yj
j
N
X
V 0 ( i )Yi ,
i
and an integration by parts shows that, under PNV , E[LY] = 0 for any vector field Y. This
(log N )3
N
implies that E[RN
.
t ] = 0, therefore |Ct |  C
N
This concludes the proof of Proposition 3.1.
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
20
4. Reconstructing the transport map via the flow
In this section we study the properties of the flow generated by the vector field YN
t defined
0
r
in (3.3). As we shall see, we will need to assume that W, V 2 C with r 15.
We consider the flow of YN
t given by
N
ẊtN = YN
t (Xt ).
XtN : RN ! RN ,
Recalling the form of YN
t (see (3.3)) it is natural to expect that we can give an expansion
for XtN . More precisely, let us define the flow of y0,t ,
(4.1)
Ẋ0,t = y0,t (X0,t ),
X0,t : R ! R,
X0,t ( ) = ,
N,1
N,N
N
and let X1,t
= (X1,t
, . . . , X1,t
) : RN ! RN be the solution of the linear ODE
N,k
Ẋ1,t
( 1, . . . ,
(4.2)
N)
N,k
= y00,t (X0,t ( k )) · X1,t
( 1, . . . ,
ˆ
X
+ zt (X0,t ( k ), y) dMN 0,t (y)
N
⇣
1 X
+
@2 zt X0,t (
N j=1
k ), X0,t ( j )
+ y1,t (X0,t (
N)
⌘
N,j
· X1,t
( 1, . . . ,
k ))
N)
X
N
with the initial condition X1,t
= 0, and MN 0,t is defined as
ˆ
ˆ
N 
X
X0,t
f (y)dMN (y) =
f (X0,t ( i ))
f dµVt
8 f 2 Cc (R).
i=1
If we set
N
X0,t
( 1, . . . ,
N)
:= X0,t ( 1 ), . . . , X0,t (
N)
,
then the following result holds.
Lemma 4.1. Assume that W, V 0 2 C r with r 15. Then the flow XtN = (XtN,1 , . . . , XtN,N ) :
N
RN ! RN is of class C r 8 and the following properties hold: Let X0,t and X1,t
be as in (4.1)
N
N
N
and (4.2) above, and define X2,t : R ! R via the identity
N
XtN = X0,t
+
1 N
1 N
X1,t + 2 X2,t
.
N
N
Then
(4.3)
N,k
sup kX1,t
kL4 (PVN )  C log N,
1kN
where
N
kXi,t
kL2 (PVN )
=
✓ˆ
N 2
|Xi,t
| dPVN
◆1/2
,
N
kX2,t
kL2 (PVN )  CN 1/2 (log N )2 ,
N
|Xi,t
|
:=
s X
j=1,...,N
N,j 2
|Xi,t
|,
i = 0, 1, 2.
In addition, there exists a constant C > 0 such that, with probability greater than 1 N
p
N,k
N,k0
(4.4)
max
|X
(
,
.
.
.
,
)
X
(
,
.
.
.
,
)|

C
log
N
N| k
1
N
1
N
k0 |.
1,t
1,t
0
1k,k N
N/C
,
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
21
r 8
Proof. Since YN
(see Lemma 3.3) it follows by Cauchy-Lipschitz theory that XtN is
t 2 C
of class C r 8 .
Using the notation ˆ = ( 1 , . . . , N ) 2 RN and
XtN,k,
N,k
N,k
X1,t
X2,t
ˆ
( )+
( ˆ ) = (1
k) +
N
N2
( ˆ ) := X0,t (
)X0,t (
k)
+ XtN,k ( ˆ )
X N,s
and defining the measure MN t as
ˆ
N 
X
XtN,s
(4.5)
f (y) dMN (y) =
f (1
N,i ˆ
i ) + sXt ( )
s)X0,t (
i=1
ˆ
f dµVt
8 f 2 Cc (R).
N
by a Taylor expansion we get an ODE for X2,t
:
(4.6)
N,k ˆ
Ẋ2,t
( )
=
ˆ
1
0
+N
⇣
⌘
N,k ˆ
y00,t XtN,k,s ( ˆ ) ds · X2,t
( )
ˆ 1h
0
+
ˆ
1
ˆ
1
0
+
0
y00,t
y01,t
ˆ
⇣
⇣
XtN,k,s ( ˆ )
XtN,k,s ( ˆ )
⌘
ds ·
+
+
+
⇣
@1 zt X0,t (
N ˆ 1
X
j=1
0
N ˆ 1
X
j=1
0
y00,t
⇣
⇣
X0,t (
k)
⌘i
N,k ˆ
ds · X1,t
( )
N,k ˆ ⌘
X2,t
( )
+
N
N,k ˆ
X1,t
( )
⇣
⌘
X N,s
@1 zt XtN,k,s ( ˆ ), y dMN t (y)
ˆ
ˆ
⌘
k ), y
⌘
⇣
@1 zt X0,t (
X
dMN 0,t (y)
·
k ), y
⇣
@2 zt XtN,k,s ( ˆ ), XtN,j,s ( ˆ )
X
dMN 0,t (y)
ds ·
⇣
N,k ˆ
X1,t
( )
N,k ˆ ⌘
X2,t
( )
+
N
N,k ˆ ⌘
X2,t
( )
+
N
⇣
⌘
N,j ˆ
@2 zt X0,t ( k ), X0,t ( j ) ds · X1,t
( )
N,k ˆ
X1,t
( )
⇣
⌘
@2 zt XtN,k,s ( ˆ ), XtN,j,s ( ˆ )
⇣
⌘
⌘
N,j ˆ
X2,t
( )
ds ·
,
N
N,k
with the initial condition X2,0
= 0. Using that
(see Lemma 3.3) we obtain
(4.7)
ky0,t kC r
2 (R)
C
kX0,t kC r
2 (R)
 C.
N
We now start to control X1,t
. First, simply by using that MN has mass bounded by 2N we
N,k
obtain the rough bound |X1,t |  C N . Inserting this bound into (4.6) one easily obtain the
N,k
bound |X2,t
|  C N 2.
We now prove finer estimates. First, by (3.17) together with the fact that X0,t and
x 7! zt (y, x) are Lipschitz (uniformly in y), it follows that there exists a finite constant C
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
such that, with probability greater than 1 N
ˆ
X
(4.8)
zt (·, ) dMN 0,t ( )
Hence it follows easily from (4.2) that
N/C
1
22
,
p
 C log N N .
p
N,k
max kX1,t
k1  C log N N
(4.9)
k
outside a set of probability bounded by N N/C .
N
N
In order to control X2,t
we first estimate X1,t
in L4 (PVN ): using (4.2) again, we get
(4.10)
⌘
d⇣
N,k
max kX1,t
kL4 (PVN )
dt k
✓
C
N,k
max kX1,t
kL4 (PVN )
k
ˆ
+1+
zt (X0,t (
X0,t
k ), y) dMN (y)
L4 (PV
N)
◆
.
N
N
To bound X1,t
in L4 (PVN ) and then to be able to estimate X2,t
in L2 (PVN ), we will use the
following estimates:
Lemma 4.2. For any k = 1, . . . , N ,
ˆ
X
(4.11)
zt (X0,t ( k ), y) dMN 0,t (y)
ˆ
(4.12)
⇣
@1 zt X0,t (
k ), y
⌘
L4 (PV
N)
 C log N,
X
dMN 0,t (y)
L4 (PV
N)
 C log N.
Proof. We write the Fourier decomposition of ⌘t (x, y) := zt (X0,t (x), X0,t (y)) to get
ˆ
ˆ
ˆ
⌘t (x, y) dMN (y) = ⌘ˆt (x, ⇠) ei⇠y dMN (y) d⇠ .
Since zt 2 C u,v for u + v  r
5 and X0,t 2 C r
|ˆ
⌘t (x, ⇠)| 
so that using (3.27) we get
ˆ
sup
⌘t (x, y) dMN (y)
x
L4 (PV
N)

2
(see (4.7)), we deduce that
C
1 + |⇠|r
ˆ
⌘ˆt (·, ⇠)
 C log N
ˆ
 C log N,
5
,
1
ˆ
ei⇠y dMN (y)
⌘ˆt (·, ⇠)
d⇠
L4 (PV
N)
1
1 + |⇠|7 d⇠
provided r > 13. The same arguments work for @1 zt provided r > 14. Since by assumption
r 15, this concludes the proof.
⇤
Inserting (4.11) into (4.10) we get
(4.13)
N,k
kX1,t
kL4 (PVN )  C log N
which proves the first part of (4.3).
8 k = 1, . . . , N,
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
23
N
We now bound the time derivative of the L2 norm of X2,t
: using that MN has mass
bounded by 2N , in (4.6) we can easily estimate
ˆ 1h ⇣
⌘
⇣
⌘i
C N,k
N,k ˆ
N,k 2
N,k
N
y00,t XtN,k,s ( ˆ )
y00,t X0,t ( k ) ds · X1,t
( )  C|X1,t
| + |X1,t
| |X2,t
|,
N
0
ˆ
1
0
ˆ
N ˆ
X
j=1
0
1
⇣
@1 zt XtN,k,s ( ˆ ), y
⌘
⌘
X
),
y
dMN 0,t (y) ds
k
✓
◆
C N,k
CX
1 N,j
N,k
N,j
 C|X1,t | + |X2,t | +
|X1,t | + |X2,t | ,
N
N j
N
X N,s
dMN t (y)
⇣
⌘
@2 zt XtN,k,s ( ˆ ), XtN,j,s ( ˆ )
ˆ
⇣
@1 zt X0,t (
⇣
@2 zt X0,t (
hence
d
kX N k2 2 V = 2
dt 2,t L (PN )
ˆ X
C
C
+
N
N,j
ds |X1,t
|
✓
◆
CX
1 N,j
N,j 2
N,j

|X1,t | + |X2,t | |X1,t | ,
N j
N
N,k 2
|X2,t
|
k
ˆ X
+C
ˆ X
k
N,k
N,k 2
|X1,t
||X2,t
| dPVN
k
ˆ X
k
dPVN
C
N,k 3
|X2,t
| dPVN +
N
+C
0
N,k 2
N,k
|X1,t
| |X2,t
|dPVN
ˆ X
ˆ X
ˆ X
C
N,k 2
N,j
+ 3
|X2,t
| |X2,t
| dPVN
N
k,j
ˆ
ˆ 1 ˆ
⇣
X
N,k
+
X2,t ·
@1 zt X0,t (
k
⌘
N,k
N,k
X2,t
· Ẋ2,t
dPVN
k
ˆ X
C
+ 2
N
k ), X0,t ( j )
k,j
k
N,k
N,k
|X1,t
||X2,t
| dPVN
N,j
N,k
N,k
|X1,t
| |X1,t
| |X2,t
| dPVN
k ), y
⌘
X
N,k
dMN 0,t (y) ds · X1,t
dPVN
ˆ X
ˆ X
C
C
N,j 2
N,k
N,k
N,j
N,j
V
+
|X1,t | |X2,t | dPN + 2
|X2,t
| |X2,t
| |X1,t
| dPVN
N
N
k,j
k,j
ˆ X
C
N,k
N,j
+
|X2,t
| |X2,t
| dPVN .
N
k,j
N,k
N,k
Using the trivial bounds |X1,t
|  C N and |X2,t
|  C N 2 , (4.12), and elementary inequalities such as, for instance,
⌘
X N,j
X⇣ N,j
N,k
N,k
N,k 4
N,k 2
|X1,t | |X1,t
| |X2,t
|
|X1,t |4 + |X1,t
| + |X2,t
| ,
k,j
k,j
TRANSPORT MAPS FOR
we obtain
(4.14)
-MATRIX MODELS AND UNIVERSALITY
24
✓
ˆ X
d
N,k 4
N 2
N 2
kX2,t kL2 (PV )  C kX2,t kL2 (PV ) +
|X1,t
| dPVN
N
N
dt
k
◆
ˆ X
X
N,k 2
N,k
N,k
V
+
|X1,t | dPN +
log N kX2,t kL2 (PVN ) kX1,t kL4 (PVN ) .
k
k
We now observe, by (4.13), that the last term is bounded by
X
N,k 2
N 2
N 2
kX2,t
kL2 (PV ) + (log N )2
kX1,t
kL4 (PV )  kX2,t
kL2 (PV ) + C N (log N )4 .
N
N
N
k
N,k
N,k
Hence, using that kX1,t
kL2 (PVN )  kX1,t
kL4 (PVN ) and (4.13) again, the right hand side of
N 2
(4.14) can be bounded by CkX2,t kL2 (PV ) + C N (log N )4 , and a Gronwall argument gives
N
N 2
kX2,t
kL2 (PV )
N
thus
 C N (log N )4 ,
N
kX2,t
kL2 (PVN )  C N 1/2 (log N )2 ,
concluding the proof of (4.3).
We now prove (4.4): using (4.2) we have
0
|Ẋ N,k ( ˆ ) Ẋ N,k ( ˆ )|
1,t
1,t
N,k ˆ
y00,t (X0,t ( k0 ))| |X1,t
( )|
N,k ˆ
N,k0 ˆ
+ |y00,t (X0,t ( k0 ))| |X1,t
( ) X1,t
( )| + |y1,t (X0,t (
ˆ ⇣
⌘
X
+
zt (X0,t ( k ), y) zt (X0,t ( k0 ), y) dMN 0,t (y)

|y00,t (X0,t ( k ))
N ˆ
⇣
1 X 1
+
@2 zt X0,t (
N j=1 0
k ), X0,t ( j )
⌘
⇣
@2 zt X0,t (
k ))
y1,t (X0,t (
k0 ), X0,t ( j )
⌘
k0 ))|
N,j ˆ
ds |X1,t
( )|.
Using that |X0,t ( k ) X0,t ( k0 )|  C| k
k0 |, the bound (4.9), the Lipschitz regularity of
0
y0,t , y1,t , zt , and @2 zt , and the fact that
ˆ
p
X
@1 zt (·, ) dMN 0,t ( )
 C log N N
1
with probability greater than 1 N
(see (3.17)), we get
p
0
N,k ˆ
N,k ˆ
N,k ˆ
N,k0 ˆ
|Ẋ1,t
( ) Ẋ1,t
( )|  C|X1,t
( ) X1,t
( )| + C log N N |
N/C
outside a set of probability less than N
N/C
k
, so (4.4) follows from Gronwall.
k0 |
⇤
5. Transport and universality
In this section we prove Theorem 1.5 on universality using the regularity properties of
the approximate transport maps obtained in the previous sections. We note that the hypotheses in the statement of the theorem are verified when V (x) = 12 x2 , and in that cases
the fluctuation estimates follow the Sine kernel in the bulk (after rescaling by N ) and the
Tracy-Widom fluctuations at the edge (after rescaling by N 2/3 ), see [VV09, RRV11].
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
25
Proof of Theorem 1.5. Let us first remark that the map T0 from Theorem 1.4 coincides with
X0,1 , where X0,t is the flow defined in (4.1). Also, notice that X1N : RN ! RN is an
V +W
approximate transport of PN
(see Lemma 2.2 and Proposition 3.1). Set X̂1N :=
V onto PN
N
N
N
N
N
X0,1
+ N1 X1,1
, with X0,t
and X1,t
as in Lemma 4.1. Since X1N X̂1N = N12 X2,1
, recalling (4.3)
1
2
and using Hölder inequality to control the L norm with the L norm, we see that
ˆ
ˆ
ˆ
1
N
N
N
N
N
g(X̂1 ) dPV
g(X1 ) dPV  krgk1 2 |X2,1
| dPN
V
N
1
N
(5.1)
 krgk1 2 kX2,1
kL2 (PV )
N
(log N )2
 Ckrgk1
.
N 3/2
V +W
This implies that also X̂1N : RN ! RN is an approximate transport of PN
. In
V onto PN
N
addition, we see that X̂1 preserves the order of the i with large probability. Indeed, first of
all X0,t : R ! R is the flow of y0,t which is Lipschitz with some constant L, so by Gronwall
we have
e
Lt
j
i
In particular,
e
L
X0,t ( i )  eLt
 X0,t ( j )
j
i
1 N,j ˆ
X ( )
N 1,t
N ),
j
i
,
8
j
i
i
<
j.
.
since
1 N,i ˆ
log N
X1,t ( )  C p |
N
N
(see (4.4)) with probability greater than 1
1
C
i
X0,1 ( i )  eL
 X0,1 ( j )
Hence, using the notation ˆ = ( 1 , . . . ,
j
 X̂1N,j ( ˆ )
N
N/C
j|
i
we get
X̂1N,i ( ˆ )  C
j
i
with probability greater than 1 N N/C .
We now make the following observation: the ordered measures P̃VN and P̃VN+W are obtained
N
N
N
as the image of PN
defined as
V and PV +W via the map R : R ! R
[R(x1 , . . . , xN )]i := min max xj .
]J=i j2J
Notice that this map is 1-Lipschitz for the sup norm.
p
Hence, if g is a function of m-variables we have kr(g R)k1  mkrgk1 , so by Lemma
2.2, Proposition 3.1, and (5.1), we get
ˆ
ˆ
⌘
p
(log N )3 ⇣
N
N
N
g R(N X̂1 ) dPV
g R dPV +W  C
kgk1 + m krgk1 .
N
Since X̂1N preserves the order with probability greater than 1 N N/C , we can replace
g R(N X̂1N ) with g(N X̂1N R) up to a very small error bounded by kgk1 N N/C . Hence,
N
N
N
since R# PN
V = P̃V and R# PV +W = P̃V +W , we deduce that, for any Lipschitz function
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
26
f : Rm ! R,
ˆ
f N(
i ), . . . , N ( i+m
i+1
ˆ
⇣
f N X̂1N,i+1 ( ˆ )
i)
dP̃NV +W
X̂1N,i ( ˆ ) , . . . , N X̂1N,i+m ( ˆ )
⌘
X̂1N,i ( ˆ )
dP̃NV
⌘
p
(log N )3 ⇣
C
kf k1 + m krf k1 .
N
Recalling that
X̂1N,j ( ˆ ) = X0,1 ( j ) +
1 N,j ˆ
X ( ),
N 1,1
we first observe that, as X0,1 is of class C 2 ,
X0,1 ( i )
X0,1 (
i+k )
0
= X0,1
( i )(
i+k )
i
+ O(|
i
i+k |
2
).
Also, by (4.4) we deduce that, out of a set of probability bounded by N N/C ,
p
N,i+k ˆ
N,i ˆ
(5.2)
|X1,1
( ) X1,1
( )|  C log N N | i+k
i |,
p
and the right hand side is bounded by C MN,m log N/ N on the set {N | i+k
i |  MN,m }
which has probability greater than 1 pN,m . Hence, we see that with probability greater
than 1 pN,m N N/C it holds
✓
◆
2
MN,m
MN,m log N
N,i ˆ
N,i+k ˆ
0
X̂1 ( ) X̂1
( ) = X0,t ( i )( i
+
,
i+k ) + O
N 3/2
N2
from which the first bound follows easily.
For the second point we observe that aV +W = X0,1 (aV ) and, arguing as before,
ˆ
f N 2/3 ( 1 aV +W ), . . . , N 2/3 ( m aV +W ) dP̃NV +W
ˆ ⇣
f N 2/3 X̂1N,1 ( ˆ ) X0,1 (aV ) , . . . , N 2/3 X̂1N,m ( ˆ )
⌘
X0,1 (aV ) dP̃NV
✓
◆
p
(log N )3
m
C
kf k1 + 1/3 krf k1 .
N
N
Since, by (4.3),
X̂1N,i (
) = X0,1 ( i ) + OL4 (PVN )
= X0,1 (aV ) +
✓
log N
N
0
X0,1
(aV )( i
we conclude as in the first point.
◆
aV ) + O(
i
2
aV ) + OL4 (PVN )
✓
◆
log N
,
N
⇤
TRANSPORT MAPS FOR
-MATRIX MODELS AND UNIVERSALITY
27
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