Large Deviations Asymptotics of Rectangular Spherical Integral

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Large Deviations Asymptotics of Rectangular Spherical Integral

Alice Guionnet, Jiaoyang Huang

To cite this version:

Alice Guionnet, Jiaoyang Huang. Large Deviations Asymptotics of Rectangular Spherical Integral.

2021. �hal-03272504�

Large Deviations Asymptotics of Rectangular Spherical Integral

Alice Guionnet ^{∗ 1} and Jiaoyang Huang ^†2

1

CNRS-ENS Lyon

2

New York University

Abstract

In this article we study the Dyson Bessel process, which describes the evolution of singular values of rectangular matrix Brownian motions, and prove a large deviation principle for its empirical particle density. We then use it to obtain the asymptotics of the so-called rectangular spherical integrals as m, n go to infinity while m/n converges.

Contents

1 Introduction 2

2 Dyson Bessel Process 6

3 Large deviations for the Dyson Brownian motion 11

4 Large deviations for the Dyson Bessel process 25

5 Applications 39

∗

aguionne@ens-lyon.fr

†

jh4427@nyu.edu

1 Introduction

In this article we shall study the asymptotics of the so-called rectangular spherical integrals, also called Berezin-Karpelevich type integrals in the literature. This type of integrals arises when one studies rectangular matrices and is the natural counterpart of the well known Harish-Chandra -Itzykson-Zuber (HCIZ) integral. The interest in spherical integrals comes from different fields. Harish-Chandra was motivated by Fourier analysis in semi-simple Lie algebras. They appear in physics as the density in matrix models such as the Ising model [16, 25, 44] or more generally matrix models with an external field [11], including the famous Kontsevich matrix model [40]. Their uses in random matrix theory appeared more recently. First it was shown that spherical integral with a rank one external field gives asymptotically the famous R-transform defined by Voiculescu in free probability [32] as an analogue of Fourier transform. This approach was generalized to the rectangular-free convolution by using rectangular spherical integrals [27] or to the multiplicative free convolution and the S-transform [10, 45].

Knowing the asymptotics of rank one spherical integrals allowed as well to investigate the large deviations for the extreme eigenvalues of random matrices. This approach was introduced in [30] where it was shown that the probability that the largest eigenvalue of a Wigner matrix takes an unexpected value is the same when the entries are Rademacher or Gaussian. This universality phenomenon was shown to hold for random matrices with i.i.d. entries whose Laplace transform is bounded by the Laplace transform of a Gaussian variable with the same covariance. For more general sub-Gaussian entries, a transition appears in the rate function between large deviations towards a very large value with a heavy tail type rate function, and deviations close to the bulk which are governed by the Gaussian rate function.

Such considerations were extended to unitary invariant ensembles [33], to the joint distribution of the largest eigenvalue and its eigenvector [9], to sum of matrices, to finitely many extreme eigenvalues [31].

Indeed, the asymptotics of spherical integral could be extended to finite rank external fields [31]. For small enough matrices, the same asymptotics were shown to extend to the case where the rank goes to infinity more slowly than the dimension [19] and to full rank matrices [18]. However, the limit differs when the rank of both matrices are of the same order and the matrices do not have small norms. Such a limit can as well be used to prove large deviation principles for the empirical measure of the eigenvalues of random matrices [5] and more generally study the asymptotics of matrix models with an external field [11, 29].

The formula for the asymptotics of HCIZ integrals was foreseen by Matytsin [42] and then proven

rigorously in [29, 34,35]. Matytsin used the description of Spherical integrals as invariant eigenfunctions

of the Laplacian. The approach of [34] is kind of dual and based on a representation of spherical integrals

as the density of a Dyson Brownian motion conditioned at time one, a representation which allows to use

large deviations techniques and martingales. In this paper, we follow the same route for the rectangular

case but prove a more general large deviation principle for conditioned Dyson Brownian motions. In fact,

the result in [34] relies on the matrix model, and only concerns the case β = 1 or 2 whereas we can deal

in this paper with all cases β > 1. The extension of [34] to the rectangular case is a natural step, which

however posed significant difficulties for the proof of the lower bound if one uses the methods of [34],

due to additional singularity of the drifts. We should also mention the heuristics proposed in this setting

in [26] following Matystin’s arguments. One key idea of this paper is to improve the large deviations

lower bound by obtaining better criteria for the uniqueness of solutions to McKean-Vlasov equations

with smooth fields inspired from [41], rather than the weaker approach developed in [15]. Another

novelty in this paper is a quantitative estimate for the convergence to Dyson Brownian motion with

very general potential by a coupling argument, see Proposition 3.5. Under more restricted assumptions,

i.e. the limiting profile has square root behavior around the edge, such quantitative estimates for the

convergence has been obtained in [?,?,?] by using the characteristic method. The quantitative estimate

for the convergence allows us to efficiently control the locations of each particles and extend our result to

Dyson Bessel processes which arises when one considers rectangular matrices and hence derive the limits

of rectangular spherical integrals. We now state more precisely our main results.

The rectangular spherical integral is given by I

n,m

(A

n

, B

n

) =

Z Z

e

^{βnRe[Tr(A∗nU BnV∗)]}

dU dV, (1.1) {e:UAVB} {e:UAVB}

where if β = 1, U ∈ O(n), V ∈ O(m) follow the Haar distribution over the orthogonal group, and A

n

∈ R

^n×m

, B

n

∈ R

^n×m

, whereas for β = 2, U ∈ U(n), V ∈ U(m) follow the Haar distribution over the unitary group, A

n

∈ C

^n×m

, B

n

∈ C

^n×m

for β = 2. We call such integrals rectangular spherical integrals and shall study their asymptotic behavior when m and n go to infinity so that the ratio m/n converges towards some 1 + α ∈ [1, ∞). This type of spherical integral arises when one studies rectangular matrices and is the natural counterpart of the well known Harish-Chandra -Itzykson-Zuber (HCIZ) integral defined when β = 2 and for two self-adjoint matrices A

n

, B

n

∈ C

^n×n

by

I

n

(A

n

, B

n

) = Z

e

^{nTr(AnU BnU∗)}

dU ,

where U follows the Haar distribution over the unitary group. This integral was shown by Harish- Chandra [37] and then Itzykson and Zuber [38] to be equal to a determinant:

I

n

(A

n

, B

n

) = c

n

det e

^naibj

16i,j6n

∆(a)∆(b) , (1.2) {tg} {tg}

where a = (a

1

, a

2

, · · · , a

n

), b = (b

1

, b

2

, · · · , b

n

) are eigenvalues of A

n

and B

n

respectively, and ∆(a) = Q

i<j

(a

i

− a

j

), ∆(b) = Q

i<j

(b

i

− b

j

) are Vandermonde determinants. In 2003, Schlittgen and Wettig [46]

considered a generalization of the above rectangular spherical integral given by Z Z

det[U V ]

^ν

e

^{τTr(A∗nU BnV∗+V D∗nU∗Cn)/2}

dU dV, (1.3) {e:inegral0} {e:inegral0}

where U, V ∈ U (n) are n × n unitary matrices following Haar distribution, A

n

, B

n

, C

n

, D

n

are determin- istic n × n matrices, and ν is a non-negative integer. They showed that the generalization of the above integral to the case of unequal dimensions of U, V leads to an integral which can be nonzero only if ν = 0, and predicted the following formula: for m > n

Z Z

e

^{τTr(A∗nU BnV∗+V D∗nU∗Cn)/2}

dU dV = τ

^n(m−1)

Q

n

i=1

(m − i)!(n − i)!

∆(x

²

)∆(y

²

) Q

n

i=1

(x

i

y

i

)

^m−n

det[I

m−n

(2τ x

i

y

j

)]

_16i,j6n

, (1.4) {e:integral1} {e:integral1}

where U ∈ U(n) is an n × n unitary matrix, V ∈ U(m) is an m × m unitary matrix, both follow the Haar distribution, B

n

, C

n

are deterministic n × m matrices, and A

n

, D

n

are deterministic m × n rectangular matrices, I

m−n

(x) is the Bessel function

I

κ

(2y) = y

^κ

∞

X

k=0

y

^2k

k!(k + κ)! ,

and x

²

= (x

²₁

, x

²₂

, · · · , x

²_n

), y

²

= (y

₁²

, y

₂²

, · · · , y

²_n

) are eigenvalues of the matrices A

n

C

_n^∗

, B

n

D

^∗_n

. This formula was proven in [28]. We get the rectangular spherical integral (1.1) from (1.4) by taking A

n

= C

n

and B

n

= D

n

. Such formulas can be obtained by using the character expansion method. Another approach is based on heat flows [11, 43]. Indeed, one can notice that Fourier functions X → e

^iTr(AX)

are the eigenfunctions of the Laplacian for any matrix A. Looking for eigenfunctions depending only on the eigenvalues of X one gets the spherical integral I

n

(A

n

, X

n

), which in turns has to be an eigen- function of the Laplace operator restricted to functions invariant under conjugation, namely the Dyson Laplace operator L = −∆(X)

⁻¹

P

i

δ

²_xi

∆(X ). Note however that (1.2) and (1.4) are not useful to derive

asymptotics as they are given in terms of a signed sum of diverging terms.

For a rectangular n × m matrix A

n

, m > n, with non trivial singular values (s

i

)

16i6n

, we denote ˆ ν

ⁿ_A

its symmetrized empirical singular values

ˆ ν

Aⁿ

= 1

2n

n

X

i=1

(δ

s_i

+ δ

−s_i

) . We denote by Σ the non commutative entropy

Σ(ν) = Z

log |x − y|dν(x)dν(y) .

Then, we prove the following asymptotics for the rectangular spherical integrals:

{main1}

Theorem 1.1. Let A

n

, B

n

∈ R

^n×m

and U ∈ O(n), V ∈ O(m) following Haar distribution over orthogo- nal group for β = 1; A

n

, B

n

∈ C

^n×m

and U ∈ U (n), V ∈ U(m) following Haar distribution over unitary group, for β = 2, where m > n and m/n → 1 + α, α > 0. We assume that the symmetrized empirical singular values ˆ ν

_Aⁿ

and ν ˆ

_Bⁿ

of A

n

and B

n

converge weakly to ˆ ν

A

and ν ˆ

B

respectively. We moreover assume that for C = A or B, we have sup

_n

ν ˆ

Cⁿ

(x

²

) < ∞, Σ(ˆ ν

C

) > −∞ and, if α 6= 0, R

ln|x|dˆ ν

C

> −∞. Then, the following limit of the rectangular spherical integral exists

lim

n

1

n

²

log I

n,m

(A

n

, B

n

) = β

2 I

^α

(ˆ ν

A

, µ ˆ

B

), I

n,m

(A

n

, B

n

) = Z

e

^{βnRe[Tr(A∗nU BnV∗)]}

dU dV.

It is given explicitly by

I

^α

(ˆ ν

A

, ν ˆ

B

) = − inf

{ˆρ_t}_06t61

Z

1 0

Z

u

²_s

ρ ˆ

s

dxds + π

²

3

Z

1 0

Z ˆ

ρ

³_s

dxds + α

²

4

Z ρ ˆ

s

(x) x

²

dxds

+ (ˆ ν

A

(x

²

− α log |x|) + ˆ ν

B

(x

²

− α log |x|)) − (Σ(ˆ ν

A

) + Σ(ˆ ν

B

)) + const,

(1.5) {e:arate} {e:arate}

where const is a constant depending on α. The infimum is taken over continuous symmetric measure valued processes ( ˆ ρ

t

(x)dx)

0<t<1

such that

t→0

lim ρ ˆ

t

(x)dx = ˆ ν

A

, lim

t→1

ρ ˆ

t

(x)dx = ˆ ν

B

. (1.6) {e:bbterm} {e:bbterm}

Moreover, u is the weak solution of the following conservation of mass equation

∂

s

ρ ˆ

s

+ ∂

x

( ˆ ρ

s

u

s

) = 0.

This theorem will be proved in Section 5.1. We show in Proposition 5.1 that in fact the non commuta- tive law of (A

n

, U B

n

V

^∗

) converges when(U, V ) follows the Gibbs measure with free energy I

n,m

(A

n

, B

n

).

As in [34], the main point is to derive a large deviation principle for the associated processes, namely Bessel Dyson processes. Indeed, let G

n

be an n × m rectangular matrix with independent real (β = 1) or complex (β = 2) Gaussian entries and set

X

n

= A

n

+ 1

√ n G

n

.

then, we claim that the large deviation principle for the symmetrized empirical singular values of X

n

gives the asymptotics of spherical integrals. In fact, denote the singular value decomposition of X

n

as X

n

= U B

n

V

^∗

. Then the joint law of (B

n

, U, V ) is given by

1 Z

n,m

Y

i

b

^{β(m−n+1)−1}_i

Y

i<j

|b

²i

− b

²j

|

^β

e

^{−βn2 (Pib2i+Pa2i})+βnRe[Tr(A^∗_nU BnV^∗)]

dU dV dB

n

. (1.7) {e:lawXX0} {e:lawXX0}

Assume that we have proven a large deviation principle for ˆ ν

ⁿ_X

with a good rate function I

ˆν_A

so that for any symmetric probability measure ˆ ν

B

n→∞

lim 1

n

²

log P (ˆ ν

Bⁿ

∈ B (ˆ ν

B

, δ)) = −I

νˆ_A

(ˆ ν

B

) + o

δ(1)

(1.8) {conv10} {conv10}

where o

δ

(1) goes to zero as δ goes to zero. By integrating (1.7) over the ball B (ˆ ν

B

, δ), we have Z

ˆ

ν_Bⁿ∈B(ˆν_B,δ)

1 Z

n,m

Y

i

b

^{β(m−n+1)−1}_i

Y

i<j

|b

²i

− b

²j

|

^β

e

^{−βn2 (}

P ib²_i+P

a²_i)+βnRe[Tr(A^∗U BV^∗)]

dU dV dB

n

= 1

Z

n,m

e

^βn

2 2 (2αR

log|x|dˆν_B+2Σ(ˆν_B)−(ˆν_A(x²)+ˆν_B(x²))+o_δ(1))

Z

ˆ

ν_Bⁿ∈B(ˆν_B,δ)

Z

e

^{βnRe[Tr(A∗U BV∗)]}

dU dV dB

n

. By rearranging, we obtain the following asymptotics of the spherical integral (following the standard arguments to prove large deviations for Beta-ensembles [6]):

n→∞

lim 1

n

²

log I

n,m

(A

n

, B

n

) = −I

νˆ_A

(ˆ ν

B

)

− β 2

2α

Z

log xdˆ ν

B

(x) + 2Σ(ˆ ν

B

) − (ˆ ν

A

(x

²

) + ˆ ν

B

(x

²

))

+ const.

To prove (1.8), we see X

n

= H(1) as the matrix valued process H (t) = A + G

n

(t)/ √

n at time one, where G

n

(t) is field with independent Brownian motions. The singular values s

1

(t) > s

2

(t) > · · · > s

n−1

(t) >

|s

n

(t)| of H (t) follow the Dyson Bessel process:

ds

i

(t) = dW

i

√ βn +



 1 2n

X

j:j6=i

1

s

i

(t) − s

j

(t) + 1 2n

X

j:j6=i

1

s

i

(t) + s

j

(t) + α

n

2s

i

(t)



 dt, 1 6 i 6 n, (1.9) {e:DBPcopy} {e:DBPcopy}

where W

1

, W

2

, · · · , W

n

are independent Brownian motions and α

n

= m − n

n + (1 − 1 β ) 1

n .

We denote the empirical particle density of (1.9) and its symmetrized version, which is also the sym- metrized empirical singular values of H (t), as

ν

ⁿt

= 1 n

n

X

i=1

δ

s_i(t)

, ν ˆ

ⁿt

= 1 2n

n

X

i=1

(δ

s_i(t)

+ δ

−s_i(t)

),

We prove a large deviation principle for {ˆ ν

tⁿ

}

06t61

, in Section 4. The rate function is given by S

_µ^α_ˆ0

({ˆ ν

t

}

_06t61

) = sup

f∈C_b^2,1

S

^α

({ˆ ν

t

, f

t

}

_06t61

), (1.10) {e:ratt} {e:ratt}

where

S

^α

({ˆ ν

t

, f

t

}

_06t61

) = ˆ ν

1

(f

1

) − µ ˆ

0

(f

0

) − Z

1

0

Z

∂

s

f

s

(x)dˆ ν

s

(x)ds − 1 2

Z

1 0

Z f

_s⁰

(x) − f

_s⁰

(y)

x − y dˆ ν

s

(x)dˆ ν

s

(y)ds

− α 2

Z

1 0

Z f

s⁰

(x)

x dˆ ν

s

(x)ds − 1 8β

Z

1 0

Z

(f

s⁰

(x) − f

s⁰

(−x))

²

d, ν ˆ

s

(x)ds.

If ˆ ν

0

6= ˆ µ

0

, S

_µ^α_ˆ0

({ˆ ν

t

}

06t61

) = ∞. We then prove the following result

{main2}

Theorem 1.2. Fix a symmetric probability measure µ ˆ

0

and an initial condition with symmetrized empir- ical measure ν ˆ

₀ⁿ

with uniformly bounded second moment converging weakly to µ ˆ

0

. Then, if α

n

converges towards α ∈ [0, ∞) when n goes to infinity so that either α

n

> 1/βn or α

n

≡ 0, the distribution of the empirical particle density {ˆ ν

_tⁿ

}

_06t61

of the Dyson Bessel process (4.3) satisfies a large deviations principle in the scale n

²

and with good rate function S

^α_ˆν0

. In particular, for any continuous symmetric measure-valued process {ˆ ν

t

}

_06t61

, we have:

δ→0

lim lim inf

n→∞

1

n

²

log P ({ˆ ν

tⁿ

}

06t61

∈ B ({ˆ ν

t

}

06t61

, δ))

= lim

δ→0

lim sup

n→∞

1

n

²

log P ({ˆ ν

tⁿ

}

06t61

∈ B ({ˆ ν

t

}

06t61

, δ)) = −S

^αˆµ₀

({ˆ ν

t

}

06t61

).

(1.11) {e:ulbb} {e:ulbb}

Remark 1.3. In Theorem 1.2, we assumed that either α

n

> 1/βn or α

n

≡ 0. This assumption is always true for β = 1, 2 and m > n. If this condition is violated, i.e. 0 < α

n

< 1/βn, the particles s

n

(t) and s

−n

(t) in (1.9) may collapse at 0. In this case, to make sense of (1.9), we need to specify the boundary condition when they collapse at 0. We will not discuss these conditions in this paper.

As a consequence, we deduce from the contraction principle [20] that (1.12) holds and more precisely Corollary 1.4. For any symmetric probability measures ˆ ν

_Aⁿ

, ˆ ν

_Bⁿ

with uniformly bounded second moment converging weakly towards ν ˆ

A

, ν ˆ

B

, under the measure (1.7) we have

n→∞

lim 1

n

²

log P (ˆ ν

Bⁿ

∈ B (ˆ ν

B

, δ)) = −I

νˆ_A

(ˆ ν

B

) + o

δ(1)

, (1.12) {conv1} {conv1}

where

I

ˆν_A

(ˆ ν

B

) = inf

ˆ

ν₁=ˆν_B

S

_ν^α_ˆA

({ˆ ν

t

}

_06t61

) .

Theorem 1.1 is deduced from Theorem 1.2 in section 5.1. The main difficulty to prove Theorem 1.2 lies in the singularity of the potential at the origin and the repulsion between the particles. To prove it, we revisit in section 3 the large deviation principle for the empirical measure of the Dyson Brownian motion of [34] and extend it to to all values of β greater or equal to one.

Acknowledgements The research of J.H. is supported by the Simons Foundation as a Junior Fellow at the Simons Society of Fellows, and NSF grant DMS-2054835. The work of A. Guionnet is partly supported by ERC Project LDRAM : ERC-2019-ADG Project 884584. We thank O. Zeitouni for many inspiring discussions about spherical integrals, including preliminary ideas about the questions addressed in this article.

Notations O(n) denotes the orthogonal group in dimension n and U(n) the unitary group in dimension n. We denote by d(·, ·) the 2-Wasserstein distance defined on the space P

2

( R ) of probability measures with finite second moment by

d(µ, ν) = inf Z

|x − y|

²

dπ(x, y)

1/2

,

where the infimum is taken over distribution on R

²

with marginal distribution µ and ν. C

^2,1_b

( R × [0, 1]) is the space of functions on R × [0, 1] with bounded first two derivatives in x and bounded derivative in t. C([0, 1], M

¹

( R )) is the space of continuous (with respect to weak topology) measure valued process.

2 Dyson Bessel Process

In this section we introduce Dyson Bessel process, which is the singular value process of rectangular matrix

brownian motions. Then in section 2.2, we will write Dyson Bessel process as a change of measure from

Dyson Brownian motion using Girsanov’s theorem.

2.1 Decomposition

The rectangular spherical integral (1.1) is related to real (β = 1) and complex (β = 2) rectangular random matrices with nonzero mean. We consider an n × m rectangular random matrices X

n

with nonzero mean,

X

n

= A

n

+ 1

√ n G

n

, (2.1) {e:defX} {e:defX}

where A

n

= E [X

n

] is deterministic, and G

n

is an n×m rectangular matrix with independent real (β = 1) or complex (β = 2) Gaussian entries. We denote the singular value decomposition of X

n

= U B

n

V

^∗

, with B

n

= diag{b

1

, b

2

, · · · , b

n

}. Then we can rewrite the law of X

n

as

r βn 2π

!

βmn

e

^{−βn2 Tr((Xn−An)(Xn−An)∗)}

dX

n

∝ Y

i

b

^{β(m−n+1)−1}_i

Y

i<j

|b

²i

− b

²j

|

^β

e

^{−βN2 (}

P ib²_i+P

a²_i)+βnRe[Tr(A^∗_nU B_nV^∗)]

dU dV dB

n

.

(2.2) {e:lawU} {e:lawU}

Therefore, conditioning on the singular values of X

n

, i.e. the matrix B

n

, the joint law of singular vectors of X

n

, i.e. U, V is given by the integrand of the rectangular spherical integral (1.1)

e

^{βNRe[Tr(A∗nU BnV∗)]}

Z

m,n^β

dU dV. (2.3) {e:lawU2} {e:lawU2}

We study the random matrices X

n

as in (2.1) via a dynamical approach. By constructing a matrix valued real/complex Brownian motions starting from A

n

, its value at time t = 1 has the same law as X

n

. Theorem 2.1 (Dyson Bessel Process). Take β > 1. Fix m > n, and let H(t) be a n × m matrix with entries given by independent real/complex Brownian motions starting from A

n

:

H (t) = A

n

+ 1

√ n G(t), (2.4) {e:defBB} {e:defBB}

The singular values s

1

(t) > s

2

(t) > · · · > s

n−1

(t) > |s

n

(t)| of H(t) satisfies the following stochastic differential equations

ds

i

(t) = dW

i

√ βn +



 1 2n

X

j:j6=i

1

s

i

(t) − s

j

(t) + 1 2n

X

j:j6=i

1

s

i

(t) + s

j

(t) + α

n

2s

i

(t)



 dt, 1 6 i 6 n, (2.5) {e:dsk} {e:dsk}

where

α

n

= m − n

n +

1 − 1

β 1

n ,

and W

1

, W

2

, · · · , W

n

are independent Brownian motions. We denote by P the law of s(t) = (s

1

(t), . . . , s

n

(t)), 0 6 t 6 1.

The eigenvalues process of λ

1

(t) > λ

2

(t) > · · · > λ

n

(t) of H(t)H

^∗

(t) has been intensively studied in the literature [12,13, 21, 22, 39], called the β-Laguerre process or β-Wishart process

dλ

i

(t) =2 √ λ

i

dW

i

(t)

√ βn +



 1 n

X

j:j6=i

λ

i

(t) + λ

j

(t) λ

i

(t) − λ

j

(t) + m

n



 dt, 1 6 i 6 n, (2.6) {e:DLWM} {e:DLWM}

where W

1

, W

2

, · · · , W

n

are independent Brownian motions. In [39], the case β = 2 Laguerre process was shown to correspond to squared Bessel processes conditioned never to collide in the sense of Doob. It is known that for β > 1 and m > n, (2.6) has a unique strong solution satisfying λ

1

(t) > λ

2

(t) > · · · >

λ

n

(t) > 0 for t > 0. Then a formal calculation gives that s

i

(t) = p

λ

i

(t) satisfies (2.5). When n = 1, s

1

is a Bessel process. We call the process (2.5) Dyson Bessel process. The same argument as in [2, Lemma 4.3.3], we can show that for β > 1, α

n

> 1/βn, and any initial condition s

1

(0) > s

2

(0) > · · · > s

n

(0) > 0, the unique strong solution of (2.5) satisfy s

1

(t) > s

2

(t) > · · · s

n

(t) > 0 for t > 0. Therefore, s

1

(t) >

s

2

(t) > · · · > s

n−1

(t) > s

n

(t) > 0 has the same law of singular values of H(t). We notice that α

n

> 1/βn is satisfied for any m > n and β = 2. In the special case that β = 1, m = n and α

n

= 0, as discussed in [17, Appendix 1], s

n

(t) can be negative, and s

1

(t) > s

2

(t) > · · · > s

n−1

(t) > |s

n

(t)| > 0 has the same law of singular values of H (t). For our study of Dyson Bessel process, we restrict ourselves to these two choices of parameters

2.2 Change of Measure

{s:changeM}

In this section, we relate the Dyson Bessel process (2.5) with the Dyson Brownian motion by a change of measure using Girsanov’s theorem. We recall the Dyson Brownian motion (DBM) is given for β > 1 by

dx

i

(t) = dW

i

(t)

√ βn + 1 2n

X

j:j6=i

dt

x

i

(t) − x

j

(t) . (2.7) {e:DBM} {e:DBM}

We denote the law of Dyson Brownian motion (2.7) as Q .

The Dyson Bessel process (2.5) can be obtained from the DBM (2.7) by a change of measure using an exponential martingale constructed from the following function

θ(s

1

, s

2

, · · · , s

n

) = β 2

X

i<j

log(s

i

+ s

j

) + α

n

n X log s

k

!

, s

1

> s

2

> · · · > s

n

. (2.8) {e:theta} {e:theta}

The above function θ has logarithmic singularity when s

n

is close to 0. Fix a small parameter a > 0, we define the stopping time τ

a

, the first time that s

n

(t) gets too close to 0,

τ

a

= inf {t > 0 : s

n

6 a}. (2.9) {e:taue} {e:taue}

Then for t 6 τ

a

, we have s

n

(t) > a, and θ(s

1

(t), s

2

(t), · · · , s

n

(t)) is bounded below uniformly.

{p:changem}

Proposition 2.2. Let F

t

be the σ algebra generated by the Brownian motions {W

i

(t)}. We take Q the law of DBM

dx

i

(t) = dW

i

(t)

√ βn + 1 2n

X

j:j6=i

dt

x

i

(t) − x

j

(t) , 1 6 i 6 n, (2.10) {e:DBMa} {e:DBMa}

and P

^a

the law of the following modified Dyson Bessel process

ds

i

(t) = dW

i

(t)

√ βn + 1 2n

X

j:j6=i

1

s

i

(t) − s

j

(t) + 1(t 6 τ

a

)



 1 2n

X

j:j6=i

1

s

i

(t) + s

j

(t) + α

n

2s

i

(t)



 dt, for 1 6 i 6 n. Then the two laws P

^a

and Q are related by a change of measure

P

^a

= e

^{L1∧τa−12hL,Li1∧τa}

Q ,

where the exponent is given by L

t∧τ_a

− 1

2 hL, Li

t∧τ_a

= θ(x

1

(u), · · · , x

n

(u))|

^t∧τ₀ ^a

− βn 2

Z

t∧τ_a 0

X

i

α

²n

4x

²_i

(u) du

− β

2 − 1 Z

t∧τ_a

0

1 4n

X

k6=`

du

(x

k

(u) + x

`

(u))

²

+ Z

t∧τ_a

0

α

n

4 X

k

du x

²_k

(u) . Remark 2.3. We remark that P

^a

depends on a > 0. For any event Ω of singular value DBM, we can can lower bound its probability in the following way

P (Ω) > P (Ω ∩ {s

n

> a}) = P

^a

(Ω ∩ {s

n

> a}).

Proof of Proposition 2.2. The first and second derivatives of θ are given by

∂

s_i

θ(s

1

, s

2

, · · · , s

n

) = β 2



 X

j:j6=i

1 s

i

+ s

j

+ α

n

n s

i



 ,

∂

s²_i

θ(s

1

, s

2

, · · · , s

n

) = − β 2



 X

j:j6=i

1

(s

i

+ s

j

)

²

+ α

n

n s

²_i



 ,

(2.11) {e:dtheta} {e:dtheta}

for 1 6 i 6 n. Since θ is C

^∞

on sets where it is bounded below, Itˆ o’s lemma gives that if x(t) = (x

1

(t), . . . , x

n

(t)),

dθ(x(t)) = dL

t

+ 1 4n

X

i6=j

∂

x_i

θ(x(t)) − ∂

x_j

θ(x(t))

x

i

(t) − x

j

(t) dt + X

i

∂

_x²_i

θ(x(t))

2βn dt, (2.12) {e:dL} {e:dL}

where the martingale term L

t

is dL

t

= X

i

∂

x_i

θ(x(t)) dW

i

(t)

√ βn = X

i





√ β 2 √

n X

j:j6=i

1

x

i

(t) + x

j

(t) + p βn α

n

2x

i

(t)



 dW

i

(t), Its quadratic variance is given by

hL, Li

t

= Z

t

0

X

i





√ β 2 √

n X

j:j6=i

1

x

i

(u) + x

j

(u) + p βn α

n

2x

i

(u)





2

du.

For the second term on the righthand side of (2.12), using (2.11) we have 1

4n X

i6=j

∂

x_i

θ(x(t)) − ∂

x_j

θ(x(t)) x

i

− x

j

= − β 8n

X

i6=j6=k

1

(x

i

+ x

k

)(x

j

+ x

k

) − βα

n

8 X

i6=j

1 x

i

x

j

= − β 8n

X

i



 X

j:j6=i

1 (x

i

+ x

j

)





2

+ β 8n

X

i6=j

1

(x

i

+ x

j

)

²

− βα

n

8 X

i

1 x

i

!

2

+ βα

n

8 X

i

1 x

²_i

.

(2.13) {e:term1} {e:term1}

For the last term on the righthand side of (2.12), using (2.11)we have X

i

∂

_x²_i

θ(x(t)) 2βn = − 1

4n X

i6=j

1

(x

i

+ x

j

)

²

− α

n

4 X

i

1

x

²_i

. (2.14) {e:term2} {e:term2}

By plugging (2.13) and (2.14) back into (2.12), we get dθ(x(t)) = dL

t

− X

i

β 8n



 X

j:j6=i

1 (x

i

(t) + x

j

(t))





2

− βα

n

8 X

i

1 x

i

(t)

!

2

+ β

2 − 1



 1 4n

X

i6=j

1

(x

i

(t) + x

j

(t))

²

+ α 4

X

i

1 x

²_i

(t)



 .

(2.15) {e:df} {e:df}

We recall the stopping time τ

a

from (2.9), then hL, Li

t∧τ_a

=

Z

t∧τ_a 0

X

i





√ β 2 √

n X

j:j6=i

1

x

i

(u) + x

j

(u) + p βn α

n

2x

i

(u)





2

du

6 Z

t∧τ_a

0

X

i





√ β 2 √

n X

j:j6=i

1 2a + p

βn α

n

2a





2

du 6 α

n

2 + 1 4

2

βn

²

(t ∧ τ

a

) a

²

,

which is uniformly bounded. Therefore, Novikov’s theorem [2, H.10] implies the following is an exponen- tial martingale

e

^{Lt∧τa−12hL,Lit∧τa}

. Using (2.15), more explicitly, we can rewrite

L

t∧τ_a

− 1

2 hL, Li

t∧τ_a

= θ(x

1

(u), · · · , x

n

(u)|

^t∧τ₀ ^a

− βn 2

Z

t∧τ_a 0

X

i

α

²_n

4x

²_i

(u) du

− β

2 − 1 Z

t∧τ_a

1 4n

X

k6=`

du

(x

k

(u) + x

`

(u))

²

+ Z

t∧τ_a

α

n

4 X

k

du x

²_k

(u) . We recall that Q is the law of DBM (2.10), and denote the rescaled Brownian motions M ,

M

i

(t) = x

i

(t) − x

i

(0) − Z

t

0

1 2n

X

j:j6=i

du x

i

(u) − x

j

(u) =

Z

t 0

dW

i

(u)

√ βn = W

i

(t)

√ βn , then Girsanov’s theorem [2, Theorem H.11] implies that

M

i

(t) − hM

i

, Li

t∧τ_a

= x

i

(t) − x

i

(0) − Z

t

0

1 2n

X

j:j6=i

du x

i

(u) − x

j

(u)

− Z

t∧τ_a

0



 1 2n

X

j:j6=i

1

x

i

(u) + x

j

(u) + α

n

2x

i

(u)



 du,

(2.16) {e:newM} {e:newM}

are independent Brownian motions under the measure P

^a

: P

^a

= e

^{L1∧τa−12hL,Li1∧τa}

Q , Therefore, P

^a

is the unique solution of the stochastic differential system

ds

i

(t) = dW

i

(t)

√ βn + 1 2n

X

j:j6=i

1

s

i

(t) − s

j

(t) + 1(t 6 τ

a

)



 1 2n

X

j:j6=i

1

s

i

(t) + s

j

(t) + α

n

2s

i

(t)



 dt.

where W

1

, W

2

, · · · , W

n

are independent Brownian motions.

3 Large deviations for the Dyson Brownian motion

{sec-DBM}

Thanks to Proposition 2.2, the law of singular value Dyson Brownian motion can be rewritten as a change of measure from the Dyson Brownian motion. The large deviations principle for Dyson Brownian motion has been proven in [34, 36] when β = 1 or 2 and the initial condition has finite 5 + ε moment for some ε > 0. In this section we give a shorter proof for the large deviations principle valid for any β > 1 and under the assumption that the initial condition has finite second moment only. The main technical improvement comes from Propositions 3.3 and 3.4 which allow to prove the lower bound in greater generality, thanks to better approximation of our processes by processes with smooth drifts

We denote the empirical particle density of the Dyson Brownian motion (2.10) as ν

_tⁿ

= 1

n

n

X

i=1

δ

x_i(t)

. (3.1) {e:empd} {e:empd}

{a:mu0}

Assumption 1. We assume the probability density µ

0

has bounded second moment. Moreover, as n goes to infinite, ν

ⁿ0

converges to µ

0

in 2-Wasserstein distance, i.e. d(µ

0

, ν

0ⁿ

) = o

n

(1).

Given a continuous measure process {ν

t

}

06t61

with ν

0

satisfying Assumption 1, we define the following dynamical entropy:

S({ν

t

, f

t

}

06t61

) =

ν

1

(f

1

) − ν

0

(f

0

) − Z

1

0

Z

∂

t

f

t

(x)dν

t

(x)dt

− 1 2

Z

1 0

Z f

_t⁰

(x) − f

_t⁰

(y)

x − y dν

t

(x)dν

t

(y)dt − 1 2β

Z

1 0

Z

(f

t⁰

(x))

²

dν

t

dt

,

(3.2) {e:rateD} {e:rateD}

where f

t

(x) ∈ C

_b^2,1

has bounded twice derivative in x and bounded derivative in t. For any measure µ

0

, if ν

0

= µ

0

, we set

S

µ0

({ν

t

}

_06t61

) = sup

f∈C^2,1

S({ν

t

, f

t

}

_06t61

). (3.3) {e:rateSmu} {e:rateSmu}

If ν

0

6= µ

0

, we set S

µ₀

({ν

t

}

06t61

) = ∞. In this section we give a new proof of the following large deviations principle for the empirical particle density of the Dyson Brownian motion (3.1)

{t:DBMLDP}

Theorem 3.1. Fix a probability density µ

0

and an initial condition with empirical distribution ν

₀ⁿ

sat- isfying Assumption 1. Then, the empirical particle density {ν

tⁿ

}

_06t61

of the Dyson Brownian motion (3.1) satisfies a large deviations principle in the scale n

²

and with good rate function S

µ₀

({ν

t

}

06t61

). In particular for any continuous measure process {ν

t

}

06t61

, it holds

lim

δ→0

lim sup

n→∞

1

n

²

log P ({ν

tⁿ

}

06t61

∈ B ({ν

t

}

06t61

, δ))

= lim

δ→0

lim inf

n→∞

1

n

²

log P ({ν

_tⁿ

}

_06t61

∈ B ({ν

t

}

_06t61

, δ)) = −S

µ0

({ν

t

}

_06t61

) .

For any measure valued proces {ν

t

}

_06t61

such that S

µ0

({ν

t

}

_06t61

) < ∞, by Riesz representation theorem, there exists a measurable function ∂

x

k

t

∈ L

²

(dν

t

(x)dt), such that for any f ∈ C

_b^2,1

ν

1

(f

1

) − ν

0

(f

0

) − Z

∂

t

f

t

(x)dν

t

(x)dt − 1 2

Z

1 0

Z

f

t⁰

(x)H (ν

t

)dν

t

(x)dt = Z

1

0

Z

f

t⁰

(x)∂

x

k

t

(x)dν

t

dt. (3.4) {e:f1f0} {e:f1f0}

Here H(ν) denotes the Hilbert transform of ν. Then we can rewrite the rate function S

µ₀

({ν

t

}

_06t61

) in (3.2) as

S

µ₀

({ν

t

}

06t61

) = sup

f∈C^2,1 b

Z

1 0

f

_t⁰

(x)∂

x

k

t

(x)dν

t

dt − 1 2β

Z

1 0

Z

(f

_t⁰

(x))

²

dν

t

dt = β 2

Z

1 0

Z

(∂

x

k

t

(x))

²

dν

t

dt,

(3.5) {e:minimizereq} {e:minimizereq}

where the equality is achieved when f

t⁰

(x) = β∂

x

k

t

(x).

We collect some properties of the rate function (3.2), which were essentially proven in [29, 34, 36].

{p:rate}

Proposition 3.2. Fix a probability measure µ

0

with finite second moment and bounded free entropy, i.e.

Σ(µ

0

) > −∞. Then, S

µ₀

is a good rate function on C([0, 1], M

¹

( R )). If S

µ₀

({ν

t

}

06t61

) < 0, then we have (i) There exists a constant C depending only on µ

0

and S

µ0

({ν

t

}

_06t61

), such that the L

2

norms of ν

t

are uniformly bounded,

Z

x

²

dν

t

(x) 6 C. (3.6) {e:L2norm} {e:L2norm}

(ii) ν

t

has a density for almost surely all 0 6 t 6 1, i.e.

dν

t

(x)

dx = ρ

t

(x).

(iii) We denote the velocity field u

t

(x) = H (ν

t

)(x)/2 + ∂

x

k

t

(x), then it satisfies the conservation of mass equation

∂

t

ρ

t

+ ∂

x

(ρ

t

u

t

) = 0, 0 6 t 6 1, (3.7) {e:masseq} {e:masseq}

in the sense of distribution. We can rewrite the dynamical entropy (3.2) as S

µ₀

({ν

t

}

06t61

) = β

2 Z

1

0

Z

(u

²t

+ H (ν

t

)

²

/4)ρ

t

(x)dxdt − 1

2 (Σ(ν

1

) − Σ(ν

0

))

= β 2

Z

1 0

Z

(u

²t

+ π

²

12 ρ

t

(x)

²

)ρ

t

(x)dxdt − 1

2 (Σ(ν

1

) − Σ(ν

0

))

.

(3.8) {Sent} {Sent}

Proof. It is proven in [34, Theorem 1.4] that S

µ₀

({ν

t

}

06t61

) is a good rate function. If S

µ₀

({ν

t

}

06t61

) <

∞, by definition we have µ

0

= ν

0

. For Item (i), we take a test function f

ε

(x) = x

²

/(1 + εx

²

) with small ε > 0. Then it is easy to see that f

_ε⁰

(x) = 2x/(1 + εx

²

)

²

and |f

_ε⁰⁰

(x)| 6 10. By the definition of the dynamical free entropy (3.2), for any 0 < t 6 1, we have

ν

t

(f

ε

) − ν

0

(f

ε

) − 1 4

Z

t 0

Z f

ε⁰

(x) − f

ε⁰

(y)

y − x dν

s

(y)dν

s

(x)ds − 1 2β

Z

t 0

Z

(f

ε⁰

(x))

²

dν

s

ds 6 S

µ₀

({ν

s

}

06s61

) < 0.

(3.9) {e:bbd} {e:bbd}

By our assumption that ν

0

= µ

0

has finite second moment, it holds that sup

_ε

ν

0

(f

ε

) < ∞. Using

|f

ε⁰⁰

(x)| 6 10, we find for t 6 1, 1 4

Z

t 0

Z f

_ε⁰

(x) − f

_ε⁰

(y)

y − x dν

s

(y)dν

s

(x)ds 6 5/2.

Therefore, there exists a constant C depending only on µ

0

and S

µ₀

({ν

t

}

06t61

), such that ν

t

(f

ε

) =

Z x

²

1 + εx

²

dν

t

6 C + 1 2β

Z

t 0

Z

(f

_ε⁰

(x))

²

dν

s

ds 6 C + 2

β Z

t

0

Z x

²

1 + εx

²

dν

s

ds = C + 2 β

Z

t 0

ν

s

(f

ε

)ds.

Gr¨ onwall’s inequality then implies that for all t 6 1

ν

t

(f

ε

) 6 e

^2/β

C.

The claim 3.6 follows by sending ε to 0 and monotone convergence theorem.

It was proven in [29, Theorem 2.1] and [36, Theorem 3.3] that if µ

0

= ν

0

has bounded 5 + ε moments, i.e.

Z

|x|

^5+ε

dν

0

< ∞, (3.10) {e:five} {e:five}

and Σ(µ

0

), Σ(µ

1

) are finite, then Item (ii) and (iii) hold. This can be extended to the case where µ

0

has only a finite second moment following the arguments of the proof of [15, Lemma 5.9]. We briefly recall the main steps of the proof. First recall that free convolution reduces the dynamical entropy (see [15]) so that if σ

ε

denotes the semi-circle law with covariance ε

S

µ0σε

({ν

t

σ

ε

}

06t61

) 6 S

µ₀

({ν

t

}

06t61

).

But on the other hand, H(ν

t

σ

ε

) is uniformly bounded by 1/ √

ε. Therefore if we denote by u

^ε

the velocity field of ν

_t^ε

= ν

t

σ

ε

,

Z

1 0

Z

(u

^ε_t

)

²

dν

_t^ε

dt 6 2 Z

1

0

Z

(u

^ε_t

− Hν

_t^ε

)

²

dν

_t^ε

dt + 2 Z

1

0

Z

(Hν

_t^ε

)

²

dν

_t^ε

dt 6 4

β S

µ₀σε

({ν

_t^ε

}

_06t61

) + 2 ε < ∞ . Hence, we can write

S

µ₀σε

({ν

_t^ε

}

_06t61

) = β 2

Z

1 0

Z

(u

^ε_t

)

²

dν

_t^ε

dt + β 2

Z

1 0

Z

(Hν

_t^ε

)

²

dν

_t^ε

dt − β Z

1

0

Z

Hν

_t^ε

u

^ε_t

dν

_t^ε

dt . (3.11) {lk} {lk}

For the second term we used the well known formula (recall that dν

t^ε

dx) Z

1

0

Z

(Hν

t^ε

)

²

dν

t^ε

= π

²

3

Z

1 0

Z ( dν

^εt

dx )

³

dx .

Finally for the last term of (3.11), we observe following [15, Lemma 5.9] that the continuity of t 7→ ν

t

implies that t 7→ Hν

t^ε

is continuous (thanks to the explicit formulas for the Hilbert transform of measures freely convoluted with the semi-circle laws given by Biane [8]). Since it is bounded and u

^ε

is in L

²

, we see that we can approximate the last term by Riemann sum. Then, recall that by definition we have

Z

t

u

^ε_s

dν

s^ε

dx ds = − Z

x

dν

_t^ε

, to conclude that

Z

1 0

Z

Hν

_t^ε

u

^ε_t

dν

^ε_t

dt = 1 2

Z

1 0

∂

t

Σ(ν

_t^ε

)dt = 1

2 (Σ(ν

1

σ

ε

) − Σ(µ

0

σ

ε

)) .

Hence, (3.8) holds for {ν

t^ε

}

06t61

. This implies that Σ(ν

1

σ

ε

) is bounded since it is bounded from above as ν

1

σ

ε

has bounded second moment and also from below since

β 2

Z

1 0

Z

(u

^ε_t

)

²

dν

_t^ε

dt+ β 2

Z

1 0

Z

(Hν

_t^ε

)

²

dν

_t^ε

dt− β

2 (Σ(ν

1

σ

ε

)−Σ(ν

0

)) 6 S

µ₀σε

({ν

t^ε

}

06t61

) 6 S

µ₀

({ν

t

}

06t61

) . In fact, because we could have done the same reasoning on the time interval [0, t], we also see that for all s 6 1

S

µ₀

({ν

t

}

_06t61

) > β 2

Z

s 0

Z

(u

^εt

)

²

dν

t^ε

dt + β 2

Z

s 0

Z

(Hν

t^ε

)

²

dν

t^ε

dt − β

2 (Σ(ν

s

σ

ε

) − Σ(ν

0

)),

which implies that Σ(ν

s

σ

ε

) is uniformly bounded. We can finally let ε going to zero to conclude. As

a consequence of (3.6), we deduce that ν

t

has finite free entropy, i.e. Σ(ν

t

) < +∞. We refer the reader

to [15] for details.

3.1 Large deviations upper bound

In this section, we prove the large deviations upper bound. We recall that the exponential tightness was already proven in this setting in the proof of [34, Theorem 2.4]: for the sake of completeness we will recall this proof but in the new setting of the Bessel Dyson processes, see section 4. We next prove the large deviations upper bound of Theorem 3.1

lim sup

δ→0

lim sup

n→∞

1

n

²

log P ({ν

_tⁿ

}

_06t61

∈ B ({ν

t

}

_06t61

, δ)) 6 −S

µ₀

({ν

t

}

_06t61

). (3.12) {e:DBMupbb} {e:DBMupbb}

Take any test function f

t

(x) ∈ C

_b^2,1

([0, 1] × R ), and use Itˆ o’s lemma to find that d X

i

f

t

(x

i

(t)) = X

i

f

_t⁰

(x

i

(t))dx

i

(t) + X

i

f

t⁰⁰

(x

i

(t))

2βn + ∂

t

f

t

(x

i

(t))

dt

= X

i





f

_t⁰⁰

(x

i

(t))

2βn + ∂

t

f

t

(x

i

(t)) + f

_t⁰

(x

i

(t)) 2n

X

j:j6=i

1 x

i

(t) − x

j

(t)



 dt + X

i

f

t⁰

(x

i

(t)) dW

i

(t)

√ βn

= dL

^f_t

+ 1 4n

X

i6=j

f

_t⁰

(x

i

(t)) − f

_t⁰

(x

j

(t))

x

i

(t) − x

j

(t) dt + X

i

f

_t⁰⁰

(x

i

(t))

2βn dt + X

i

∂

t

f

t

(x

i

(t))dt,

(3.13) {e:lin0} {e:lin0}

where the martingale term is given by dL

^f_t

= X

i

f

_t⁰

(x

i

(t)) dW

i

(t)

√ βn , hL

^f

, L

^f

i

t

= 1 βn

Z

t 0

X

i

(f

_t⁰

(x

i

(t)))

²

dt. (3.14) {e:MLt0} {e:MLt0}

We recall the empirical particle density {ν

tⁿ

}

06t61

from (3.1). With it, we can rewrite (3.13) as Z

f

t

(x)dν

_tⁿ

− Z

f

t

(x)dν

₀ⁿ

= L

^f_t

n + 1

4 Z

t

0

Z

x6=y

f

_s⁰

(x) − f

_s⁰

(y)

x − y dν

_sⁿ

(x)dν

_sⁿ

(y)ds + 1 2βn

Z

t 0

Z

f

_s⁰⁰

(x)dν

_sⁿ

(x)ds + Z

t

0

Z

∂

s

f

s

(x)dν

_sⁿ

(x)ds

= L

^f_t

n +

Z

t 0

1 4

Z f

s⁰

(x) − f

s⁰

(y)

x − y dν

_sⁿ

(x)dν

_sⁿ

(y) + Z

∂

s

f

s

(x)dν

_sⁿ

(x) + 1 n

1 2β − 1

4 Z

f

_t⁰⁰

(x)dν

_sⁿ

(x)

ds.

(3.15) {e:df220} {e:df220}

As L

^f

is bounded uniformly for f ∈ C

_b^2,1

, we can construct an exponential martingale using the martingale dL

^f_t

from (3.14)

D

t

= e

^nLft−n

2 2 hL^f,L^fi_t

, E [D

t

] = E [D

0

] = 1. (3.16) {e:Dt0} {e:Dt0}

Using (3.15) we can rewrite

nL

^f_t

− n

²

2 hL

^f

, L

^f

i

t

= n

²

S

tⁿ

({ν

tⁿ

, f

t

}

_06t61

), where

S

tⁿ

({ν

sⁿ

, f

s

}

06s6t

) = Z

f

t

dν

ⁿt

− Z

f

0

dν

0ⁿ

− 1 4

Z

t 0

Z f

s⁰

(x) − f

s⁰

(y)

x − y dν

sⁿ

(x)dν

ⁿs

(y)ds − Z

t

0

Z

∂

s

f

s

(x)dν

sⁿ

(x)ds

− Z

t

0

Z 1 n

1 2β + 1

4

f

t⁰⁰

(x)dν

sⁿ

ds − 1 2β

Z

t 0

Z

(f

t⁰

(x))

²

dν

sⁿ

ds.

We also define

S

ⁿ

({ν

tⁿ

, f

t

}

06t61

) = 1 n

²

nL

^f₁

2 − n

²

2 hL

^f

, L

^f

i

1

!

= S

1ⁿ

({ν

ⁿt

, f

t

}

06t61

). (3.17) {e:defSn} {e:defSn}

Then for {ν

tⁿ

}

06t61

∈ B ({ν

t

}

06t61

, δ), we have by uniform (in n > 1) continuity of ν 7→ S

ⁿ

({ν

t

, f

t

}

06t61

) for any f ∈ C

_b^2,1

([0, 1] × R ),

S

ⁿ

({ν

tⁿ

, f

t

}

06t61

) = S

ⁿ

({ν

t

, f

t

}

06t61

) + o

n

(1).

We can use the exponential martingale (3.16) to obtain the large deviations upper bound as follows.

P ({ν

ⁿ_t

}

_06t61

∈ B ({ν

t

}

_06t61

, δ)) = E

"

1({ν

_tⁿ

}

_06t61

∈ B ({ν

t

}

_06t61

, δ)) e

^{n2Sn({νtn,ft}06t61)}

e

^{n2Sn({νtn,ft}06t61)}

#

= E

h 1({ν

sⁿ

}

_06t61

∈ B ({ν

s

}

_06t61

, δ))e

^{n2Sn({νtn,ft}06t61)}

i e

^o(n2)

e

^{n2Sn({νt,ft}06t61)}

6 E

h

e

^{n2Sn({νtn,ft}06t61)}

i e

^o(n2)

e

^{n2Sn({νt,ft}06t61)}

= e

^{−n2Sn({νt,ft}06t61)+on(1))}

.

(3.18) {e:upp} {e:upp}

The large deviations upper bound (3.12) follows from rearranging (3.18), and taking the infimum over f ∈ C

^2,1_b

.

3.2 Large deviations Lower Bound

In the rest of this section, we prove the large deviations lower bound of Theorem 3.1, namely we show that for any continuous measure-valued process {ν

t

}

06t61

, we have

lim inf

δ→0

lim inf

n→∞

1

n

²

log P ({ν

tⁿ

}

06t61

∈ B ({ν

t

}

06t61

, δ)) > −S

µ₀

({ν

t

}

06t61

). (3.19) {e:DBMlow} {e:DBMlow}

The proof itself will be used to derive the large deviations for Dyson Bessel processes as it allows to control the positions of the extreme particules, see Proposition 3.5, key to control the singularity at the origin of the Dyson Bessel process.

The proof consists of two steps, in the first step we approximate {ν

t

}

06t61

by a sequence of measure- valued process with benign properties.

{p:approximation}

Proposition 3.3. Fix a probability measure µ

0

with finite second moment. Then, any measure-valued process {ν

t

}

06t61

with S

µ₀

({ν

t

}

06t61

) < ∞ can be approximated by a sequence of measure-valued pro- cesses {ν

_t^ε

}

_06t61

satisfying

• ν

_t^ε

has uniformly bounded density ρ

^ε_t

, supp(ν

_t^ε

) is a single interval for all times t ∈ [0, 1], and

ε→0

lim sup

06t61

d(ν

t

, ν

t^ε

) = 0.

• The dynamical entropy satisfies

ε→0

lim S

ν^ε₀

({ν

t^ε

}

06t61

) = S

µ₀

({ν

t

}

06t61

). (3.20) {convS} {convS}

• The density {ρ

^εt

(x)}

06t61

of the measure-valued process {ν

t^ε

}

06t61

is smooth in both x, t, and the corresponding drift ∂

x

k

^ε_t

(x) as defined by

∂

t

ρ

^εt

+ ∂

x

(ρ

^εt

u

^εt

) = 0, u

^εt

(x) = 1

2 H(ν

^εt

)(x) + ∂

x

k

^εt

(x), is also smooth in both x, t.

Above, smooth means differentiable and with continuous derivative (we shall not need more the proof yields eventually the existence of more derivatives).

{p:lowerb}

Proposition 3.4. Fix a probability measure µ

0

satisfying Assumption 1 and δ > 0. Let µ ˜

0

be a com- pactly supported probability measure such that d(µ

0

, µ ˜

0

) 6 δ/3. Let {˜ ν

t

(x)}

06t61

be a compactly supported measure-valued process with a smooth density ρ ˜ in both x, t such that ˜ ν

0

= ˜ µ

0

. Assume that the corre- sponding drift ∂

x

˜ k

t

defined by

∂

t

ρ ˜

t

(x) + ∂

x

( ˜ ρ

t

u ˜

t

) = 0, u ˜

t

(x) = 1

2 H (˜ ν

t

)(x) + ∂

x

k ˜

t

(x), (3.21) {driftk} {driftk}

is also smooth in both x, t. Then, the following large deviations lower bound holds lim inf

n→∞

1

n

²

log P ({ν

_tⁿ

}

_06t61

∈ B ({˜ ν

t

}

_06t61

, δ)) > −S

µ˜₀

({˜ ν

t

}

_06t61

) + o

δ

(1).

Proof of Large deviations lower bound (3.19). We approximate {ν

t

}

_06t61

by {ν

_t^ε

}

_06t61

as in Proposition 3.3 and take {˜ ν

t

(x)}

06t61

equal {ν

^εt

}

06t61

in Proposition 3.4 with sufficiently small ε. Then it follows

n→∞

lim 1

n

²

log P ({ν

ⁿt

}

06t61

∈ B ({ν

t

}

06t61

, δ)) > lim

ε→0

lim

n→∞

1

n

²

log P ({ν

tⁿ

} ∈ B ({ν

t^ε

}

06t61

, δ/2))

> lim

ε→0

−S

ν^ε₀

({ν

t^ε

}

06t61

)) + o

δ

(1) > −S

µ₀

({ν

t

}

06t61

)) + o

δ

(1),

where in the last inequality we used (3.20). The large deviations lower bound follows by taking δ → 0.

Proof of Proposition 3.3. We fix three parameters ε

3

ε

2

ε

1

1. The construction of ν

t^ε

consists of the following three steps. Note that S is lower semi-continuous hence we only need to show that

lim sup

ε→0

S

ν^ε₀

({ν

_t^ε

}

_06t61

) 6 S

µ₀

({ν

t

}

_06t61

).

Step 1 (Free Convolution). We replace ν

t

by ν

_t⁽¹⁾

= ν

t

σ

ε1

, its free convolution with a small semi-circle distribution of size ε

1

. Then we have d(ν

t

, ν

_t⁽¹⁾

) = o

ε₁

(1). More importantly, ν

_t⁽¹⁾

(dx) = ρ

⁽¹⁾_t

(x)dx has density bounded by O(1/ √

ε

1

), and it is proven in [15] that

S

ν⁽¹⁾₀

({ν

_t⁽¹⁾

}

06t61

) 6 S

µ₀

({ν

t

}

06t61

). (3.22) {bor} {bor}

By our assumption S

µ0

({ν

t

}

_06t61

) < ∞, Proposition 3.2 implies that ν

1

has bounded second moment and finite free entropy −∞ < Σ(ν

1

) < ∞. The same bound holds for its free convolution with semi-circle distribution, i.e. −∞ < Σ(ν

₁⁽¹⁾

) < ∞. Moreover, the second moments of ρ

⁽¹⁾

and u

⁽¹⁾

under ρ

⁽¹⁾_t

(x)dxdt are bounded independently of ε

1

by Proposition 3.2 and (3.22).

Step 2 (Truncation). If {ν

_t⁽¹⁾

}

_06t61

is not compactly supported, in this step we truncate it to have compact support. Let a(t), b(t) be such that

Z

a(t)

∞

ρ

⁽¹⁾_t

(x)dx = ε

1

/2, Z

∞

b(t)

ρ

⁽¹⁾_t

(x)dx = ε

1

/2 .