Gaussian Multiplicative Chaos for Symmetric Isotropic Matrices

Abstract

Motivated by isotropic fully developed turbulence, we define a theory of symmetric matrix valued isotropic Gaussian multiplicative chaos. Our construction extends the scalar theory developed by J.P. Kahane in 1985.

Appendix

1.1 A.1 Discussion About the Construction of Kernels

In this subsection, we discuss in further detail the construction of the kernel K as summarized in Remark 2. In dimension 1 and 2, it is plain to see that

$$ \ln_+\frac{L}{|x|}=\int_0^{+\infty}\bigl (t-|x|\bigr)_+ \nu_L(dt) $$

(19)

where the measure ν _L is given by (δ _u stands for the Dirac mass at u):

$$\nu_L(dt)=\mathbf{1}_{[0,L]}(t)\frac{dt}{t^2}+ \frac{1}{L}\delta_L(dt). $$

Hence, for every μ>0, we have

$$\ln_+\frac{L}{|x|}=\frac{1}{\mu}\ln_+\frac{L^\mu}{|x|^\mu}=\int _0^{+\infty}\bigl(t-|x|^\mu\bigr)_+ \nu_{L^\mu}(dt). $$

We are therefore led to consider μ>0 such that the function x↦(1−|x|^μ)₊ is positive definite, the so-called Kuttner-Golubov problem (see [16]).

For d=1, it is straightforward to check that (1−|x|)₊ is positive definite. We can thus consider a Gaussian process X ^ϵ with covariance kernel given by

$$K_\epsilon(x)=\gamma^2\int_{\epsilon}^{L}\bigl(t-|x|\bigr)_+ \nu_L(dt). $$

Notice that

$$ \forall x\neq0,\quad\gamma^2\ln_+ \frac{L}{|x|}=\lim_{\epsilon \to0}K_\epsilon(x) $$

(20)

and

$$ \forall\epsilon<|x|\leqslant L,\quad K_\epsilon(x)= \gamma^2\int_{|x|}^{L}\bigl(t-|x|\bigr)_+ \nu_L(dt)=\gamma^2\ln_+\frac{L}{|x|} . $$

(21)

In dimension 2, we can use the same strategy since Pasenchenko [23] proved that the mapping x↦(1−|x|^1/2)₊ is positive definite over ℝ². We can thus consider a Gaussian process X ^ϵ with covariance kernel given by

$$K_\epsilon(x)=2\gamma^2\int_{\epsilon ^{1/2}}^{L^{1/2}} \bigl(t-|x|^{1/2}\bigr)_+\nu_{L^{1/2}}(dt), $$

sharing the same properties (20) and (21).

In dimension 3, it is not known whether the mapping $x\mapsto\ln _{+}\frac{L}{|x|}$ admits an integral representation of the type explained above. Nevertheless it is positive definite so that we can use the convolution techniques developed in [27]. In dimension 4, it is not positive definite [27] so that another construction is required. We explain the methods in [25]. We set the dimension d to be larger than d⩾3. Let us denote by S the sphere of ℝ^d and σ the surface measure on the sphere such that σ(S)=1. Remind that this measure is invariant under rotations. We define the function

$$ \forall x\in\mathbb{R}^d,\quad F(x)= \gamma^2\int_S\ln_+\frac {L}{|\langle x,s\rangle|} \sigma(ds). $$

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It is plain to see that F is an isotropic function. Let us compute it over a neighborhood of 0: for |x|⩽L, we can write x=|x|e _x with e _x ∈S. Then we have

$$F(x)=\gamma^2\int_S\ln\frac{L}{|x||\langle e_x,s\rangle|} \sigma(ds)=\lambda^2\ln\frac{L}{|x|}+\int_S \ln\frac{1}{|\langle e_x,s\rangle|}\sigma(ds). $$

Notice that the integral $\int_{S}\ln\frac{1}{|\langle e_{x},s\rangle |}\sigma(ds)$ is finite (use Lemma 3 below for instance) and does not depend on x by invariance under rotations of the measure σ. By using the decomposition (19), we can thus consider a Gaussian process X ^ϵ with covariance kernel given by

$$K_\epsilon(x)= \gamma^2\int_S\int _{\epsilon}^{L}\bigl(t-\bigl|\langle x,s\rangle\bigr|\bigr)_+ \nu_L(dt)\sigma(ds), $$

sharing the properties

$$ \forall x\neq0,\quad\lim_{\epsilon\to0}K_\epsilon(x)=F(x) $$

(23)

and

$$ \forall\epsilon<|x|\leqslant L,\quad K_\epsilon(x)=F(x)= \lambda^2\ln\frac{L}{|x|}+C $$

(24)

for some constant C.

1.2 A.2 Auxiliary Results

We give a proof of the following standard result.

Lemma 3

If (Z _i )_1⩽i⩽N are i.i.d. standard Gaussian random variables then the vector

$$V=\frac{1}{\sqrt{\sum_{i=1}^NZ_i^2}} (Z_1,\dots,Z_N) $$

is distributed as the Haar measure on the sphere of ℝ^N. In particular, the density of the first entry of a random vector uniformly distributed on the sphere is given by:

$$\frac{ \varGamma(\frac{N}{2})}{\varGamma(\frac{1}{2})\varGamma (\frac {N-1}{2})}y^{-\frac{1}{2}} (1-y )^{\frac{N-3}{2}} \mathbf{1}_{[0,1]}(y) \,dy. $$

Proof

By using the invariance under rotations of the law of the Gaussian vector (Z _i )_1⩽i⩽N , the law of V is invariant under rotations and is supported by the sphere. By uniqueness of the Haar measure, V is distributed as the Haar measure. We have to compute the density of $\zeta_{1}=\frac{Z_{1}^{2}}{ \sum_{i=1}^{N}Z_{i}^{2} }$. Notice that

$$\zeta_1 =\frac{Y}{Y+Z} $$

where Y, Z are independent random variables with their respective laws being chi-squared distributions of parameters 1 and N−1. Therefore

 □

Next we characterize all the symmetric Gaussian random matrices.

Lemma 4

Let X be a symmetric and isotropic centered Gaussian random matrix of size N×N. Then the diagonal terms (X ₁₁,…,X _NN ) have a covariance matrix of the form σ ²(1+c)I _N −cσ ² P for some σ ²⩾0 and $c\in{]}{-}1,\frac{1}{N-1}]$, where P is the N×N matrix whose all entries are 1. The off-diagonal terms are i.i.d. with variance $\sigma^{2}\frac{1+c}{2}$ and are independent of the diagonal terms.

Proof

If X admits a density with respect to the Lebesgue measure dM over the set of symmetric matrices (see [1, Chap. 4]), then the density of M is given by

$$e^{-f(M)} \,dM, $$

where f is a homogeneous polynomial of degree 2. By isotropy, f must be a symmetric function of the eigenvalues of M. Therefore it takes on the form

$$f(M)=\alpha\operatorname{tr}\bigl(M^2\bigr)+\beta\operatorname{tr}(M)^2 $$

for some α,β∈ℝ. In this case, the result follows easily.

If M does not admit a density with respect to the Lebesgue measure over the set of symmetric matrices, we can add an independent “small GOE”, i.e. we consider M+ϵM′ where M′ is a matrix of the GOE ensemble with a normalized variance independent of M. The matrix M+ϵM′ admits a density so that we can apply the above result. Then we pass to the limit as ϵ→0. □

1.3 A.3 Some Integral Formulae

Let α,c>0. We want to compute the integral

$$ \int_{\mathbb{R}^N} e^{-\alpha(\sum_{i=1}^{N} \lambda_{i})^2-\frac {1}{2(1+c)}\sum_{i=1}^{N} \lambda_{i}^2} \prod_{i<j} | \lambda_j-\lambda_i | d \lambda. $$

We write the integrand in the form (7):

$$ e^{-\alpha(\sum_{i=1}^{N} \lambda_{i})^2-\frac{1}{2\sigma _d^2(1+\bar{c})}\sum_{i=1}^{N} \lambda_{i}^2} \prod_{i<j} |\lambda_j- \lambda_i| $$

where $\sigma_{d}^{2}(1+\bar{c})=(1+c)$ and $\alpha=\frac{\bar{c}}{2 \sigma_{d}^{2}(1+\bar{c})}\frac{1}{(1+\bar{c}(1-N))}$. In that case, we have $\bar{c}=\frac{2 \alpha(1+c)}{1+2 \alpha(1+c)(N-1)}$ and $1+\bar{c}(1-N)= \frac{1}{1+2 \alpha(1+c)(N-1)}$. We deduce

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We also want to compute the integral

$$ \int_{\mathbb{R}^N} e^{-\alpha(\sum_{i=1}^{N} \lambda_{i})^2-\frac {1}{2(1+c)}\sum_{i=1}^{N} \lambda_{i}^2} \prod_{2 \leqslant i<j} | \lambda_j-\lambda_i | d \lambda. $$

We have

for $\sigma_{d}^{2}(1+\bar{c})=1+c$ and $\bar{c}=\frac{2 \alpha (1+c)}{2 \alpha(1+c)(N-1)+1}$ (or equivalently, $1+\bar{c}(2-N)=\frac {1+2\alpha(1+c)}{1+2\alpha(1+c)(N-1)}$ and $1+\bar{c}=\frac {1+2\alpha(1+c)N}{1+2\alpha(1+c)(N-1)}$). This leads to the following

$$ \int_{\mathbb{R}^{N-1}} e^{ -\frac{\bar{c}}{2 \sigma_d^2(1+\bar {c})}\frac {1}{(1+\bar{c}(2-N))}(\sum_{i=2}^{N} \lambda_{i})^2 -\frac {1}{2\sigma_d^2(1+\bar{c})} \sum_{i=2}^{N} \lambda_{i}^2 } \underset{2 \leqslant i<j}{\prod} |\lambda_j-\lambda_i| d \lambda_2 \cdots d \lambda_N = \bar{Z}_{N-1} $$

where

$$\bar{Z}_{N-1}=(N-1)! (2\pi)^{(N-1)/2} \Biggl(\prod _{k=1}^{N-1} \frac{\varGamma(k/2)}{\varGamma(1/2)}\Biggr) \sigma_{d}^{N(N-1)/2} (1+\bar {c})^{(N-2)(N+1)/4}\sqrt{1+ \bar{c}(2-N)}. $$

In conclusion, we get:

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1.4 A.4 Heuristic Derivation of the Conjecture

Let ℓ<1. We can roughly write as ℓ→0 (where ≈ means equivalent to a random constant of order 1)

$$ M\bigl( B(0,\ell) \bigr) \approx\ell^d \frac{e^{ \gamma\sqrt{ \ln\frac {1}{\ell} } \varOmega- \frac{\gamma^2}{2} \ln\frac{1}{\ell}}}{ \gamma^{N-1} (\ln\frac{1}{\ell})^{(N-1)/2} }, $$

where Ω is a random matrix whose density is given by (6) with $\sigma_{d}^{2}=1$, and thus we get (we forget terms of order 1)

where $\alpha=\frac{c}{2 (1+c)}\frac{1}{(1+c(1-N))}$ Thus, if q>0

$$ E \bigl[ \operatorname{tr} M\bigl( B(0,\ell) \bigr)^q \bigr] \approx\frac{\ell^{ (d + \frac {\gamma^2}{2})q }}{ (\ln\frac{1}{\ell})^{(q-1)(N-1)/2} } $$