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Let {n} be a large natural number, and let {M_n} be a matrix drawn from the Gaussian Unitary Ensemble (GUE), by which we mean that {M_n} is a Hermitian matrix whose upper triangular entries are iid complex gaussians with mean zero and variance one, and whose diagonal entries are iid real gaussians with mean zero and variance one (and independent of the upper triangular entries). The eigenvalues {\lambda_1(M_n) \leq \ldots \leq \lambda_n(M_n)} are then real and almost surely distinct, and can be viewed as a random point process {\Sigma^{(n)} := \{\lambda_1(M_n),\ldots,\lambda_n(M_n)\}} on the real line. One can then form the {k}-point correlation functions {\rho_k^{(n)}: {\bf R}^k \rightarrow {\bf R}^+} for every {k \geq 0}, which can be defined by duality by requiring

\displaystyle  \mathop{\bf E} \sum_{i_1,\ldots,i_k \hbox{ distinct}} F( \lambda_{i_1}(M_n),\ldots,\lambda_{i_k}(M_n))

\displaystyle  = \int_{{\bf R}^k} \rho_k^{(n)}(x_1,\ldots,x_k) F(x_1,\ldots,x_k)\ dx_1 \ldots dx_k

for any test function {F: {\bf R}^k \rightarrow {\bf R}^+}. For GUE, which is a continuous matrix ensemble, one can also define {\rho_k^{(n)}(x_1,\ldots,x_k)} for distinct {x_1<\ldots<x_k} as the unique quantity such that the probability that there is an eigenvalue in each of the intervals {[x_1,x_1+\epsilon],\ldots,[x_k,x_k+\epsilon]} is {(\rho_k^{(n)}(x_1,\ldots,x_k)+o(1))\epsilon^k} in the limit {\epsilon\rightarrow 0}.

As is well known, the GUE process is a determinantal point process, which means that {k}-point correlation functions can be explicitly computed as

\displaystyle  \rho^{(n)}_k(x_1,\ldots,x_k) = \det( K^{(n)}(x_i,x_j) )_{1 \leq i,j \leq k}

for some kernel {K^{(n)}: {\bf R} \times {\bf R} \rightarrow {\bf C}}; explicitly, one has

\displaystyle  K^{(n)}(x,y) := \sum_{k=0}^{n-1} P_k(x) e^{-x^2/4}P_k(y) e^{-y^2/4}

where {P_0, P_1,\ldots} are the (normalised) Hermite polynomials; see this previous blog post for details.

Using the asymptotics of Hermite polynomials (which then give asymptotics for the kernel {K^{(n)}}), one can take a limit of a (suitably rescaled) sequence of GUE processes to obtain the Dyson sine process, which is a determinantal point process {\Sigma} on the real line with correlation functions

\displaystyle  \rho_k(x_1,\ldots,x_k) = \det( K(x_i,x_j) )_{1 \leq i,j \leq k} \ \ \ \ \ (1)

where {K} is the Dyson sine kernel

\displaystyle  K(x,y) := \frac{\sin(\pi(x-y))}{\pi(x-y)}. \ \ \ \ \ (2)

A bit more precisely, for any fixed bulk energy {-2 < u < 2}, the renormalised point processes {\rho_{sc}(u) \sqrt{n} ( \Sigma^{(n)} - \sqrt{n} u )} converge in distribution in the vague topology to {\Sigma} as {n \rightarrow \infty}, where {\rho_{sc}(u) := \frac{1}{2\pi} (4-u^2)^{1/2}_+} is the semi-circular law density.

On the other hand, an important feature of the GUE process {\Sigma^{(n)} = \{\lambda_1,\ldots,\lambda_n\}} is its stationarity (modulo rescaling) under Dyson Brownian motion

\displaystyle  d\lambda_i = dB_i + \sum_{j \neq i} \frac{dt}{\lambda_i-\lambda_j}

which describes the stochastic evolution of eigenvalues of a Hermitian matrix under independent Brownian motion of its entries, and is discussed in this previous blog post. To cut a long story short, this stationarity tells us that the self-similar {n}-point correlation function

\displaystyle  \rho^{(n)}_n(t,x) := t^{-n/2} \rho^{(n)}_n(x/\sqrt{t})

obeys the Dyson heat equation

\displaystyle  \partial_t \rho^{(n)}_n = \frac{1}{2} \sum_{i=1}^n \partial_{x_i}^2 \rho^{(n)}_n - \sum_{1 \leq i,j \leq n;i\neq j} \partial_{x_i} \frac{\rho^{(n)}_n}{x_i-x_j}

(see Exercise 11 of the previously mentioned blog post). Note that {\rho^{(n)}_n} vanishes to second order whenever two of the {x_i} coincide, so there is no singularity on the right-hand side. Setting {t=1} and using self-similarity, we can rewrite this equation in time-independent form as

\displaystyle  -\frac{1}{2} \sum_{i=1}^n \partial_i (x_i \rho^{(n)}_n) = \frac{1}{2} \sum_{i=1}^n \partial_{x_i}^2 \rho^{(n)}_n - \sum_{1 \leq i,j \leq n;i\neq j} \partial_{x_i} \frac{\rho^{(n)}_n}{x_i-x_j}.

One can then integrate out all but {k} of these variables (after carefully justifying convergence) to obtain a system of equations for the {k}-point correlation functions {\rho^{(n)}_k}:

\displaystyle  -\frac{1}{2} \sum_{i=1}^k \partial_i (x_i \rho^{(n)}_k) = \frac{1}{2} \sum_{i=1}^k \partial_{x_i}^2 \rho^{(n)}_k - \sum_{1 \leq i,j \leq k;i\neq j} \partial_{x_i} \frac{\rho^{(n)}_k}{x_i-x_j}

\displaystyle  - \sum_{i=1}^k \partial_{x_i} \int_{\bf R} \frac{\rho^{(n)}_{k+1}(x_1,\ldots,x_{k+1})}{x_i-x_{k+1}}\ dx_{k+1},

where the integral is interpreted in the principal value case. This system is an example of a BBGKY hierarchy.

If one carefully rescales and takes limits (say at the energy level {u=0}, for simplicity), the left-hand side turns out to rescale to be a lower order term, and one ends up with a hierarchy for the Dyson sine process:

\displaystyle  0 = \frac{1}{2} \sum_{i=1}^k \partial_{x_i}^2 \rho_k - \sum_{1 \leq i,j \leq k;i\neq j} \partial_{x_i} \frac{\rho_k}{x_i-x_j} \ \ \ \ \ (3)

\displaystyle  - \sum_{i=1}^k \partial_{x_i} \int_{\bf R} \frac{\rho_{k+1}(x_1,\ldots,x_{k+1})}{x_i-x_{k+1}}\ dx_{k+1}.

Informally, these equations show that the Dyson sine process {\Sigma = \{ \lambda_i: i \in {\bf Z} \}} is stationary with respect to the infinite Dyson Brownian motion

\displaystyle  d\lambda_i = dB_i + \sum_{j \neq i} \frac{dt}{\lambda_i-\lambda_j}

where {dB_i} are independent Brownian increments, and the sum is interpreted in a suitable principal value sense.

I recently set myself the exercise of deriving the identity (3) directly from the definition (1) of the Dyson sine process, without reference to GUE. This turns out to not be too difficult when done the right way (namely, by modifying the proof of Gaudin’s lemma), although it did take me an entire day of work before I realised this, and I could not find it in the literature (though I suspect that many people in the field have privately performed this exercise in the past). In any case, I am recording the computation here, largely because I really don’t want to have to do it again, but perhaps it will also be of interest to some readers.

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Let {n} be a large integer, and let {M_n} be the Gaussian Unitary Ensemble (GUE), i.e. the random Hermitian matrix with probability distribution

\displaystyle  C_n e^{-\hbox{tr}(M_n^2)/2} dM_n

where {dM_n} is a Haar measure on Hermitian matrices and {C_n} is the normalisation constant required to make the distribution of unit mass. The eigenvalues {\lambda_1 < \ldots < \lambda_n} of this matrix are then a coupled family of {n} real random variables. For any {1 \leq k \leq n}, we can define the {k}-point correlation function {\rho_k( x_1,\ldots,x_k )} to be the unique symmetric measure on {{\bf R}^k} such that

\displaystyle  \int_{{\bf R}^k} F(x_1,\ldots,x_k) \rho_k(x_1,\ldots,x_k) = {\bf E} \sum_{1 \leq i_1 < \ldots < i_k \leq n} F( \lambda_{i_1}, \ldots, \lambda_{i_k} ).

A standard computation (given for instance in these lecture notes of mine) gives the Ginebre formula

\displaystyle  \rho_n(x_1,\ldots,x_n) = C'_n (\prod_{1 \leq i < j \leq n} |x_i-x_j|^2) e^{-\sum_{j=1}^n |x_j|^2/2}.

for the {n}-point correlation function, where {C'_n} is another normalisation constant. Using Vandermonde determinants, one can rewrite this expression in determinantal form as

\displaystyle  \rho_n(x_1,\ldots,x_n) = C''_n \det(K_n(x_i,x_j))_{1 \leq i, j \leq n}

where the kernel {K_n} is given by

\displaystyle  K_n(x,y) := \sum_{k=0}^{n-1} \phi_k(x) \phi_k(y)

where {\phi_k(x) := P_k(x) e^{-x^2/4}} and {P_0, P_1, \ldots} are the ({L^2}-normalised) Hermite polynomials (thus the {\phi_k} are an orthonormal family, with each {P_k} being a polynomial of degree {k}). Integrating out one or more of the variables, one is led to the Gaudin-Mehta formula

\displaystyle  \rho_k(x_1,\ldots,x_k) = \det(K_n(x_i,x_j))_{1 \leq i, j \leq k}. \ \ \ \ \ (1)

(In particular, the normalisation constant {C''_n} in the previous formula turns out to simply be equal to {1}.) Again, see these lecture notes for details.

The functions {\phi_k(x)} can be viewed as an orthonormal basis of eigenfunctions for the harmonic oscillator operator

\displaystyle  L \phi := (-\frac{d^2}{dx^2} + \frac{x^2}{4})\phi;

indeed it is a classical fact that

\displaystyle  L \phi_k = (k + \frac{1}{2}) \phi_k.

As such, the kernel {K_n} can be viewed as the integral kernel of the spectral projection operator {1_{(-\infty,n+\frac{1}{2}]}(L)}.

From (1) we see that the fine-scale structure of the eigenvalues of GUE are controlled by the asymptotics of {K_n} as {n \rightarrow \infty}. The two main asymptotics of interest are given by the following lemmas:

Lemma 1 (Asymptotics of {K_n} in the bulk) Let {x_0 \in (-2,2)}, and let {\rho_{sc}(x_0) := \frac{1}{2\pi} (4-x_0^2)^{1/2}_+} be the semicircular law density at {x_0}. Then, we have

\displaystyle  K_n( x_0 \sqrt{n} + \frac{y}{\sqrt{n} \rho_{sc}(x_0)}, x_0 \sqrt{n} + \frac{z}{\sqrt{n} \rho_{sc}(x_0)} )

\displaystyle  \rightarrow \frac{\sin(\pi(y-z))}{\pi(y-z)} \ \ \ \ \ (2)

as {n \rightarrow \infty} for any fixed {y,z \in {\bf R}} (removing the singularity at {y=z} in the usual manner).

Lemma 2 (Asymptotics of {K_n} at the edge) We have

\displaystyle  K_n( 2\sqrt{n} + \frac{y}{n^{1/6}}, 2\sqrt{n} + \frac{z}{n^{1/6}} )

\displaystyle  \rightarrow \frac{Ai(y) Ai'(z) - Ai'(y) Ai(z)}{y-z} \ \ \ \ \ (3)

as {n \rightarrow \infty} for any fixed {y,z \in {\bf R}}, where {Ai} is the Airy function

\displaystyle  Ai(x) := \frac{1}{\pi} \int_0^\infty \cos( \frac{t^3}{3} + tx )\ dt

and again removing the singularity at {y=z} in the usual manner.

The proof of these asymptotics usually proceeds via computing the asymptotics of Hermite polynomials, together with the Christoffel-Darboux formula; this is for instance the approach taken in the previous notes. However, there is a slightly different approach that is closer in spirit to the methods of semi-classical analysis, which was briefly mentioned in the previous notes but not elaborated upon. For sake of completeness, I am recording some notes on this approach here, although to focus on the main ideas I will not be completely rigorous in the derivation (ignoring issues such as convegence of integrals or of operators, or (removable) singularities in kernels caused by zeroes in the denominator).

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