Physics and Mathematics of Disordered Systems

Posts Tagged ‘eigenvalue

Random matrix theory and the Coulomb gas

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Today I have the pleasure of presenting you a guest post by Ricardo, a good physicist friend of mine in Paris, who is working on random matrix theory. Enjoy!

After writing a nice piece of hardcore physics to my science blog (in Portuguese, I am sorry), Alex asked me to come by and repay the favor. I am happy to write a few lines about the basis of my research in random matrices, and one of the first nice surprises we have while learning the subject.

In this post, I intent to present you some few neat tricks I learned while tinkering with Random Matrix Theory (RMT). It is a pretty vast subject, whose ramifications extend to nuclear physics, information theory, particle physics and, surely, mathematics as a whole. One of the main questions on this subject is: given a matrix M whose entries are taken randomly from a known distribution, what would the distribution for its eigenvalues be?

The applications of such a question are countless. We might be interested in the energy levels of a random hamiltonian, the information transmitted through a random noisy channel, the transmission probability of an electron through an open quantum system; every eigenvalue problem, when the effects of the environment are taken into account, must answer this most fundamental question in RMT. To do so, first we restrict our analysis to symmetric (resp. hermitian) matrices. We have two reasons for it: firstly, most applications in physics involve this kind of matrix (hamiltonians, S-matrices), secondly, they are much easier to analyze, since they can always be diagonalized by a orthogonal (resp. unitary) matrix.

To enable a complete analysis, I must introduce an extra assumption: our matrix must have a probability density function (p.d.f) invariant with respect to orthogonal (resp. unitary) conjugation. This is not such a narrow class of matrices as one might think, we can construct this kind of matrix with nearly any infinitely divisible probability. Roughly, we want to be able to write:

\displaystyle P(M) = P(UDU^{-1}) = P(D),

where D is a diagonal matrix.

However, we should not get too excited with this property, as to write P(D) = P(\{\lambda_1,\ldots ,\lambda_n\}) right away. We are always talking about a probability density function, the computation of the actual probability requires an integral over a certain space, and the passage from the matrix to its eigenvalues requires a variable change, hence we must calculate the Jacobian of such a transformation.

Let us take the simplest and perhaps most useful example: the Gaussian real (resp. complex) symmetric (resp. hermitian) matrix whose entries are i.i.d. normal variables with zero mean and variance 1. We can verify that this matrix is invariant with respect to orthogonal conjugation (actually, we can prove that the Gaussian matrix is the only matrix invariant with respect to orthogonal conjugation and whose entries are independent). Hence, we expect the eigenvalue probability to be:

\displaystyle P(\{\lambda_1,\ldots ,\lambda_n\})=C_N \prod_{j=1}^N e^{-\frac{\beta}{2}\lambda^2}.J(\{\lambda_1,\ldots ,\lambda_n\}).

The value for \beta will be discussed soon, C_N is a normalization constant. I will not derive the Jacobian term and I will refer to [1] for the complete demonstration, not only of this expression, but also of the uniqueness theorem mentioned above. Even though the entries of our matrix are independent, the p.d.f. of its eigenvalues its highly coupled due to the Jacobian:

\displaystyle J(\{\lambda_1,\ldots ,\lambda_n\})=\prod_{j\neq i}|\lambda_i-\lambda_j|^\beta

\beta is one of three possible values: if our initial matrix is real, \beta=1. If it is a complex matrix, \beta=2. If M is a quaternion matrix, \beta=4. This is called Dyson’s threefold way.

Since we have the eigenvalues p.d.f., it should be easy to calculate all quantities involved in a eigenvalue problem. We could be interested in the probability of having an eigenvalue larger than a threshold [2], or the chance of not finding any eigenvalue at all inside an interval. The calculation necessary would be simply the N-dimensional integral of the p.d.f. multiplied by a function, but, oh dear, what a probability function!

The Jacobian will make most variables changes futile, all attempts of uncoupling the equations are faced with the product (\lambda_j-\lambda_i). One standard approach would be to interpret the Jacobian as a Vandermonde determinant (up to a sign), rearrange rows to compose polynomials and adjust them as to create orthogonal functions, whose integral would preserve only the “diagonal” terms of our calculations. I will not explain this technique in detail, even though it is quite beautiful. I will travel another path.

Let us write the whole probability as an exponential. We already have the Gaussian term as an exponential, let us raise the Jacobian to the exponent in the most standard way, we write:

\displaystyle P(\{\lambda_j\})=C_N e^{-\frac{\beta}{2}\sum_{j=1}^N\lambda^2}\prod_{j\neq i}|\lambda_i-\lambda_j|^\beta=C_Ne^{-\beta\mathcal{H}},

where \mathcal{H}=\frac{1}{2}\sum_j \lambda_j^2 - \sum_{i\neq j}\log|\lambda_i-\lambda_j|. If we write C_N=\frac{1}{Z}, the analogy is complete, we can treat this probability density function as a system analyzed by the Boltzmann Canonical Ensemble! Hence, we import all our techniques and tools from statistical mechanics to solve this ugly, nasty piece of probability density function.

It is interesting to remark that this Hamiltonian has a natural physical interpretation. If you recall your electrodynamics course, you might find, deep in your memory, the derivation of the Coulomb potential for two point charges in 2D: if we say their position is \lambda_i and \lambda_j, we have V \propto \log|\lambda_i-\lambda_j|. Our Hamiltonian is the description of a 2D gas under a quadratic potential, or, as we call it, Dyson’s Coulomb gas. We can actually infer the equilibrium position of these electrons, if we consider the quadratic potential and the mutual logarithmic repulsion:

It is no surprise that, for the large N limit, the p.d.f. of the electrons, and hence that of the eigenvalues, is the Wigner Semicircle Law. To deduce it, the next natural steps are (I am not going to detail them any further): replace \lambda_j by its density: \rho(x)=\frac{1}{N}\sum_j \delta(x-\lambda_j), write the Hamiltonian in terms of the density, replace the integral over the eigenvalues by a functional integral over the possible densities \rho, use a saddle-point approximation to argue that only the “best” density will matter for our calculations, derive functionally the Hamiltonian and solve a Tricomi equation to obtain the optimal value for \rho, the equilibrium position of the electrons. Surely, we suppose that N is large for the saddle-point approximation, results for small N require a different approach.

To wrap it up, we can avoid treating a highly coupled integral by sending the nasty coupled part to the exponent, interpret the system as a Canonical Ensemble, and treat it in the thermodynamical limit to find the density of eigenvalues. This beautiful treatment was popularized by Dyson and revisited many times in the last decade. RMT is a very active subject, for physicists and mathematicians,  and I hope that, with these few lines, you have been able to see a little but of what I see on it, and to appreciate, for a change, a place where physics has come to help mathematics on a hard day of work.

[1] M. L. Mehta, Random Matrices, Third Edition, Elsevier, 2004.

[2] S. N. Majumdar, C. Nadal, A. Scardicchio, P. Vivo, How many eigenvalues of a Gaussian random matrix are positive?, Phys. Rev. E 83, 041105 (2011)


Written by Ricardo Marino

October 19, 2012 at 11:42 am