Eigenvalues of random Hamiltonian matrices - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T16:13:09Z http://mathoverflow.net/feeds/question/120397 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120397/eigenvalues-of-random-hamiltonian-matrices Eigenvalues of random Hamiltonian matrices Austen 2013-01-31T10:34:09Z 2013-05-15T08:04:58Z <p>A real $2n\times 2n$ Hamiltonian matrix has the general form</p> <p>$$H=\begin{pmatrix} A &amp; B \cr C &amp; -A^T \end{pmatrix}$$</p> <p>where $A$, $B$ and $C$ are $n\times n$ matrices, and $B$ and $C$ are symmetric. Are there any results regarding the eigenvalue distribution of an ensemble of such matrices? For example, the above condition is equivalent to the symmetry of $JH$ with</p> <p>$$J=\begin{pmatrix} 0 &amp; 1 \cr -1 &amp; 0 \end{pmatrix},$$</p> <p>so it would be natural to consider the Gaussian orthogonal ensemble for $JH$. Generically, the eigenvalues come in quadruples: $\lambda, -\lambda, \bar\lambda, -\bar\lambda$</p> http://mathoverflow.net/questions/120397/eigenvalues-of-random-hamiltonian-matrices/120501#120501 Answer by Carlo Beenakker for Eigenvalues of random Hamiltonian matrices Carlo Beenakker 2013-02-01T08:43:27Z 2013-02-01T14:20:19Z <p>The exponent $M=e^H\$ of a <A HREF="http://en.wikipedia.org/wiki/Hamiltonian_matrix" rel="nofollow">Hamiltonian matrix</A> $H$ is a <A HREF="http://en.wikipedia.org/wiki/Symplectic_matrix" rel="nofollow">symplectic matrix</A>. So you might equivalently ask for the distribution of the eigenvalues $\xi=e^{\lambda}\$ of $M$. There is an extensive literature on this in random matrix theory, I give some pointers below. </p> <p>The random symplectic matrix $M$ appears as the transfer matrix for the wave equation of a disordered medium, with many applications in optics and electronics. The most natural ensemble for the transfer matrix is inherited from the circular ensemble of the scattering matrix $S$ for the same wave equation. There is a one-to-one algebraic relation between the symplectic transfer matrix $M$ and the unitary scattering matrix $S$. The random matrix ensemble for $S$ is the familiar circular ensemble.</p> <p>Here are some references to papers on random transfer matrix ensembles. (Notice that the word <em>Hamiltonian</em> has a different meaning in these papers.) </p> <ol> <li><p>S. Bachmann, M. Butz, W. De Roeck, <A HREF="http://arxiv.org/abs/1204.0823" rel="nofollow">Disordered quantum wires: microscopic origins of the DMPK theory and Ohm's law</A> (2012).</p></li> <li><p>M. Caselle, U. Magnea, Random matrix theory and symmetric spaces (2004).</p></li> <li><p>J. An, Z. Wang, K. Yan, A generalization of random matrix ensembles (2005).</p></li> <li>P. Devillard, <A HREF="http://dx.doi.org/10.1007/BF01020873" rel="nofollow">Statistics of transfer matrices for disordered quantum thin metallic slabs</A> (1991).</li> </ol> http://mathoverflow.net/questions/120397/eigenvalues-of-random-hamiltonian-matrices/130458#130458 Answer by Carlo Beenakker for Eigenvalues of random Hamiltonian matrices Carlo Beenakker 2013-05-13T08:23:22Z 2013-05-15T08:04:58Z <p>In the course of a <A HREF="http://arxiv.org/abs/1305.2924" rel="nofollow">physics project</A> in my group, I have had an opportunity to learn more about the eigenvalue statistics of Hamiltonian matrices. (Our physics problem actually involved skew-Hamiltonian matrices, so I made a small detour, joined by Jonathan Edge &amp; Jan Dahlhaus.)</p> <p>The ensemble is the one you suggested: $2n\times 2n$ real matrices $H$ with Hamiltonian symmetry and normally distributed elements. It is convenient to rescale the eigenvalues $\varepsilon_k$ of $H$ by a factor $\sqrt{2n}$, and separate the real and imaginary parts:</p> <p>$(2n)^{-1/2}\varepsilon_k=x_k+iy_k$.</p> <p>The eigenvalue density in the complex plane $x+iy$ consists of three parts: a two-dimensional density $\rho_{c}(x,y)$ of the complex eigenvalues, a one-dimensional density $\rho_{r}(x)$ of the real eigenvalues and another one-dimensional density $\rho_{i}(y)$ of the imaginary eigenvalues.</p> <p>Based on numerical experiments, I can offer three conjectures:</p> <p>1) For large $n$, the rescaled complex eigenvalues $x_k+iy_k$ uniformly cover a disc of unit radius, </p> <p>$\lim_{n\rightarrow\infty}n^{-1}\rho_{c}(x,y)=2/\pi$ for $x^2+y^2&lt;1$.</p> <p>2) For large $n$, the rescaled real eigenvalues $x_k$ uniformly cover the interval <code>$-1&lt;x&lt;1$</code>, with density </p> <p><code>$\lim_{n\rightarrow\infty}n^{-1/2}\rho_{r}(x)=1/\sqrt{\pi}$</code>. </p> <p>Therefore the expectation value of the number $n_{r}$ of real eigenvalues satisfies $\lim_{n\rightarrow\infty}n^{-1/2}E[n_{r}]=2/\sqrt{\pi}$.</p> <p>3) Also the rescaled imaginary eigenvalues $iy_{k}$ have a uniform density in the large-$n$ limit, in the interval <code>$-1&lt;y&lt;1$</code>, but this density is less than the density of the real eigenvalues. The expectation value of the number $n_{i}$ of imaginary eigenvalues satisfies <code>$\lim_{n\rightarrow\infty}n^{-1/2}E[n_{i}]={\rm constant}\approx 0.72$</code>.</p> <p>Conjectures 1 and 2 were <A HREF="http://www-math.mit.edu/~edelman/homepage/papers/circular.pdf" rel="nofollow">proven</A> by Edelman and collaborators in the absence of Hamiltonian symmetry, so when all $(2n)^{2}$ real matrix elements of $H$ are chosen from independent normal distributions. [This is known as the (real) Ginibre ensemble.] Our numerics suggests that, for large matrices, the Hamiltonian symmetry only affects the (rescaled) eigenvalue distribution within a distance of order $n^{-1/2}$ from the imaginary axis.</p> <p>By way of illustration, I include a plot of the eigenvalues $\varepsilon$ of $200$ real matrices of size $100\times 100$ (so $n=50$), with normally distributed matrix elements, both with the Hamiltonian symmetry (left) and without (right). These eigenvalues are shown without rescaling, so they cover a disc of radius $\sqrt{2n}=10$.</p> <p><IMG SRC="http://www.lorentz.leidenuniv.nl/beenakker/MO/Hamiltonian_Ginibre.png"></p>