ON THE DISTRIBUTION OF THE LARGEST REAL EIGENVALUE FOR THE REAL GINIBRE ENSEMBLE

Let $\sqrt{N}+\lambda_{max}$ be the largest real eigenvalue of a random $N\times N$ matrix with independent $N(0,1)$ entries (the `real Ginibre matrix'). We study the large deviations behaviour of the limiting $N\rightarrow \infty$ distribution $P[\lambda_{max}<t]$ of the shifted maximal real eigenvalue $\lambda_{max}$. In particular, we prove that the right tail of this distribution is Gaussian: for $t>0$, P[\lambda_{max} This is a rigorous confirmation of the corresponding result of Forrester and Nagao. We also prove that the left tail is exponential: for $t<0$, $$P[\lambda_{max}<t]= e^{\frac{1}{2\sqrt{2\pi}}\zeta\left(\frac{3}{2}\right)t+O(1)},$$ where $\zeta$ is the Riemann zeta-function. Our results have implications for interacting particle systems. The edge scaling limit of the law of real eigenvalues for the real Ginibre ensemble is a rescaling of a fixed time distribution of annihilating Brownian motions (ABM's) with the step initial condition. Therefore, the tail behaviour of the distribution of $X_s^{(max)}$ - the position of the rightmost annihilating particle at fixed time $s>0$ - can be read off from the corresponding answers for $\lambda_{max}$ using $X_s^{(max)}\stackrel{D}{=} \sqrt{4s}\lambda_{max}$.Comment: 20 pages, expanded introduction, added reference

Kang-Redner Anomaly in Cluster-Cluster Aggregation

The large time, small mass, asymptotic behavior of the average mass distribution \pb is studied in a $d$-dimensional system of diffusing aggregating particles for $1\leq d \leq 2$. By means of both a renormalization group computation as well as a direct re-summation of leading terms in the small reaction-rate expansion of the average mass distribution, it is shown that \pb \sim \frac{1}{t^d} (\frac{m^{1/d}}{\sqrt{t}})^{e_{KR}} for $m \ll t^{d/2}$, where $e_{KR}=\epsilon +O(\epsilon ^2)$ and $\epsilon =2-d$. In two dimensions, it is shown that \pb \sim \frac{\ln(m) \ln(t)}{t^2} for $m \ll t/ \ln(t)$. Numerical simulations in two dimensions supporting the analytical results are also presented.Comment: 11 pages, 6 figures, Revtex

Persistence properties of a system of coagulating and annihilating random walkers

We study a d-dimensional system of diffusing particles that on contact either annihilate with probability 1/(q-1) or coagulate with probability (q-2)/(q-1). In 1-dimension, the system models the zero temperature Glauber dynamics of domain walls in the q-state Potts model. We calculate P(m,t), the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly (m-1) coagulations. Using perturbative renormalization group analysis for d < 2, we show that, if the number of coagulations m is much less than the typical number M(t), then P(m,t) ~ m^(z/d) t^(-theta), with theta=d Q + Q(Q-1/2) epsilon + O(epsilon^2), z=(2Q-1) epsilon + (2 Q-1) (Q-1)(1/2+A Q) epsilon^2 +O(epsilon^3), where Q=(q-1)/q, epsilon =2-d and A =-0.006. M(t) is shown to scale as t^(d/2-delta), where delta = d (1 -Q)+(Q-1)(Q-1/2) epsilon+ O(epsilon^2). In two dimensions, we show that P(m,t) ~ ln(t)^(Q(3-2Q)) ln(m)^((2Q-1)^2) t^(-2Q) for m << t^(2 Q-1). The 1-dimensional results corresponding to epsilon=1 are compared with results from Monte Carlo simulations.Comment: 12 pages, revtex, 5 figure

Information theory of massively parallel probe storage channels

Motivated by the concept of probe storage, we study the problem of information retrieval using a large array of N nano-mechanical probes, N ~ 4000. At the nanometer scale it is impossible to avoid errors in the positioning of the array, thus all signals retrieved by the probes of the array at a given sampling moment are affected by the same amount of random position jitter. Therefore a massively parallel probe storage device is an example of a noisy communication channel with long range correlations between channel outputs due to the global positioning errors. We find that these correlations have a profound effect on the channel's properties. For example, it turns out that the channel's information capacity does approach 1 bit per probe in the limit of high signal-to-noise ratio, but the rate of the approach is only polynomial in the channel noise strength. Moreover, any error correction code with block size N >> 1 such that codewords correspond to the instantaneous outputs of the all probes in the array exhibits an error floor independently of the code rate. We illustrate this phenomenon explicitly using Reed-Solomon codes the performance of which is easy to simulate numerically. We also discuss capacity-achieving error correction codes for the global jitter channel and their complexity