The noise in the circular law and the Gaussian free field

Fill an n x n matrix with independent complex Gaussians of variance 1/n. As n approaches infinity, the eigenvalues {z_k} converge to a sum of an H^1-noise on the unit disk and an independent H^{1/2}-noise on the unit circle. More precisely, for C^1 functions of suitable growth, the distribution of sum_{k=1}^n (f(z_k)-E f(z_k)) converges to that of a mean-zero Gaussian with variance given by the sum of the squares of the disk H^1 and the circle H^{1/2} norms of f. Moreover, with p_n the characteristic polynomial, log|p_n|- E log|p_n| tends to the planar Gaussian free field conditioned to be harmonic outside the unit disk. Finally, for polynomial test functions f, we prove that the limiting covariance structure is universal for a class of models including Haar distributed unitary matrices.Comment: 30 pages, 5 figures. Revised introduction. New section

Some impressions of a visit to parts of the South Island, June 1962

In June, 1962, at the invitation of the Tussock Grasslands and Mountain Lands Institute of New Zealand, I inspected parts of the South Island (Appendix 1), to make comparisons between high mountain areas of Australia and tussock grassland and mountain areas of New Zealand (Appendix 2) and thereby gain a clearer understanding of New Zealand problems. The inspections were arranged and conducted by the Director of the Institute, Mr L. W. McCaskill, usually in conjunction with other workers, runholders and administrators concerned with high country problems. Despite the necessarily selective nature of the visit, both as regards places and people, a reasonable cross-section of country, problems and opinions was encountered which, with recollections of an earlier visit in 1951, permitted some impressions to be formed. What is the solution to the deteriorated condition of New Zealand tussock grasslands and mountain lands, as manifest in many ways such as soil erosion, stream aggradation, flooding, weed and pest invasion, and declining stock-carrying capacity? Since there is a common denominator to most of these areas-tussock grassland-universal solution is sometimes expected. But the environment is so diverse, especially as regards topography, altitude and associated climate that no one solution can be possible and the illusion is best forgotten. There are many problems and each may require a separate solution. There is little point is discussing the many day-to-day problems with which New Zealand workers are already fully familiar, such as the need for cheaper effective fencing, and feral animal and weed control. The basic question is the determination of correct land use and this is the issue which is considered here

Nonlinear Schroedinger equation with two symmetric point interactions in one dimension

We consider a time-dependent one-dimensional nonlinear Schroedinger equation with a symmetric potential double well represented by two delta interactions. Among our results we give an explicit formula for the integral kernel of the unitary semigroup associated with the linear part of the Hamiltonian. Then we establish the corresponding Strichartz-type estimate and we prove local existence and uniqueness of the solution to the original nonlinear problem

When is a bottleneck a bottleneck?

Bottlenecks, i.e. local reductions of capacity, are one of the most relevant scenarios of traffic systems. The asymmetric simple exclusion process (ASEP) with a defect is a minimal model for such a bottleneck scenario. One crucial question is "What is the critical strength of the defect that is required to create global effects, i.e. traffic jams localized at the defect position". Intuitively one would expect that already an arbitrarily small bottleneck strength leads to global effects in the system, e.g. a reduction of the maximal current. Therefore it came as a surprise when, based on computer simulations, it was claimed that the reaction of the system depends in non-continuous way on the defect strength and weak defects do not have a global influence on the system. Here we reconcile intuition and simulations by showing that indeed the critical defect strength is zero. We discuss the implications for the analysis of empirical and numerical data.Comment: 8 pages, to appear in the proceedings of Traffic and Granular Flow '1

Decay versus survival of a localized state subjected to harmonic forcing: exact results

We investigate the survival probability of a localized 1-d quantum particle subjected to a time dependent potential of the form $rU(x)\sin{\omega t}$ with $U(x)=2\delta (x-a)$ or $U(x)= 2\delta(x-a)-2\delta (x+a)$. The particle is initially in a bound state produced by the binding potential $-2\delta (x)$. We prove that this probability goes to zero as $t\to\infty$ for almost all values of $r$, $\omega$, and $a$. The decay is initially exponential followed by a $t^{-3}$ law if $\omega$ is not close to resonances and $r$ is small; otherwise the exponential disappears and Fermi's golden rule fails. For exceptional sets of parameters $r,\omega$ and $a$ the survival probability never decays to zero, corresponding to the Floquet operator having a bound state. We show similar behavior even in the absence of a binding potential: permitting a free particle to be trapped by harmonically oscillating delta function potential

Hierarchical pinning models, quadratic maps and quenched disorder

We consider a hierarchical model of polymer pinning in presence of quenched disorder, introduced by B. Derrida, V. Hakim and J. Vannimenius in 1992, which can be re-interpreted as an infinite dimensional dynamical system with random initial condition (the disorder). It is defined through a recurrence relation for the law of a random variable {R_n}_{n=1,2,...}, which in absence of disorder (i.e., when the initial condition is degenerate) reduces to a particular case of the well-known Logistic Map. The large-n limit of the sequence of random variables 2^{-n} log R_n, a non-random quantity which is naturally interpreted as a free energy, plays a central role in our analysis. The model depends on a parameter alpha>0, related to the geometry of the hierarchical lattice, and has a phase transition in the sense that the free energy is positive if the expectation of R_0 is larger than a certain threshold value, and it is zero otherwise. It was conjectured by Derrida et al. (1992) that disorder is relevant (respectively, irrelevant or marginally relevant) if 1/2<alpha<1 (respectively, alpha<1/2 or alpha=1/2), in the sense that an arbitrarily small amount of randomness in the initial condition modifies the critical point with respect to that of the pure (i.e., non-disordered) model if alpha is larger or equal to 1/2, but not if alpha is smaller than 1/2. Our main result is a proof of these conjectures for the case alpha different from 1/2. We emphasize that for alpha>1/2 we find the correct scaling form (for weak disorder) of the critical point shift.Comment: 26 pages, 2 figures. v3: Theorem 1.6 improved. To appear on Probab. Theory Rel. Field

Central limit theorem for multiplicative class functions on the symmetric group

Hambly, Keevash, O'Connell and Stark have proven a central limit theorem for the characteristic polynomial of a permutation matrix with respect to the uniform measure on the symmetric group. We generalize this result in several ways. We prove here a central limit theorem for multiplicative class functions on symmetric group with respect to the Ewens measure and compute the covariance of the real and the imaginary part in the limit. We also estimate the rate of convergence with the Wasserstein distance.Comment: 23 pages; the mathematics is the same as in the previous version, but there are several improvments in the presentation, including a more intuitve name for the considered function

Star Unfolding Convex Polyhedra via Quasigeodesic Loops

We extend the notion of star unfolding to be based on a quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple (non-overlapping), planar polygon: cut along one shortest path from each vertex of P to Q, and cut all but one segment of Q.Comment: 10 pages, 7 figures. v2 improves the description of cut locus, and adds references. v3 improves two figures and their captions. New version v4 offers a completely different proof of non-overlap in the quasigeodesic loop case, and contains several other substantive improvements. This version is 23 pages long, with 15 figure

Transmission Properties of the oscillating delta-function potential

We derive an exact expression for the transmission amplitude of a particle moving through a harmonically driven delta-function potential by using the method of continued-fractions within the framework of Floquet theory. We prove that the transmission through this potential as a function of the incident energy presents at most two real zeros, that its poles occur at energies $n\hbar\omega+\varepsilon^*$ ($0<Re(\varepsilon^*)<\hbar\omega$), and that the poles and zeros in the transmission amplitude come in pairs with the distance between the zeros and the poles (and their residue) decreasing with increasing energy of the incident particle. We also show the existence of non-resonant "bands" in the transmission amplitude as a function of the strength of the potential and the driving frequency.Comment: 21 pages, 12 figures, 1 tabl

Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process

Consider the zero set of the random power series f(z)=sum a_n z^n with i.i.d. complex Gaussian coefficients a_n. We show that these zeros form a determinantal process: more precisely, their joint intensity can be written as a minor of the Bergman kernel. We show that the number of zeros of f in a disk of radius r about the origin has the same distribution as the sum of independent {0,1}-valued random variables X_k, where P(X_k=1)=r^{2k}. Moreover, the set of absolute values of the zeros of f has the same distribution as the set {U_k^{1/2k}} where the U_k are i.i.d. random variables uniform in [0,1]. The repulsion between zeros can be studied via a dynamic version where the coefficients perform Brownian motion; we show that this dynamics is conformally invariant.Comment: 37 pages, 2 figures, updated proof