Asymptotic Uniqueness of Best Rational Approximants to Complex Cauchy Transforms in L2{L}^2 of the Circle

For all n large enough, we show uniqueness of a critical point in best rational approximation of degree n, in the L^2-sense on the unit circle, to functions f, where f is a sum of a Cauchy transform of a complex measure \mu supported on a real interval included in (-1,1), whose Radon-Nikodym derivative with respect to the arcsine distribution on its support is Dini-continuous, non-vanishing and with and argument of bounded variation, and of a rational function with no poles on the support of \mu.Comment: 28 page

Meromorphic Approximants to Complex Cauchy Transforms with Polar Singularities

We study AAK-type meromorphic approximants to functions $F$, where $F$ is a sum of a rational function $R$ and a Cauchy transform of a complex measure $\lambda$ with compact regular support included in $(-1,1)$, whose argument has bounded variation on the support. The approximation is understood in $L^p$-norm of the unit circle, $p\geq2$. We obtain that the counting measures of poles of the approximants converge to the Green equilibrium distribution on the support of $\lambda$ relative to the unit disk, that the approximants themselves converge in capacity to $F$, and that the poles of $R$ attract at least as many poles of the approximants as their multiplicity and not much more.Comment: 39 pages, 4 figure

SLE and Virasoro representations: localization

We consider some probabilistic and analytic realizations of Virasoro highest-weight representations. Specifically, we consider measures on paths connecting points marked on the boundary of a (bordered) Riemann surface. These Schramm-Loewner Evolution (SLE)- type measures are constructed by the method of localization in path space. Their partition function (total mass) is the highest-weight vector of a Virasoro representation, and the action is given by Virasoro uniformization. We review the formalism of Virasoro uniformization, which allows to define a canonical action of Virasoro generators on functions (or sections) on a - suitably extended - Teichm\"uller space. Then we describe the construction of families of measures on paths indexed by marked bordered Riemann surfaces. Finally we relate these two notions by showing that the partition functions of the latter generate a highest-weight representation - the quotient of a reducible Verma module - for the former.Comment: 59 pages. To appear in Comm. Math. Phy

Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction

Virtually all questions that one can ask about the behavioral and structural complexity of a stochastic process reduce to a linear algebraic framing of a time evolution governed by an appropriate hidden-Markov process generator. Each type of question---correlation, predictability, predictive cost, observer synchronization, and the like---induces a distinct generator class. Answers are then functions of the class-appropriate transition dynamic. Unfortunately, these dynamics are generically nonnormal, nondiagonalizable, singular, and so on. Tractably analyzing these dynamics relies on adapting the recently introduced meromorphic functional calculus, which specifies the spectral decomposition of functions of nondiagonalizable linear operators, even when the function poles and zeros coincide with the operator's spectrum. Along the way, we establish special properties of the projection operators that demonstrate how they capture the organization of subprocesses within a complex system. Circumventing the spurious infinities of alternative calculi, this leads in the sequel, Part II, to the first closed-form expressions for complexity measures, couched either in terms of the Drazin inverse (negative-one power of a singular operator) or the eigenvalues and projection operators of the appropriate transition dynamic.Comment: 24 pages, 3 figures, 4 tables; current version always at http://csc.ucdavis.edu/~cmg/compmech/pubs/sdscpt1.ht

An Lp Analog to AAK Theory for p⩾2

AbstractWe develop an Lp analog to AAK theory on the unit circle that interpolates continuously between the case p=∞, which classically solves for best uniform meromorphic approximation, and the case p=2, which is equivalent to H2-best rational approximation. We apply the results to the uniqueness problem in rational approximation and to the asymptotic behaviour of poles of best meromorphic approximants to functions with two branch points. As pointed out by a referee, part of the theory extends to every p∈[1, ∞] when the definition of the Hankel operator is suitably generalized; this we discuss in connection with the recent manuscript by V. A. Prokhorov, submitted for publication

The Ising Partition Function: Zeros and Deterministic Approximation

We study the problem of approximating the partition function of the ferromagnetic Ising model in graphs and hypergraphs. Our first result is a deterministic approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters $\beta$ (the interaction) and $\lambda$ (the external field), except for the case $\vert{\lambda}\vert=1$ (the "zero-field" case). A randomized algorithm (FPRAS) for all graphs, and all $\beta,\lambda$, has long been known. Unlike most other deterministic approximation algorithms for problems in statistical physics and counting, our algorithm does not rely on the "decay of correlations" property. Rather, we exploit and extend machinery developed recently by Barvinok, and Patel and Regts, based on the location of the complex zeros of the partition function, which can be seen as an algorithmic realization of the classical Lee-Yang approach to phase transitions. Our approach extends to the more general setting of the Ising model on hypergraphs of bounded degree and edge size, where no previous algorithms (even randomized) were known for a wide range of parameters. In order to achieve this extension, we establish a tight version of the Lee-Yang theorem for the Ising model on hypergraphs, improving a classical result of Suzuki and Fisher.Comment: clarified presentation of combinatorial arguments, added new results on optimality of univariate Lee-Yang theorem