Paul Balduf#paperOfTheDay is "Correlation functions and zeros of a Gaussian power series and Pfaffians" from 2013.
This paper is a generalisation of the study of random polynomials: They consider random power series, i.e. polynomials with infinitely many terms. These have (almost always) a radius of convergence of unity, so that it only makes sense to study them in the domain (-1,1). There is an accumulation of zeros (=roots) close to the boundaries of this interval.
Given that the coefficients of the power series are random, so are the locations of zeros. The positive locations form an infinite sequence of random numbers, a point process. As such, one can ask about the mean, variance, and all other correlation functions. The main result of the article is that these quantities are given by a Pfaffian (which is an algebraic object similar to a determinant) of some explicitly known matrices.
I got interested in this observation because Pfaffians also show up in #quantumFieldTheory . For example, Isserlis theorem (sometimes called Wicks theorem by physicists) says that the expectation of a product of Gaussian variables is the Pfaffian of their covariances. Or, Pfaffians show up as the integrands in #FeynmanIntegral s in topological field theories.
#mathematics #probabilityTheory
https://projecteuclid.org/journals/electronic-journal-of-probability/volume-18/issue-none/Correlation-functions-for-zeros-of-a-Gaussian-power-series-and/10.1214/EJP.v18-2545.full
This paper is a generalisation of the study of random polynomials: They consider random power series, i.e. polynomials with infinitely many terms. These have (almost always) a radius of convergence of unity, so that it only makes sense to study them in the domain (-1,1). There is an accumulation of zeros (=roots) close to the boundaries of this interval.
Given that the coefficients of the power series are random, so are the locations of zeros. The positive locations form an infinite sequence of random numbers, a point process. As such, one can ask about the mean, variance, and all other correlation functions. The main result of the article is that these quantities are given by a Pfaffian (which is an algebraic object similar to a determinant) of some explicitly known matrices.
I got interested in this observation because Pfaffians also show up in #quantumFieldTheory . For example, Isserlis theorem (sometimes called Wicks theorem by physicists) says that the expectation of a product of Gaussian variables is the Pfaffian of their covariances. Or, Pfaffians show up as the integrands in #FeynmanIntegral s in topological field theories.
#mathematics #probabilityTheory
https://projecteuclid.org/journals/electronic-journal-of-probability/volume-18/issue-none/Correlation-functions-for-zeros-of-a-Gaussian-power-series-and/10.1214/EJP.v18-2545.full
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