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We introduce the class of strongly Rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class contains e.g. product measures, uniform random spanning tree measures, and large classes of determinantal probability measures and distributions for symmetric exclusion… (More)

- Julius Borcea, Petter Br¨andén
- 2008

In 1952 Lee and Yang proposed the program of analyzing phase transitions in terms of zeros of partition functions. Linear operators preserving non-vanishing properties are an essential tool in this program as well as various contexts in complex analysis, probability theory, combinatorics, matrix theory. We characterize all linear operators on finite or… (More)

- JULIUS BORCEA
- 2008

A multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts. We classify all linear partial differential operators in the Weyl algebra An that preserve stability. An important tool that we develop in the process is the higher dimensional generalization of Pólya-Schur's notion of multiplier sequence. We… (More)

- JULIUS BORCEA
- 2006

We characterize all linear operators on finite or infinite-dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region Ω ⊆ C for arbitrary closed circular domains Ω (i.e., images of the closed unit disk under a Möbius transformation) and their boundaries. This provides a natural framework for dealing with… (More)

- Julius Borcea, Petter Br¨andén
- 2008

In the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing in products of open circular domains. For such sets this completes the multivariate generalization of the classification program initiated by Pólya-Schur for univariate real polynomials. We build on… (More)

- JULIUS BORCEA
- 2007

The study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of double-periodic solutions of its Weier-strass form has a long history. Such solutions appear at integer values of the spectral parameter and their respective eigenvalues serve as the ends of bands in the boundary value problem for the corresponding… (More)

- JULIUS BORCEA
- 2008

A multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts. We characterize all finite order linear differential operators that preserve stability. An important technical tool that we develop in the process is the multivariate generalization of the classical notion of multiplier sequence. We give a… (More)

- JULIUS BORCEA, J. BORCEA
- 2005

A notion of weighted multivariate majorization is defined as a preorder on sequences of vectors in Euclidean space induced by the Choquet ordering for atomic probability measures. We characterize this preorder both in terms of stochastic matrices and convex functions and use it to describe the distribution of equilibrium points of logarithmic potentials… (More)

Univariate polynomials with only real roots, while special, do occur often enough that their properties can lead to interesting conclusions in diverse areas. Due mainly to the recent work of two young mathematicians, Julius Borcea and Petter Brändén, a very successful multivariate generalization of this method has been developed. The first part of this… (More)

- JULIUS BORCEA
- 2006

For n × n matrices A and B define η(A, B) = X S det(A[S]) det`BˆS ′ ˜´, where the summation is over all subsets of {1,. .. , n}, S ′ is the complement of S, and A[S] is the principal submatrix of A with rows and columns indexed by S. We prove that if A ≥ 0 and B is Hermitian then (1) the polynomial η(zA, −B) has all real roots (2) the latter polynomial has… (More)