%0 Journal Article %J Symmetry, Integrability and Geometry. Methods and Applications %D 2018 %T Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane %A Marco Bertola %A José Gustavo Elias Rebelo %A Tamara Grava %X

We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated to a certain normal matrix model. The model depends on a parameter and the asymptotic distribution of the eigenvalues undergoes a transition for a special value of the parameter, where it develops a corner-type singularity. In the double scaling limit near the transition we determine the asymptotic behaviour of the orthogonal polynomials in terms of a solution of the Painlev´e IV equation. We determine the Fredholm determinant associated to such solution and we compute it numerically on the real line, showing also that the corresponding Painlev´e transcendent is pole-free on a semiaxis.

%B Symmetry, Integrability and Geometry. Methods and Applications %I National Academy of Sciences of Ukraine %V 14 %G eng %R 10.3842/SIGMA.2018.091 %0 Journal Article %J J. Phys. A %D 2015 %T The partition function of the extended $r$-reduced Kadomtsev-Petviashvili hierarchy %A Marco Bertola %A Di Yang %B J. Phys. A %V 48 %P 195205, 20 %G eng %U http://dx.doi.org/10.1088/1751-8113/48/19/195205 %R 10.1088/1751-8113/48/19/195205 %0 Journal Article %J J. Math. Phys. %D 2009 %T The partition function of the two-matrix model as an isomonodromic τ function %A Marco Bertola %A Marchal, O. %B J. Math. Phys. %V 50 %P 013529, 17 %G eng %U http://0-dx.doi.org.mercury.concordia.ca/10.1063/1.3054865 %R 10.1063/1.3054865 %0 Journal Article %J Math. Phys. Anal. Geom. %D 2006 %T The PDEs of biorthogonal polynomials arising in the two-matrix model %A Marco Bertola %A B. Eynard %B Math. Phys. Anal. Geom. %V 9 %P 23–52 %G eng %0 Journal Article %J J. Phys. A %D 2003 %T Partition functions for matrix models and isomonodromic tau functions %A Marco Bertola %A B. Eynard %A Harnad, J. %B J. Phys. A %V 36 %P 3067–3083 %G eng