@article {bertola2018painleve, title = {Painlev{\'e} IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane}, journal = {Symmetry, Integrability and Geometry. Methods and Applications}, volume = {14}, year = {2018}, publisher = {National Academy of Sciences of Ukraine}, abstract = {

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{\textasciiacute}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{\textasciiacute}e transcendent is pole-free on a semiaxis.

}, doi = {10.3842/SIGMA.2018.091}, author = {Marco Bertola and Jos{\'e} Gustavo Elias Rebelo and Tamara Grava} } @article {MR3342766, title = {The partition function of the extended $r$-reduced Kadomtsev-Petviashvili hierarchy}, journal = {J. Phys. A}, volume = {48}, number = {19}, year = {2015}, pages = {195205, 20}, issn = {1751-8113}, doi = {10.1088/1751-8113/48/19/195205}, url = {http://dx.doi.org/10.1088/1751-8113/48/19/195205}, author = {Marco Bertola and Di Yang} } @article {Bertola:IZtau, title = {The partition function of the two-matrix model as an isomonodromic τ function}, journal = {J. Math. Phys.}, volume = {50}, number = {1}, year = {2009}, pages = {013529, 17}, issn = {0022-2488}, doi = {10.1063/1.3054865}, url = {http://0-dx.doi.org.mercury.concordia.ca/10.1063/1.3054865}, author = {Marco Bertola and Marchal, O.} } @article {Bertola:PDEBOPs, title = {The PDEs of biorthogonal polynomials arising in the two-matrix model}, journal = {Math. Phys. Anal. Geom.}, volume = {9}, number = {1}, year = {2006}, pages = {23{\textendash}52}, issn = {1385-0172}, author = {Marco Bertola and B. Eynard} } @article {Bertola:PartitionJPA, title = {Partition functions for matrix models and isomonodromic tau functions}, journal = {J. Phys. A}, volume = {36}, number = {12}, year = {2003}, note = {Random matrix theory}, pages = {3067{\textendash}3083}, issn = {0305-4470}, author = {Marco Bertola and B. Eynard and Harnad, J.} }