Generalized multicritical one-matrix models


We show that there exists a simple generalization of Kazakov’s multicritical one-matrix model, which interpolates between the various multicritical points of the model. The associated multicritical potential takes the form of a power series with a heavy tail, leading to a cut of the potential and its derivative at the real axis, and reduces to a polynomial at Kazakov’s multicritical points. From the combinatorial point of view the generalized model allows polygons of arbitrary large degrees (or vertices of arbitrary large degree, when considering the dual graphs), and it is the weight assigned to these large order polygons which brings about the interpolation between the multicritical points in the one-matrix model.

Nucl. Phys. B 913 (2016) 357-380


  1. Kortchemski, Igor, and Loïc Richier. “Condensation in critical Cauchy Bienaym\‘e-Galton-Watson trees.” arXiv preprint arXiv:1804.10183 (2018).
  2. Bondesan, Roberto, Sergio Caracciolo, and Andrea Sportiello. “Critical behaviour of spanning forests on random planar graphs.” Journal of Physics A: Mathematical and Theoretical 50.7 (2017): 074003.
  3. Budd, Timothy, and Nicolas Curien. “Geometry of infinite planar maps with high degrees.” Electronic Journal of Probability 22 (2017).
  4. Richier, Loïc. “Limits of the boundary of random planar maps.” Probability Theory and Related Fields (2017): 1-39.
  5. Ambjørn, J., L. Chekhov, and Y. Makeenko. “Perturbed generalized multicritical one-matrix models.” Nuclear Physics B 928 (2018): 1-20.
  6. Richier, Loïc. “Géométrie et percolation sur des cartes à bord aléatoires.” PhD diss., ENS Lyon, 2017.
  7. Budd, Timothy, and Curien, Nicolas. “Simple peeling of planar maps with application to site percolation.” arXiv preprint arXiv:1909.10217 (2019).
  8. Bouttier, Jérémie, “Cartes planaires et partitions aléatoires”, Habilitation à diriger des recherches, Université Paris-Sud (2019).
  9. Anninos, Dionysios, and Beatrix Mühlmann. “Notes on matrix models (matrix musings).” Journal of Statistical Mechanics: Theory and Experiment 2020, no. 8 (2020): 083109.
  10. Anninos, Dionysios, and Beatrix Mühlmann. “Matrix integrals & finite holography.” arXiv preprint arXiv:2012.05224