## Pavel Bleher

Chancellor's Professor, Mathematical Sciences

Chancellor's Professor, Mathematical Sciences

Prof. Bleher's principal research interests focus in four directions: statistical physics, quantum integrable systems, theory of random polynomials, and random matrix models. Bleher is a world expert in the rigorous theory of phase transitions in systems with random interactions and in the relations between the quantum spectra of integrable quantum models and the properties of the underlying systems of classical mechanics. His long-term collaborators in this area include Jean Ruiz and Valentin Zagrebnov (Center for Theoretical Physics, Luminy-Marseille, France), Freeman Dyson and Jean Bourgain (Institute for Advanced Study, Princeton), Joel Lebowitz (Rutgers University), and Yakov Sinai and Denis Kosygin (Princeton University).

The theories of random polynomials and random matrix models represent Bleher's most recent areas of research. Together with Bernard Shiffman and Steven Zelditch (of Johns Hopkins University), he solved the problem of universality and scaling for systems of random multivariate polynomials or, more generally, random sections of powers of line bundles over a compact Kahler manifold. This problem has significant applications to the theory of quantum chaos.

On random matrices, Pavel Bleher has been collaborating with Alexander Its, the other member of the Mathematical Physics Group, since both of them came to the Department in 1993-1994. In 1995, Bleher and Its launched a joint project on the Riemann-Hilbert approach to the study of the universalities in the theory of random matrix models and orthogonal polynomials. This approach allowed them to solve several long-standing problems concerning the asymptotic analysis of the orthogonal polynomials with exponential weights (e.g., the so-called Nevai problem). Simultaneously, these works of Bleher and Its have been followed by a flurry of activity in the area with an increasing involvement of new researchers with very different backgrounds and experiences, some of whom have already become Bleher's new collaborators.

- 1970 B. S. (Mathematics), Department of Mathematics of the Moscow State University, Moscow, USSR
- 1971 M. S. (Mathematics), The Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences, Moscow, USSR
- 1974 Ph. D. (Mathematical Physics), The Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences, Moscow, USSR
- Dissertation: `` Investigation of the second order phase transitions in some models of ferromagnetism’’ (supervisor: Ya. G. Sinai)

- 1984 Habilitation (Mathematics), Vilnius University, Vilnius, Lithuania
- Dissertation ``Limit theorems with asymptotics of large deviations for strongly dependent random variables’’

- Pavel Bleher, Mikhail Lyubich, and Roland Roeder. Lee-Yang-Fisher zeros for DHL and 2D rational dynamics, II. Global Pluripotential Interpretation.
- Pavel Bleher Critical phenomena in the Dyson hierarchical model and renormalization group.
- Pavel Bleher, Mikhail Lyubich, and Roland Roeder. Lee-Yang Zeros for the DHL and 2D Rational Dynamics, I. Foliation of the Physical Cylinder.

- Bleher, Pavel; Kuijlaars, Arno B.J. Orthogonal polynomials in the normal matrix model with a cubic potential . Adv. Math. 230 (2012), 1272–1321.
- Bleher, Pavel; Deano, Alfredo. Topological Expansion in the Cubic Random Matrix Model. Int. Math. Res. Not. IMRN (2012), doi:10.1093/imrn/rns126, 57 pages.
- Pavel M. Bleher, Youkow Homma, Lyndon L. Ji, Roland K. W. Roeder, and Jeffrey Shen Nearest Neighbor Distances on a Circle: Multidimensional Case. Journal of Statistical Physics, 146(2): 446-465, 2012.
- Bleher, Pavel; Liechty, Karl. Uniform asymptotics for discrete orthogonal polynomials with respect to varying exponential weights on a regular infinite lattice. Int. Math. Res. Not. IMRN (2011), no. 2, 342–386.
- Bleher, P.; Delvaux, S.; Kuijlaars, A. B. J. Random matrix model with external source and a constrained vector equilibrium problem. Comm. Pure Appl. Math. 64 (2011), no. 1, 116–160.
- Bleher, Pavel; Liechty, Karl. Exact solution of the six-vertex model with domain wall boundary conditions: antiferroelectric phase. Comm. Pure Appl. Math. 63 (2010), no. 6, 779–829.
- Bleher, Pavel; Liechty, Karl. Exact solution of the six-vertex model with domain wall boundary conditions. Critical line between ferroelectric and disordered phases. J. Stat. Phys. 134 (2009), no. 3, 463–485.
- Bleher, Pavel; Liechty, Karl. Exact solution of the six-vertex model with domain wall boundary conditions. Ferroelectric phase. Comm. Math. Phys. 286 (2009), no. 2, 777–801.
- Bleher, Pavel M.; Kuijlaars, Arno B. J. Large n limit of Gaussian random matrices with external source III. Double scaling limit. Comm. Math. Phys. 270 (2007), no. 2, 481–517.
- Bleher, Pavel; Mallison, Robert, Jr. Zeros of sections of exponential sums. Int. Math. Res. Not. (2006), Art. ID 38937, 49 pp.
- Bleher, Pavel M.; Fokin, Vladimir V. Exact solution of the six-vertex model with domain wall boundary conditions. Disordered phase. Comm. Math. Phys. 268 (2006), no. 1, 223–284.
- Bleher, Pavel M.; Kuijlaars, Arno B. J. Integral representations for multiple Hermite and multiple Laguerre polynomials. Ann. Inst. Fourier (Grenoble) 55 (2005), no. 6, 2001–2014.
- Bleher, Pavel M.; Its, Alexander R. Asymptotics of the partition function of a random matrix model. Ann. Inst. Fourier (Grenoble) 55 (2005), no. 6, 1943–2000.
- Aptekarev, Alexander I.; Bleher, Pavel M.; Kuijlaars, Arno B. J. Large n limit of Gaussian random matrices with external source II. Comm. Math. Phys. 259 (2005), no. 2, 367–389.
- Bleher, Pavel; Kuijlaars, Arno B. J. Large n limit of Gaussian random matrices with external source I. Comm. Math. Phys. 252 (2004), no. 1-3, 43–76.
- Bleher, Pavel; Di, Xiaojun. Correlations between zeros of non-Gaussian random polynomials. Int. Math. Res. Not. (2004), no. 46, 2443–2484.
- Bleher, P. M.; Kuijlaars, A. B. J. Random matrices with external source and multiple orthogonal polynomials. Int. Math. Res. Not. (2004), no. 3, 109–129.
- Bleher, Pavel; Eynard, Bertrand. Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations. Random matrix theory. J. Phys. A 36 (2003), no. 12, 3085–3105.
- Bleher, Pavel; Its, Alexander. Double scaling limit in the random matrix model: the Riemann-Hilbert approach. Comm. Pure Appl. Math. 56 (2003), no. 4, 433–516.
- Bleher, Pavel; Ridzal, Denis. SU(1,1) random polynomials. J. Statist. Phys. 106 (2002), no. 1-2, 147–171.
- Bleher, P.; Ruiz, J.; Schonmann, R. H.; Shlosman, S.; Zagrebnov, V. Rigidity of the critical phases on a Cayley tree. Mosc. Math. J. 1 (2001), no. 3, 345–363, 470.
- Bleher, Pavel; Shiffman, Bernard; Zelditch, Steve. Correlations between zeros and supersymmetry. Dedicated to Joel L. Lebowitz. Comm. Math. Phys. 224 (2001), no. 1, 255–269.
- Bleher, Pavel; Shiffman, Bernard; Zelditch, Steve. Universality and scaling of zeros on symplectic manifolds. Random matrix models and ths, 31–69, Math. Sci. Res. Inst. Publ. , 40, Cambridge Univ. Press, Cambridge, 2001.
- Random matrix models and their applications. Edited by Pavel Bleher and Alexander Its. Mathematical Sciences Research Institute Publications, 40. Cambridge University Press, Cambridge, 2001. x+438 pp. ISBN: 0-521-80209-1
- Bleher, Pavel; Shiffman, Bernard; Zelditch, Steve. Universality and scaling of correlations between zeros on complex manifolds. Invent. Math. 142 (2000), no. 2, 351–395.
- Bleher, Pavel; Shiffman, Bernard; Zelditch, Steve. Poincaré-Lelong approach to universality and scaling of correlations between zeros. Comm. Math. Phys. 208 (2000), no. 3, 771–785.

- The winner of the 8th International Olympiad, Berlin, 1965