décima Escuela de Probabilidad y Procesos Estocásticos

Cursos

Carlos Lardizabal, Universidade Federal do Rio Grande do Sul, Brasil

An introduction to quantum walks and their orthogonal polynomials

Abstract. In this mini-course we present an introduction to the study of basic statistics of quantum random walks on the line in terms of their associated orthogonal polynomials. We do this in two directions, namely by considering a) walks described by unitary operators (so that we have a closed system), and b) open quantum walks and quantum Markov chains, so that we have a dissipative system. In both cases we will compare the mathematical structure of such settings with the one seen in the classical theory of random walks, both in terms of scalar and of matrix-valued orthogonal polynomials.

References (on orthogonal polynomials)

  1. T. S. Chihara. An introduction to orthogonal polynomials (Chapters 1 and 2).

  2. M. D. de la Iglesia. Orthogonal polynomials in the spectral analysis of Markov processes. Birth-death models and diffusion. Encyclopedia of Mathematics and its Applications, Cambridge Univ. Press, 2021 (Sections 1.1-1.3, 2.1 and 2.2).

  3. H. Dette, B. Reuther, W. J. Studden and M. Zygmunt. Matrix measures and random walks with a block tridiagonal transition matrix. SIAM J. Matrix Anal. Appl. (2006) 29:117-142.

References (on quantum information theory and quantum walks)

  1. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000) (Sections 2.1 and 2.2).

  2. S. E. Venegas-Andraca. Quantum Walks for Computer Scientists. Morgan & Claypool Publish. (2008).

  3. M. J. Cantero, F. A. Grünbaum, L. Moral, L. Velázquez. Matrix-Valued Szegö Polynomials and Quantum Random Walks. Comm. Pure Appl. Math. Vol. LXIII, 0464-0507 (2010).
  4. S. Attal, F. Petruccione, C. Sabot, I. Sinayskiy. Open quantum random walks. J. Stat. Phys. 147, 832-852 (2012).

Luz Roncal, Basque Center for Applied Mathematics (BCAM), España

Birth-death processes and the semidiscrete heat equation

Abstract. For n an integer and t>0, the semidiscrete heat equation:

d/dt u(n,t)=u(n+1,t)-2u(n,t)+u(n-1,t)

with u(n,0) =\delta_{nm} for every fixed integer m (here \delta_{nm} is the Kronecker delta), may be understood in the context of the so-called birth-and-death processes. The goals of this course are the following:

  1. Derive the semidiscrete heat equation as a probabilistic model related to generalized Poisson processes.
  2. Obtain the solutions, which represent probability distributions, by analytic techniques using Laplace transforms. Such solutions will be expressed in terms of special functions, namely modified Bessel functions of first kind.
  3. Explore the connection with randomized random walks.

Time permitting, we will study some properties of the heat semigroup associated with the solution of the semidiscrete heat equation and show several directions of research in harmonic analysis and partial differential equations on the lattice.

References

  1. W. Feller, An introduction to probability theory and its applications, Vol. I and II, Wiley, New York, 1968 and 1971.
  2. F. A. Grünbaum, The bispectral problem: an overview, in Special functions 2000: current perspective and future directions, (Tempe, AZ), 129--140, NATO Sci. Ser. II Math. Phys. Chem. 30, Kluwer Acad. Publ., Dordrecht, 2001.
  3. F. A. Grünbaum and P. Iliev, Heat kernel expansions on the integers, Math. Phys. Anal. Geom. 5 (2002), 183-200.
  4. Ó. Ciaurri, A.T. Gillespie, L. Roncal, J.L. Torrea, J.L. Varona, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math. 132 (2017), 109-131.
  5. F. W. J. Olver and L. C. Maximon, Bessel Functions, NIST handbook of mathematical functions, (edited by F. W. F. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark), Chapter 10, National Institute of Standards and Technology, Washington, DC, and Cambridge University Press, Cambridge, 2010. Available online here.