Two-dimensional Quantum Random Walk

Yuliy Baryshnikov, Wil Brady, Andrew Bressler, Robin Pemantle

Research output: Contribution to journalArticlepeer-review


We analyze several families of two-dimensional quantum random walks. The feasible region (the region where probabilities do not decay exponentially with time) grows linearly with time, as is the case with one-dimensional QRW. The limiting shape of the feasible region is, however, quite different. The limit region turns out to be an algebraic set, which we characterize as the rational image of a compact algebraic variety. We also compute the probability profile within the limit region, which is essentially a negative power of the Gaussian curvature of the same algebraic variety. Our methods are based on analysis of the space-time generating function, following the methods of Pemantle and Wilson (J. Comb. Theory, Ser. A 97(1):129-161, 2002).

Original languageEnglish (US)
Pages (from-to)78-107
Number of pages30
JournalJournal of Statistical Physics
Issue number1
StatePublished - Jan 2011
Externally publishedYes


  • Amoeba
  • Fourier-Laplace
  • Gauss map
  • Rational generating function
  • Residue
  • Saddle point
  • Stationary phase

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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