### Abstract

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 language | English (US) |
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Pages (from-to) | 78-107 |

Number of pages | 30 |

Journal | Journal of Statistical Physics |

Volume | 142 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2011 |

### Keywords

- 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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## Cite this

*Journal of Statistical Physics*,

*142*(1), 78-107. https://doi.org/10.1007/s10955-010-0098-2