Ising model in planar lacunary and fractal lattices: A path counting approach

We consider the time-independent scattering theory for time evolution operators of one-dimensional two-state quantum walks. The scattering matrix associated with the position-dependent quantum walk naturally appears in the asymptotic behavior at the spatial infinity of generalized eigenfunctions. The asymptotic behavior of generalized eigenfunctions is a consequence of an explicit expression of the Green function associated with the free quantum walk. When the position-dependent quantum walk is a finite rank perturbation of the free quantum walk, we derive a kind of combinatorial construction of the scattering matrix by counting paths of quantum walkers. We also mention some remarks on the tunneling effect.

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Transient Probability Functions: A Sample Path Approach

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3349 ◽

2003 ◽

Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽

Author(s):

Michael L. Green ◽

Alan Krinik ◽

Carrie Mortensen ◽

Gerardo Rubino ◽

Randall J. Swift

Keyword(s):

Markov Processes ◽

Solution Method ◽

Sample Path ◽

Dual Processes ◽

New Approach ◽

Probability Functions ◽

Counting Approach ◽

International Audience ◽

Path Counting ◽

Transient Probability

International audience A new approach is used to determine the transient probability functions of Markov processes. This new solution method is a sample path counting approach and uses dual processes and randomization. The approach is illustrated by determining transient probability functions for a three-state Markov process. This approach also provides a way to calculate transient probability functions for Markov processes which have specific sample path characteristics.

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A Graph-Path Counting Approach for Learning Head Output Connected Relations

Computational Collective Intelligence - Lecture Notes in Computer Science ◽

10.1007/978-3-319-45246-3_37 ◽

2016 ◽

pp. 387-396 ◽

Cited By ~ 1

Author(s):

Nuran Peker ◽

Alev Mutlu

Keyword(s):

Counting Approach ◽

Path Counting

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Return probability of quantum walks with final-time dependence

Quantum Information and Computation ◽

10.26421/qic11.9-10-3 ◽

2011 ◽

Vol 11 (9&10) ◽

pp. 761-773

Author(s):

Yusuke Ide ◽

Norio Konno ◽

Takuya Machida ◽

Etsuo Segawa

Keyword(s):

Random Walk ◽

Quantum Walk ◽

Quantum Walks ◽

Time Dependent ◽

One Dimension ◽

Final Time ◽

Counting Approach ◽

Return Probability ◽

Time Quantum ◽

Path Counting

We analyze final-time dependent discrete-time quantum walks in one dimension. We compute asymptotics of the return probability of the quantum walk by a path counting approach. Moreover, we discuss a relation between the quantum walk and the corresponding final-time dependent classical random walk.

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