scholarly journals Return probability and self-similarity of the Riesz walk

2021 ◽  
Vol 21 (5&6) ◽  
pp. 405-422
Author(s):  
Ryota Hanaoka ◽  
Norio Konno
Keyword(s):  
Random Walk ◽  
Unit Circle ◽  
Complex Plane ◽  
Quantum Walk ◽  
The Self ◽  
One Dimension ◽  
Continuous Measure ◽  
Self Similarity ◽  
Return Probability ◽  
Continuous Measures

The quantum walk is a counterpart of the random walk. The 2-state quantum walk in one dimension can be determined by a measure on the unit circle in the complex plane. As for the singular continuous measure, results on the corresponding quantum walk are limited. In this situation, we focus on a quantum walk, called the Riesz walk, given by the Riesz measure which is one of the famous singular continuous measures. The present paper is devoted to the return probability of the Riesz walk. Furthermore, we present some conjectures on the self-similarity of the walk.

Return probability of quantum walks with final-time dependence

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.

scholarly journals Quantum walks and elliptic integrals

2010 ◽  
Vol 20 (6) ◽  
pp. 1091-1098 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Random Walk ◽  
Generating Function ◽  
Elliptic Integral ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Elliptic Integrals ◽  
Two Dimensional ◽  
One Dimensional ◽  
Similar Expression ◽  
Return Probability

Pólya showed in his 1921 paper that the generating function of the return probability for a two-dimensional random walk can be written in terms of an elliptic integral. In this paper we present a similar expression for a one-dimensional quantum walk.

scholarly journals Limit theorems for quantum walks with memory

2010 ◽  
Vol 10 (11&12) ◽  
pp. 1004-1017
Author(s):  
Norio Konno ◽  
Takuya Machida
Keyword(s):  
Limit Theorems ◽  
Quantum Walk ◽  
Weak Limit ◽  
Quantum Walks ◽  
One Dimension ◽  
Counting Method ◽  
Return Probability ◽  
Weak Limit Theorem ◽  
Path Counting ◽  
With Memory

Recently Mc Gettrick introduced and studied a discrete-time 2-state quantum walk (QW) with a memory in one dimension. He gave an expression for the amplitude of the QW by path counting method. Moreover he showed that the return probability of the walk is more than 1/2 for any even time. In this paper, we compute the stationary distribution by considering the walk as a 4-state QW without memory. Our result is consistent with his claim. In addition, we obtain the weak limit theorem of the rescaled QW. This behavior is strikingly different from the corresponding classical random walk and the usual 2-state QW without memory as his numerical simulations suggested.

scholarly journals Quantum walk on a comb with infinite teeth

2022 ◽  
Author(s):  
François David ◽  
Thordur Jonsson
Keyword(s):  
Random Walk ◽  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Random Walk ◽  
Return Probability ◽  
Starting Point ◽  
Time Quantum

Abstract We study continuous time quantum random walk on a comb with infinite teeth and show that the return probability to the starting point decays with time t as t−1. We analyse the diffusion along the spine and into the teeth and show that the walk can escape into the teeth with a finite probability and goes to infinity along the spine with a finite probability. The walk along the spine and into the teeth behaves qualitatively as a quantum random walk on a line. This behaviour is quite different from that of classical random walk on the comb.

scholarly journals CAN AGN FEEDBACK BREAK THE SELF-SIMILARITY OF GALAXIES, GROUPS, AND CLUSTERS?

2014 ◽  
Vol 783 (1) ◽  
pp. L10 ◽  
Author(s):  
M. Gaspari ◽  
F. Brighenti ◽  
P. Temi ◽  
S. Ettori
Keyword(s):  
The Self ◽  
Self Similarity

scholarly journals On the self-similarity of diffracting gaseous detonations and the critical channel width problem

2021 ◽  
Vol 33 (6) ◽  
pp. 066106
Author(s):  
M. I. Radulescu ◽  
R. Mével ◽  
Q. Xiao ◽  
S. Gallier
Keyword(s):  
The Self ◽  
Channel Width ◽  
Self Similarity ◽  
Gaseous Detonations

scholarly journals A Discontinuity of the Energy of Quantum Walk in Impurities

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1134
Author(s):  
Kenta Higuchi ◽  
Takashi Komatsu ◽  
Norio Konno ◽  
Hisashi Morioka ◽  
Etsuo Segawa
Keyword(s):  
Unit Circle ◽  
Stationary State ◽  
Discrete Time ◽  
Quantum Walk ◽  
Time Limit ◽  
The Other ◽  
Unitary Matrix ◽  
Initial State ◽  
Time Quantum ◽  
Long Time

We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region {0,1,⋯,M−1} and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow ωn at time n(|ω|=1). From this expression, we compute the scattering on the surface of −1 and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.

scholarly journals A Community-Structure-Based Method for Estimating the Fractal Dimension, and its Application to Water Networks for the Assessment of Vulnerability to Disasters

2021 ◽  
Vol 35 (4) ◽  
pp. 1197-1210
Author(s):  
C. Giudicianni ◽  
A. Di Nardo ◽  
R. Greco ◽  
A. Scala
Keyword(s):  
Fractal Dimension ◽  
Community Structure ◽  
Distribution Systems ◽  
Water Distribution ◽  
Scaling Law ◽  
Vulnerability Index ◽  
The Self ◽  
Node Degree ◽  
Specific Response ◽  
Self Similarity

AbstractMost real-world networks, from the World-Wide-Web to biological systems, are known to have common structural properties. A remarkable point is fractality, which suggests the self-similarity across scales of the network structure of these complex systems. Managing the computational complexity for detecting the self-similarity of big-sized systems represents a crucial problem. In this paper, a novel algorithm for revealing the fractality, that exploits the community structure principle, is proposed and then applied to several water distribution systems (WDSs) of different size, unveiling a self-similar feature of their layouts. A scaling-law relationship, linking the number of clusters necessary for covering the network and their average size is defined, the exponent of which represents the fractal dimension. The self-similarity is then investigated as a proxy of recurrent and specific response to multiple random pipe failures – like during natural disasters – pointing out a specific global vulnerability for each WDS. A novel vulnerability index, called Cut-Vulnerability is introduced as the ratio between the fractal dimension and the average node degree, and its relationships with the number of randomly removed pipes necessary to disconnect the network and with some topological metrics are investigated. The analysis shows the effectiveness of the novel index in describing the global vulnerability of WDSs.

Sign in / Sign up

Export Citation Format

Share Document