Out-of-equilibrium random walks

2020 ◽  
Vol 52 (3) ◽  
pp. 772-797
Author(s):  
Leonardo A. Videla
Keyword(s):  
Stochastic Process ◽  
Random Walks ◽  
Complete Graphs ◽  
Knowledge Process ◽  
Random Walker ◽  
Underlying Graph ◽  
Graph Sequence

AbstractWe study the long-term behaviour of a random walker embedded in a growing sequence of graphs. We define a (generally non-Markovian) real-valued stochastic process, called the knowledge process, that represents the ratio between the number of vertices already visited by the walker and the current size of the graph. We mainly focus on the case where the underlying graph sequence is the growing sequence of complete graphs.

scholarly journals Image segmentation through the scale-space random walker

2021 ◽  
Author(s):  
Richard Rzeszutek
Keyword(s):  
Random Walks ◽  
Graphics Processing Unit ◽  
Scale Space ◽  
Gaussian Kernel ◽  
Processing Unit ◽  
Random Walker ◽  
User Input ◽  
Segmentation Boundary ◽  
Vector Matrix ◽  
Graphics Processing

This thesis proposes an extension to the Random Walks assisted segmentation algorithm that allows it to operate on a scale-space. Scale-space is a multi-resolution signal analysis method that retains all of the structures in an image through progressive blurring with a Gaussian kernel. The input of the algorithm is setup so that Random Walks will operate on the scale-space, rather than the image itself. The result is that the finer scales retain the detail in the image and the coarser scales filter out the noise. This augmented algorithm is referred to as "Scale-Space Random Walks" (SSRW) and it is shown in both artifical and natural images to be superior to Random Walks when an image has been corrupted by noise. It is also shown that SSRW can impove the segmentation when texture, such as the artifical edges created by JPEG compression, has made the segmentation boundary less accurate. This thesis also presents a practical application of the SSRW in an assisted rotoscoping tool. The tool is implemented as a plugin for a popular commerical compositing application that leverages the power of a Graphics Processing Unit (GPU) to improve the algorithm's performance so that it is near-realtime. Issues such as memory handling, user input and performing vector-matrix algebra are addressed.

scholarly journals Convergence theorems for graph sequences

2014 ◽  
Vol 24 (08) ◽  
pp. 1233-1251 ◽  
Author(s):  
Felix Pogorzelski
Keyword(s):  
Banach Space ◽  
Integrated Density Of States ◽  
Finite Range ◽  
Amenable Groups ◽  
Finite Graphs ◽  
Graph Sequence ◽  
Discrete Structures ◽  
Long Term Behavior ◽  
Uniformly Bounded

In this paper, we deal with a notion of Banach space-valued mappings defined on a set consisting of finite graphs with uniformly bounded vertex degree. These functions will be endowed with certain boundedness and additivity criteria. We examine their normalized long-term behavior along a particular class of graph sequences. Using techniques developed by Elek, we show convergence in the topology of the Banach space if the corresponding graph sequence possesses a hyperfinite structure. These considerations extend and complement the corresponding results for amenable groups. As an application, we verify the uniform approximation of the integrated density of states for bounded, finite range operators on discrete structures. Further, we extend results concerning an abstract version of Fekete's lemma for cancellative, amenable groups and semigroups to the geometric situation of convergent graph sequences.

scholarly journals FLUCTUATIONS OF QUANTUM RANDOM WALKS ON CIRCLES

2005 ◽  
Vol 03 (03) ◽  
pp. 535-549 ◽  
Author(s):  
NORIO INUI ◽  
YOSHINAO KONISHI ◽  
NORIO KONNO ◽  
TAKAHIRO SOSHI
Keyword(s):  
Standard Deviation ◽  
Random Walks ◽  
Initial Conditions ◽  
Classical Case ◽  
System Size ◽  
Temporal Fluctuation ◽  
Quantum Random Walks ◽  
Random Walker ◽  
Large System Size ◽  
Temporal Standard Deviation

Temporal fluctuations in the Hadamard walk on circles are studied. A temporal standard deviation of probability that a quantum random walker is positive at a given site is introduced to manifest striking differences between quantum and classical random walks. An analytical expression of the temporal standard deviation on a circle with odd sites is shown and its asymptotic behavior is considered for large system size. In contrast with classical random walks, the temporal fluctuation of quantum random walks depends on the position and initial conditions, since temporal standard deviation of the classical case is zero for any site. It indicates that the temporal fluctuation of the Hadamard walk can be controlled.

scholarly journals Image segmentation through the scale-space random walker

2021 ◽  
Author(s):  
Richard Rzeszutek
Keyword(s):  
Random Walks ◽  
Graphics Processing Unit ◽  
Scale Space ◽  
Gaussian Kernel ◽  
Processing Unit ◽  
Random Walker ◽  
User Input ◽  
Segmentation Boundary ◽  
Vector Matrix ◽  
Graphics Processing

This thesis proposes an extension to the Random Walks assisted segmentation algorithm that allows it to operate on a scale-space. Scale-space is a multi-resolution signal analysis method that retains all of the structures in an image through progressive blurring with a Gaussian kernel. The input of the algorithm is setup so that Random Walks will operate on the scale-space, rather than the image itself. The result is that the finer scales retain the detail in the image and the coarser scales filter out the noise. This augmented algorithm is referred to as "Scale-Space Random Walks" (SSRW) and it is shown in both artifical and natural images to be superior to Random Walks when an image has been corrupted by noise. It is also shown that SSRW can impove the segmentation when texture, such as the artifical edges created by JPEG compression, has made the segmentation boundary less accurate. This thesis also presents a practical application of the SSRW in an assisted rotoscoping tool. The tool is implemented as a plugin for a popular commerical compositing application that leverages the power of a Graphics Processing Unit (GPU) to improve the algorithm's performance so that it is near-realtime. Issues such as memory handling, user input and performing vector-matrix algebra are addressed.

A Formal Specification of the Memorization Process

2011 ◽  
pp. 157-170
Author(s):  
Natalia López ◽  
Manuel Núñez ◽  
Fernando L. Pelayo
Keyword(s):  
Stochastic Process ◽  
Probability Distribution ◽  
Formal Specification ◽  
Cognitive Model ◽  
Distribution Functions ◽  
Cognitive Systems ◽  
Buffer Memory ◽  
Stochastic Time ◽  
Stochastic Process Algebra

In this chapter we present the formal language, stochastic process algebra (STOPA), to specify cognitive systems. In addition to the usual characteristics of these formalisms, this language features the possibility of including stochastic time. This kind of time is useful to represent systems where the delays are not controlled by fixed amounts of time, but are given by probability distribution functions. In order to illustrate the usefulness of our formalism, we will formally represent a cognitive model of the memory. Following contemporary theories of memory classification (see Squire et al., 1993; Solso, 1999) we consider sensory buffer, short-term, and long-term memories. Moreover, borrowing from Y. Wang and Y. Wang (2006), we also consider the so-called action buffer memory.

RANDOM WALKS, SELF-AVOIDING WALKS AND 1/f NOISE ON FINITE LATTICES

2004 ◽  
Vol 04 (03) ◽  
pp. L413-L424 ◽  
Author(s):  
FERDINAND GRÜNEIS
Keyword(s):  
Random Walks ◽  
Power Law ◽  
Lattice Structure ◽  
Finite Lattice ◽  
The Self ◽  
Random Walker ◽  
Finite Lattices ◽  
Self Avoiding Walks

We investigate the probabilities for a return to the origin at step n of a random walker on a finite lattice. As a consistent measure only the first returns to the origin appear to be of relevance; these include paths with self-intersections and self-avoiding polygons. Their return probabilities are power-law distributed giving rise to 1/f b noise. Most striking is the behavior of the self-avoiding polygons exhibiting a slope b=0.83 for d=2 and b=0.93 for d=3 independent on lattice structure.

scholarly journals A limit law of the return probability for a quantum walk on a hexagonal lattice

2015 ◽  
Vol 13 (07) ◽  
pp. 1550054 ◽  
Author(s):  
Takuya Machida
Keyword(s):  
Random Walks ◽  
Discrete Time ◽  
Hexagonal Lattice ◽  
Quantum Walk ◽  
Quantum Walks ◽  
The Other ◽  
View Point ◽  
Initial State ◽  
Random Walker ◽  
Return Probability

A return probability of random walks is one of the interesting subjects. As it is well known, the return probability strongly depends on the structure of the space where the random walker moves. On the other hand, the return probability of quantum walks, which are quantum models corresponding to random walks, has also been investigated to some extend lately. In this paper, we take care of a discrete-time three-state quantum walk on a hexagonal lattice from the view point of mathematics. The mathematical result shows a limit of the return probability when the walker starts off at the origin. The result of the limit tells us about a possibility of localization at the position and a dependence of localization on the initial state.

scholarly journals Long Term Behaviour of a Reversible System of Interacting Random Walks

2019 ◽  
Vol 175 (1) ◽  
pp. 71-96 ◽  
Author(s):  
Svante Janson ◽  
Vadim Shcherbakov ◽  
Stanislav Volkov
Keyword(s):  
Random Walks ◽  
Reversible System ◽  
Interacting Random Walks

Progressive Screening: Long-Term Contracting with a Privately Known Stochastic Process

2012 ◽  
Vol 80 (1) ◽  
pp. 1-34 ◽  
Author(s):  
R. Boleslavsky ◽  
M. Said
Keyword(s):  
Stochastic Process
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