# Random Walks with Invariant Loop Probabilities: Stereographic Random Walks

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## Abstract

**:**

## 1. Introduction

## 2. Heterogeneous Processes with Invariant Loop Probabilities

## 3. A Geometric View of the Problem

## 4. Probability Functions

## 5. First- and Last-Time Events

## 6. Conclusions and Future Work

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

UB | Universitat de Barcelona |

UBICS | Universitat de Barcelona Institute of Complex Systems |

RW | Random walk |

DFT | Discrete Fourier Transform |

AGAUR | Agència de Gestió d’Ajuts Universitaris i de Recerca |

AEI | Agencia Estatal de Investigación |

FEDER | Fondo Europeo de Desarrollo Regional |

UE | Unión Europea |

## Appendix A

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**Figure 1.**Projection of a hyperbolic geometry. The points at the hyperbola are placed at the positions ${x}_{n}=rsinh\left(2n\xi \right)$, ${y}_{n}=rcosh\left(2n\xi \right)$, $n\in \mathbb{Z}$. The segment in red corresponds to the stereographic projection of the hyperbola, with the points located at the horizontal positions ${d}_{n}=2rtanh\left(n\xi \right)$.

**Figure 2.**Projection of an elliptic geometry. The points in the circle are placed at regular angular distances, ${x}_{n}=rsin$($2n$ $\theta $), ${y}_{n}$ = $rcos$($2n$ $\theta $), $n\in \{-N,-N+1,\dots ,N-1,N\}$. The red line corresponds to the stereographic projection of the circle, with the points sited at locations ${d}_{n}=2rtan\left(n\theta \right)$. ($N=3$ in this figure).

**Figure 3.**The probability function ${p}_{n,t}$. We depict the probability of finding the system at position n after: (

**a**) $t=40$ steps; (

**b**) $t=80$ steps; and (

**c**) $t=120$ steps; if ${X}_{0}=0$ and $N=16$. As the values of t are even quantities, only even values of n are shown. The solid curve corresponds to Equation (31), the red dashed curve to Equation (34), and histograms were obtained from 100,000 numerical simulations of the process, with the binning (here and hereafter) chosen to include only one attainable site in each category.

**Figure 4.**The probability function ${p}_{n,t;m}$. We depict the probability of finding the system at position n after: (

**a**) $t=10$ steps; (

**b**) $t=150$ steps; and (

**c**) $t=1000$ steps; if $m=8$ and $N=16$. As the values of t are even quantities, only even values of n are shown. The solid curve corresponds to Equation (40), the red dashed curve to Equation (36), and histograms were obtained from 100,000 numerical simulations of the process.

**Figure 5.**Expected values of ${p}_{n,t;m}$. In (

**a**), we consider the evolution of ${\mu}_{t;m}$ for $m=8$. The solid black curve is the exact evolution predicted by ${p}_{n,t;m}$, Equation (45), the blue dotted curve depicts the linear behavior predicted by Equation (46), and the red dashed line coincides with the origin. In (

**b**), we show the bounded growth ${\sigma}_{t;m}^{2}$, for $m=8$. Again, the solid black curve represents the exact Formula (48), the blue dotted curve corresponds to approximate expression (50), while the red dashed line stems from Equation (49). In both cases, $N=16$, and the solid circles were obtained from 100,000 numerical simulations of the process.

**Figure 6.**Probability function ${f}_{t,n;m}$. We depict the probability of that the first visit of the process to site n starting from m occurs after t steps: (

**a**) $n=4$ and $m=0$; (

**b**) $n=8$ and $m=0$; and (

**c**) $n=0$ and $m=8$. In all cases $N=16$. As $n-m$ is even, only even values of t are shown. The solid curve corresponds to Equation (58) or (59), and histograms were obtained from 100,000 numerical simulations of the process.

**Figure 7.**The probability function ${f}_{2t,n;n}$. We depict the probability of that the first return of the process to site n occurs after $2t$ steps: (

**a**) $n=0$; (

**b**) $n=8$. In both cases, $N=16$. The solid curve corresponds to Equation (60), the blue dotted curve depicts Equation (61), and the red dashed lines correspond to Equations (65) and (66), respectively. Histograms were obtained from 100,000 numerical simulations of the process.

**Figure 8.**The probability function ${g}_{2t,n;2T,n}$. We depict the probability that the last return of the process to site n after $2T$ steps occur at time $2t$: (

**a**) $n=0$ and (

**b**) $n=8$. In both cases, $T=75$ and $N=16$. The solid curve corresponds to Equation (67) in both panels. In panel (

**a**), the blue dotted curve shows Equation (68), and the red dashed line corresponds to the approximate Equation (71). In panel (

**b**), the green dotted line follows the heuristic Equation (72). Histograms were obtained from 100,000 numerical simulations of the process.

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**MDPI and ACS Style**

Montero, M.
Random Walks with Invariant Loop Probabilities: Stereographic Random Walks. *Entropy* **2021**, *23*, 729.
https://doi.org/10.3390/e23060729

**AMA Style**

Montero M.
Random Walks with Invariant Loop Probabilities: Stereographic Random Walks. *Entropy*. 2021; 23(6):729.
https://doi.org/10.3390/e23060729

**Chicago/Turabian Style**

Montero, Miquel.
2021. "Random Walks with Invariant Loop Probabilities: Stereographic Random Walks" *Entropy* 23, no. 6: 729.
https://doi.org/10.3390/e23060729