# Aspects of a Phase Transition in High-Dimensional Random Geometry

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## Abstract

**:**

## 1. Introduction

## 2. Dichotomies of Random Points

## 3. Phase Transitions in Portfolio Optimization under the Variance and the Maximal Loss Risk Measure

#### 3.1. Risk Measures

#### 3.2. Vanishing of the Estimated Variance

#### 3.3. The Maximal Loss

## 4. Related Problems

#### 4.1. Binary Classifications with a Perceptron

#### 4.2. Zero-Sum Games with Random Pay-Off Matrices

#### 4.3. Non-Negative Solutions to Large Systems of Linear Equations

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Replica Calculation of Maximal Loss

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**Figure 1.**Two colorings of three points in two dimensions. In the

**left**one, black and white points can be separated by a line through the origin; this coloring therefore represents a dichotomy. For the

**right**one, no such separating line exists.

**Figure 2.**Probability ${P}_{\mathrm{d}}(T,N)$ that T randomly colored points in a general position in N-dimensional space form a dichotomy as a function of the ratio $\alpha $ between T and N for different values of N. The transition between the limiting values $P=1$ at $\alpha =1$ and $P=0$ at large $\alpha $ becomes increasingly sharp when N grows.

**Figure 3.**

**Left**: The Maximal Loss $\mathrm{ML}={\kappa}_{c}$ as a function of $\alpha $. The analytical results (solid line) are compared to simulation results (circles) with $N=200$ averaged over 100 samples. The symbol size corresponds to the statistical error.

**Right**: Same as left with largely extended axis of ML.

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Prüser, A.; Kondor, I.; Engel, A.
Aspects of a Phase Transition in High-Dimensional Random Geometry. *Entropy* **2021**, *23*, 805.
https://doi.org/10.3390/e23070805

**AMA Style**

Prüser A, Kondor I, Engel A.
Aspects of a Phase Transition in High-Dimensional Random Geometry. *Entropy*. 2021; 23(7):805.
https://doi.org/10.3390/e23070805

**Chicago/Turabian Style**

Prüser, Axel, Imre Kondor, and Andreas Engel.
2021. "Aspects of a Phase Transition in High-Dimensional Random Geometry" *Entropy* 23, no. 7: 805.
https://doi.org/10.3390/e23070805