Multiple Phase Transitions in Long Range First Passage PercolationRuixin Gui, Yue Liu, Ran Zhou, Ashish K. Pandey (Project Leader), Partha Dey (Faculty Mentor)

Illinois Geometry Lab

IGL Open House, May 7, 2015

Introduction

First Passage Percolation is a mathematical model used to de-scribe paths reachable in a random medium within a givenamount of time.

We are looking at a long range version of it (denoted by LRFPP),which appears in many cases, e.g. biological infection, formationof crystal and dispersal of fire.

In this project we try to use computer program to simulate theLRFPP model under different specific settings and observe thecorresponding growth behaviors.

Main Considerations

We consider the Long RangeFirst Passage Percolation modelon the infinite integer lattice ofdimension 1, 2 and 3.

To each edge e of the lat-tice we assign an independentrandom passage-time ωe/r(|e|),where |e| is the distance be-tween the endpoints of the edgee and ωe’s are Exponential rateone random variables.

Based on these passage times,we study the asymptotic growthof the associated t-ball (the setof vertices which can be reachedwithin time t from the origin) astime t tends to infinity.

picture of a burnt paper.

Algorithm

Statement of Problem

We took r(k) = k−α where α is a key parameter and used an dynamic algorithm to simulate the infection processin dimension d = 1, 2, 3 with: d < α < 2d , α = 2d and α > 2d , in which we saw some interesting patterns [1].

Result

Dimension Two:

Dimension Three:

Description

α = 3.5 (d = 2) and 5.5 (d = 3), because of the existence of long jump, we observed that after a small amountof time, there are many clusters, distributed in a large range. As a result, the corresponding boundaries are veryrough.

α = 4.5 (d = 2), and 8 (d = 3), because of the linear growth, we observed that there is only one cluster withrelatively small diameter. As a result, the corresponding boundaries are smooth.

Comparing α = 4.5 (d = 2), and 8 (d = 3) with α = 5.5 (d = 2) and α = 9 (d = 3), we observed that theboundaries get smoother as alpha increases.

Prediction

1-Dimension 2-Dimension 3-Dimension Growth

1 < α < 2 2 < α < 4 3 < α < 6 Stretched Exponential Growth: Diameter(Bαt ) ≈ exp(tβ), 0 < β < 1

2 < α < 3 4 < α < 5 6 < α < 7 Polynomial Growth: Diameter(Bαt ) ≈ tβ, β > 1

3 < α <∞ 5 < α <∞ 7 < α <∞ Linear Growth: Diameter(Bαt ) ≈ t

Application

This model arises in many areas.We will explain the model in con-text of spread of an epidemic. Based on theoretical prediction ofthe growth set of B(t) within time t and correlated graph pattern,we can classify a random disease growth pattern to a certain cat-egory. With the scope of alpha, we can roughly predict the areathat the disease can reach to in the future, and finally, realize ourmore realistic goal, to control the disease.

Speed of an epidemic that begins in Los Angeles [2]

Further questions

Predict the optimal path structure.

Estimate the fractal dimension or boundary fluctuation of theboundary of the growth set.

Conjecture: In dimension d ,when α > 2d + 1 the fluctuationshould behave like nd/(α−2d) when α ≤ f (d) and stays of thesame order when α > f (d).

Predict the distribution of the optimal time for small value of α.

Histogram of time required to reach radius 10 in 2-D and 3-D.

References

Shirshendu Chatterjee, Partha S. Dey. Multiple phase transitionsin long-range first-passage percolation on square lattices.

Travelling epidemics: Human mobility patterns and their impacton the spread of epidemics

These posters are made with the support of University of Illinois at Urbana-Champaign Public Engagement Office