*Darrell Duffie*

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691138961
- eISBN:
- 9781400840519
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691138961.003.0005
- Subject:
- Economics and Finance, Financial Economics

This chapter describes a simple model of the “percolation” of information of common interest through an over-the-counter market with many agents. It also includes an explicit solution for the ...
More

This chapter describes a simple model of the “percolation” of information of common interest through an over-the-counter market with many agents. It also includes an explicit solution for the cross-sectional distribution of posterior beliefs at each time. It begins with the basic information structure for the economy and the setting for search and random matching. It then shows how to solve the model for the dynamics of the cross-sectional distribution of information. The remainder of the chapter is devoted to market settings and to extensions of the model that handle public releases of information, the receipt of new private information over time, and the release of information among groups of more than two agents at a time.Less

This chapter describes a simple model of the “percolation” of information of common interest through an over-the-counter market with many agents. It also includes an explicit solution for the cross-sectional distribution of posterior beliefs at each time. It begins with the basic information structure for the economy and the setting for search and random matching. It then shows how to solve the model for the dynamics of the cross-sectional distribution of information. The remainder of the chapter is devoted to market settings and to extensions of the model that handle public releases of information, the receipt of new private information over time, and the release of information among groups of more than two agents at a time.

*Paul Charbonneau*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691176840
- eISBN:
- 9781400885497
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691176840.003.0004
- Subject:
- Computer Science, Programming

This chapter explores a lattice-based system where complex structures can arise from pure randomness: percolation, typically described as the passage of liquid through a porous or granular medium. In ...
More

This chapter explores a lattice-based system where complex structures can arise from pure randomness: percolation, typically described as the passage of liquid through a porous or granular medium. In its more abstract form, percolation is an exemplar of criticality, a concept in statistical physics related to phase transitions. A classic example of criticality is liquid water boiling into water vapor, or freezing into ice. The chapter first provides an overview of percolation in one and two dimensions before discussing the use of a tagging algorithm for identifying and sizing clusters. It then considers fractal clusters on a lattice at the percolation threshold, scale invariance of power-law behavior, and critical behavior of natural systems. The chapter includes exercises and further computational explorations, along with a suggested list of materials for further reading.Less

This chapter explores a lattice-based system where complex structures can arise from pure randomness: percolation, typically described as the passage of liquid through a porous or granular medium. In its more abstract form, percolation is an exemplar of criticality, a concept in statistical physics related to phase transitions. A classic example of criticality is liquid water boiling into water vapor, or freezing into ice. The chapter first provides an overview of percolation in one and two dimensions before discussing the use of a tagging algorithm for identifying and sizing clusters. It then considers fractal clusters on a lattice at the percolation threshold, scale invariance of power-law behavior, and critical behavior of natural systems. The chapter includes exercises and further computational explorations, along with a suggested list of materials for further reading.

*Richard Evan Schwartz*

- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780691181387
- eISBN:
- 9780691188997
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691181387.001.0001
- Subject:
- Mathematics, Educational Mathematics

Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane ...
More

Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. This book provides a combinatorial model for orbits of outer billiards on kites. The book relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called “the plaid model,” has a self-similar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics. The book includes an extensive computer program that allows readers to explore the materials interactively and each theorem is accompanied by a computer demonstration.Less

Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. This book provides a combinatorial model for orbits of outer billiards on kites. The book relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called “the plaid model,” has a self-similar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics. The book includes an extensive computer program that allows readers to explore the materials interactively and each theorem is accompanied by a computer demonstration.

*Paul Charbonneau*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691176840
- eISBN:
- 9781400885497
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691176840.003.0006
- Subject:
- Computer Science, Programming

This chapter explores how a “natural” process generates dynamically something that is conceptually similar to a percolation cluster by using the case of forest fires. It first provides an overview of ...
More

This chapter explores how a “natural” process generates dynamically something that is conceptually similar to a percolation cluster by using the case of forest fires. It first provides an overview of the forest-fire model, which is essentially a probabilistic cellular automata, before discussing its numerical implementation using the Python code. It then describes a representative simulation showing the triggering, growth, and decay of a large fire in a representative forest-fire model simulation on a small 100 x 100 lattice. It also considers the behavior of the forest-fire model as well as its self-organized criticality and concludes with an analysis of the advantages and limitations of wildfire management. The chapter includes exercises and further computational explorations, along with a suggested list of materials for further reading.Less

This chapter explores how a “natural” process generates dynamically something that is conceptually similar to a percolation cluster by using the case of forest fires. It first provides an overview of the forest-fire model, which is essentially a probabilistic cellular automata, before discussing its numerical implementation using the Python code. It then describes a representative simulation showing the triggering, growth, and decay of a large fire in a representative forest-fire model simulation on a small 100 x 100 lattice. It also considers the behavior of the forest-fire model as well as its self-organized criticality and concludes with an analysis of the advantages and limitations of wildfire management. The chapter includes exercises and further computational explorations, along with a suggested list of materials for further reading.