### Abstract

A random walk in a piecewise homogeneous medium can exhibit a variety of asymptotic behaviors. In particular, it may lodge strictly in one region or divide in probability among several. This will depend upon the parameters describing (a) the walk, (b) the interregion boundary, and (c) the initial location of the walk. We analyze from this point of view a special four-dimensional walk on an integer lattice with two homogeneous regions separated by a hyperplane of codimension 1. The walk represents a continuing sequence of clinical trials of two drugs of unknown success probabilities and the two regions represent the Bayes-derived criterion as to which drug to try next. The demarcation in the parameter space of success probabilities and initial coordinates between one- and two-region asymptotics is mapped out analytically in several special cases and supporting numerical evidence given in the general case.

Original language | English (US) |
---|---|

Pages (from-to) | 755-783 |

Number of pages | 29 |

Journal | Journal of Statistical Physics |

Volume | 30 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1983 |

### Fingerprint

### Keywords

- clinical trials
- integer lattice
- Phase transition
- piecewise homogeneous
- random walk

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*30*(3), 755-783. https://doi.org/10.1007/BF01009686

**Phase transition in a four-dimensional random walk with application to medical statistics.** / Percus, O. E.; Percus, Jerome.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 30, no. 3, pp. 755-783. https://doi.org/10.1007/BF01009686

}

TY - JOUR

T1 - Phase transition in a four-dimensional random walk with application to medical statistics

AU - Percus, O. E.

AU - Percus, Jerome

PY - 1983/3

Y1 - 1983/3

N2 - A random walk in a piecewise homogeneous medium can exhibit a variety of asymptotic behaviors. In particular, it may lodge strictly in one region or divide in probability among several. This will depend upon the parameters describing (a) the walk, (b) the interregion boundary, and (c) the initial location of the walk. We analyze from this point of view a special four-dimensional walk on an integer lattice with two homogeneous regions separated by a hyperplane of codimension 1. The walk represents a continuing sequence of clinical trials of two drugs of unknown success probabilities and the two regions represent the Bayes-derived criterion as to which drug to try next. The demarcation in the parameter space of success probabilities and initial coordinates between one- and two-region asymptotics is mapped out analytically in several special cases and supporting numerical evidence given in the general case.

AB - A random walk in a piecewise homogeneous medium can exhibit a variety of asymptotic behaviors. In particular, it may lodge strictly in one region or divide in probability among several. This will depend upon the parameters describing (a) the walk, (b) the interregion boundary, and (c) the initial location of the walk. We analyze from this point of view a special four-dimensional walk on an integer lattice with two homogeneous regions separated by a hyperplane of codimension 1. The walk represents a continuing sequence of clinical trials of two drugs of unknown success probabilities and the two regions represent the Bayes-derived criterion as to which drug to try next. The demarcation in the parameter space of success probabilities and initial coordinates between one- and two-region asymptotics is mapped out analytically in several special cases and supporting numerical evidence given in the general case.

KW - clinical trials

KW - integer lattice

KW - Phase transition

KW - piecewise homogeneous

KW - random walk

UR - http://www.scopus.com/inward/record.url?scp=34250146281&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34250146281&partnerID=8YFLogxK

U2 - 10.1007/BF01009686

DO - 10.1007/BF01009686

M3 - Article

VL - 30

SP - 755

EP - 783

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3

ER -