### Abstract

Simple exact expressions are derived for all the Lyapunov exponents of certain N-dimensional stochastic linear dynamical systems. In the case of the product of independent random matrices, each of which has independent Gaussian entries with mean zero and variance 1/N, the exponents have an exponential distribution as N→∞. In the case of the time-ordered product integral of exp[N^{-1/2}dW], where the entries of the N×N matrix W(t) are independent standard Wiener processes, the exponents are equally spaced for fixed N and thus have a uniform distribution as N→∞.

Original language | English (US) |
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Pages (from-to) | 121-126 |

Number of pages | 6 |

Journal | Communications in Mathematical Physics |

Volume | 103 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1986 |

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### ASJC Scopus subject areas

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

### Cite this

**The distribution of Lyapunov exponents : Exact results for random matrices.** / Newman, Charles M.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 103, no. 1, pp. 121-126. https://doi.org/10.1007/BF01464284

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TY - JOUR

T1 - The distribution of Lyapunov exponents

T2 - Exact results for random matrices

AU - Newman, Charles M.

PY - 1986/3

Y1 - 1986/3

N2 - Simple exact expressions are derived for all the Lyapunov exponents of certain N-dimensional stochastic linear dynamical systems. In the case of the product of independent random matrices, each of which has independent Gaussian entries with mean zero and variance 1/N, the exponents have an exponential distribution as N→∞. In the case of the time-ordered product integral of exp[N-1/2dW], where the entries of the N×N matrix W(t) are independent standard Wiener processes, the exponents are equally spaced for fixed N and thus have a uniform distribution as N→∞.

AB - Simple exact expressions are derived for all the Lyapunov exponents of certain N-dimensional stochastic linear dynamical systems. In the case of the product of independent random matrices, each of which has independent Gaussian entries with mean zero and variance 1/N, the exponents have an exponential distribution as N→∞. In the case of the time-ordered product integral of exp[N-1/2dW], where the entries of the N×N matrix W(t) are independent standard Wiener processes, the exponents are equally spaced for fixed N and thus have a uniform distribution as N→∞.

UR - http://www.scopus.com/inward/record.url?scp=0001393892&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001393892&partnerID=8YFLogxK

U2 - 10.1007/BF01464284

DO - 10.1007/BF01464284

M3 - Article

AN - SCOPUS:0001393892

VL - 103

SP - 121

EP - 126

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -