Journal of Statistical Physics

, Volume 147, Issue 1, pp 206–223 | Cite as

Survival Probability of Mutually Killing Brownian Motions and the O’Connell Process

Article
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Abstract

Recently O’Connell introduced an interacting diffusive particle system in order to study a directed polymer model in 1+1 dimensions. The infinitesimal generator of the process is a harmonic transform of the quantum Toda-lattice Hamiltonian by the Whittaker function. As a physical interpretation of this construction, we show that the O’Connell process without drift is realized as a system of mutually killing Brownian motions conditioned that all particles survive forever. When the characteristic length of interaction killing other particles goes to zero, the process is reduced to the noncolliding Brownian motion (the Dyson model).

Keywords

Mutually killing Brownian motions Survival probability Quantum Toda lattice Whittaker functions The Dyson model 
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Notes

Acknowledgements

The present author would like to thank T. Sasamoto and T. Imamura for useful discussion on the present work. A part of the present work was done during the participation of the present author in École de Physique des Houches on “Vicious Walkers and Random Matrices” (May 16–27, 2011). The author thanks G. Schehr, C. Donati-Martin, and S. Péché for well-organization of the school. This work is supported in part by the Grant-in-Aid for Scientific Research (C) (No. 21540397) of Japan Society for the Promotion of Science.

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Physics, Faculty of Science and EngineeringChuo UniversityTokyoJapan

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