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The critical branching random walk in a random environment dies out


 
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1. Title Title of document The critical branching random walk in a random environment dies out
 
2. Creator Author's name, affiliation, country Olivier Garet; Université de Lorraine; France
 
2. Creator Author's name, affiliation, country Régine Marchand; Université de Lorraine; France
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) branching random walk; random environment; survival; critical behavior; renormalization; block construction
 
3. Subject Subject classification 60K35; 82B43
 
4. Description Abstract We study the possibility for branching random walks in random environment (BRWRE) to survive. The particles perform simple symmetric random walks on the $d$-dimensional integer lattice, while at each time unit, they split into independent copies according to time-space i.i.d. offspring distributions.  As noted by Comets and Yoshida, the BRWRE is naturally associated with the directed polymers in random environment (DPRE), for which the quantity $\Psi$ called the free energy is well studied.  Comets and Yoshida proved that there is no survival when $\Psi<0$ and that survival is possible when $\Psi>0$. We proved here that, except for degenerate cases, the BRWRE always die when $\Psi=0$. This solves a conjecture of Comets and Yoshida.
 
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7. Date (YYYY-MM-DD) 2013-01-31
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/2438
 
10. Identifier Digital Object Identifier 10.1214/ECP.v18-2438
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 18
 
12. Language English=en en
 
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