Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Localization for Branching Random Walks in Random Environment   We consider branching random walks in $d$-dimensional integer lattice with
time-space i.i.d. offspring distributions. This model is known to exhibit a
phase transition: If $d \ge 3$ and the environment is "not too random", then,
the total population grows as fast as its expectation with strictly positive
probability. If,on the other hand, $d \le 2$, or the environment is ``random
enough", then the total population grows strictly slower than its expectation
almost surely. We show the equivalence between the slow population growth and a
natural localization property in terms of "replica overlap". We also prove a
certain stronger localization property, whenever the total population grows
strictly slower than its expectation almost surely.

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The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
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The categories of this paper are: atom-ph
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: dg-ga
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The categories of this paper are: solv-int
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The categories of this paper are: cmp-lg
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The categories of this paper are: supr-con
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The categories of this paper are: hep-th
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The categories of this paper are: acc-phys
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The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
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The categories of this paper are: nlin