Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Phase Transitions for Random Walk Asymptotics on Free Products of Groups   Suppose we are given finitely generated groups $\Gamma_1,...,\Gamma_m$
equipped with irreducible random walks. Thereby we assume that the expansions
of the corresponding Green functions at their radii of convergence contain only
logarithmic or algebraic terms as singular terms up to sufficiently large order
(except for some degenerate cases). We consider transient random walks on the
free product {$\Gamma_1 \ast ... \ast\Gamma_m$} and give a complete
classification of the possible asymptotic behaviour of the corresponding
$n$-step return probabilities. They either inherit a law of the form
$\varrho^{n\delta} n^{-\lambda_i} \log^{\kappa_i}n$ from one of the free
factors $\Gamma_i$ or obey a $\varrho^{n\delta} n^{-3/2}$-law, where
$\varrho<1$ is the corresponding spectral radius and $\delta$ is the period of
the random walk. In addition, we determine the full range of the asymptotic
behaviour in the case of nearest neighbour random walks on free products of the
form $\Z^{d_1}\ast ... \ast \Z^{d_m}$. Moreover, we characterize the possible
phase transitions of the non-exponential types $n^{-\lambda_i}\log^{\kappa_i}n$
in the case $\Gamma_1\ast\Gamma_2$.

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The categories of this paper are: gr-qc
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The categories of this paper are: supr-con
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
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The categories of this paper are: adap-org
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The categories of this paper are: cs
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The categories of this paper are: stat
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The categories of this paper are: funct-an
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The categories of this paper are: econ
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The categories of this paper are: q-bio
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The categories of this paper are: eess
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The categories of this paper are: quant-ph