Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Kinetics of Diffusion-Controlled Annihilation with Sparse Initial
  Conditions   We study diffusion-controlled single-species annihilation with sparse initial
conditions. In this random process, particles undergo Brownian motion, and when
two particles meet, both disappear. We focus on sparse initial conditions where
particles occupy a subspace of dimension $\delta$ that is embedded in a larger
space of dimension $d$. We find that the co-dimension $\Delta=d-\delta$ governs
the behavior. All particles disappear when the co-dimension is sufficiently
small, $\Delta\leq 2$; otherwise, a finite fraction of particles indefinitely
survive. We establish the asymptotic behavior of the probability $S(t)$ that a
test particle survives until time $t$. When the subspace is a line, $\delta=1$,
we find inverse logarithmic decay, $S\sim (\ln t)^{-1}$, in three dimensions,
and a modified power-law decay, $S\sim (\ln t)\,t^{-1/2}$, in two dimensions.
In general, the survival probability decays algebraically when $\Delta <2$, and
there is an inverse logarithmic decay at the critical co-dimension $\Delta=2$.

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The categories of this paper are: hep-lat
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The categories of this paper are: alg-geom
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-th
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The categories of this paper are: eess
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The categories of this paper are: q-fin
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: math-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: q-alg
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The categories of this paper are: physics