Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the existence of accessible paths in various models of fitness
  landscapes   We present rigorous mathematical analyses of a number of well-known
mathematical models for genetic mutations. In these models, the genome is
represented by a vertex of the $n$-dimensional binary hypercube, for some $n$,
a mutation involves the flipping of a single bit, and each vertex is assigned a
real number, called its fitness, according to some rules. Our main concern is
with the issue of existence of (selectively) accessible paths; that is,
monotonic paths in the hypercube along which fitness is always increasing. Our
main results resolve open questions about three such models, which in the
biophysics literature are known as house of cards (HoC), constrained house of
cards (CHoC) and rough Mount Fuji (RMF). We prove that the probability of there
being at least one accessible path from the all-zeroes node $\mathbf {v}^0$ to
the all-ones node $\mathbf {v}^1$ tends respectively to 0, 1 and 1, as $n$
tends to infinity. A crucial idea is the introduction of a generalization of
the CHoC model, in which the fitness of $\mathbf {v}^0$ is set to some
$\alpha=\alpha_n\in[0,1]$. We prove that there is a very sharp threshold at
$\alpha_n=\frac{\ln n}{n}$ for the existence of accessible paths from $\mathbf
{v}^0$ to $\mathbf {v}^1$. As a corollary we prove significant concentration,
for $\alpha$ below the threshold, of the number of accessible paths about the
expected value (the precise statement is technical; see Corollary 1.4). In the
case of RMF, we prove that the probability of accessible paths from $\mathbf
{v}^0$ to $\mathbf {v}^1$ existing tends to $1$ provided the drift parameter
$\theta=\theta_n$ satisfies $n\theta_n\rightarrow\infty$, and for any fitness
distribution which is continuous on its support and whose support is connected.

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The categories of this paper are: econ
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex
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The categories of this paper are: solv-int
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The categories of this paper are: funct-an
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The categories of this paper are: hep-th
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
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The categories of this paper are: eess
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The categories of this paper are: supr-con
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The categories of this paper are: adap-org