Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Counting walks with large steps in an orthant   In the past fifteen years, the enumeration of lattice walks with steps
takenin a prescribed set S and confined to a given cone, especially the
firstquadrant of the plane, has been intensely studied. As a result, the
generating functions ofquadrant walks are now well-understood, provided the
allowed steps aresmall, that is $S \subset \{-1, 0,1\}^2$. In particular,
having smallsteps is crucial for the definition of a certain group of
bi-rationaltransformations of the plane. It has been proved that this group is
finite ifand only if the corresponding generating function is D-finite (that
is, it satisfies a lineardifferential equation with polynomial coefficients).
This group is also thekey to the uniform solution of 19 of the 23 small step
models possessing afinite group.In contrast, almost nothing is known for walks
with arbitrary steps. In thispaper, we extend the definition of the group, or
rather of the associatedorbit, to this general case, and generalize the above
uniform solution ofsmall step models. When this approach works, it invariably
yields a D-finitegenerating function. We apply it to many quadrant problems,
including some infinite families.After developing the general theory, we
consider the $13\ 110$ two-dimensionalmodels with steps in $\{-2,-1,0,1\}^2$
having at least one $-2$ coordinate. Weprove that only 240 of them have a
finite orbit, and solve 231 of them withour method. The 9 remaining models are
the counterparts of the 4 models of thesmall step case that resist the uniform
solution method (and which are knownto have an algebraic generating function).
We conjecture D-finiteness for their generatingfunctions, but only two of them
are likely to be algebraic. We also provenon-D-finiteness for the $12\ 870$
models with an infinite orbit, except for16 of them.

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The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: dg-ga
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The categories of this paper are: quant-ph
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The categories of this paper are: q-fin
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The categories of this paper are: physics
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The categories of this paper are: cs
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The categories of this paper are: math-ph
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The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-th