Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Parisi formula completed   The Parisi formula for the free energy of the Sherrington-Kirkpatrick model
is completed to a closed-form generating functional. We first find an integral
representation for a solution of the Parisi differential equation and represent
the free energy as a functional of order parameters. Then we set stationarity
equations for local maxima of the free energy determining the order-parameter
function on interval $[0,1]$. Finally we show without resorting to the replica
trick that the solution of the stationarity equations leads to a marginally
stable thermodynamic state.

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The categories of this paper are: econ
7
The categories of this paper are: hep-th
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The categories of this paper are: eess
35
The categories of this paper are: mtrl-th
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The categories of this paper are: stat
14
The categories of this paper are: math-ph
34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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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: acc-phys
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The categories of this paper are: funct-an
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The categories of this paper are: nlin
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The categories of this paper are: hep-ex
32
The categories of this paper are: atom-ph