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Large deviations for integer partitions. (English) Zbl 0972.60006

For an integer \(n\), let \(Q_n\) be the equidistribution on the set of all partitions of \(n\) and \(Q_n^s\) be the equidistribution on the set of all strict partitions of \(n\). A partition \(n_1 \geq n_2 \geq \cdots \geq n_k\) of \(n = \sum_{i=1}^k n_i\) can be identified with a sequence \((r_k)_{k=1,2, \ldots }\) where \(r_k = m\) if exactly \(m\) of the \(n_i\)’s equal \(k\). (A partition is strict if all the \(n_i\)’s are different or, equivalently, all the \(r_k\)’s have value \(0\) or \(1\).) Define the corresponding Young diagram \(\varphi(t) = \sum_{k = [t]}^\infty r_k \), \(t > 0\). Consider the rescaled version \(\tilde\varphi(t) = \varphi([\sqrt{n}t])/\sqrt{n}\). The authors derive large deviation principles of speed \(\sqrt{n}\) for the distributions of \(\tilde\varphi(t)\) under \(Q_n\) and under \(Q_n^s\) in the space of left continuous functions with right limits. They give explicit formulas for the (good) rate functions. The convergence of the distributions of \(\tilde\varphi(t)\) under \(Q_n\) and under \(Q_n^s\) is a consequence of these large deviation principles. A key step in the proof is a different representation of \(Q_n\) and \(Q_n^s\) by means of a conditioning of independent random variables. The methods are not limited to the case of random partitions but extended to so-called \`\` multiplicative statistics\'\'where similar representations are available.

MSC:

60F10 Large deviations
05A17 Combinatorial aspects of partitions of integers
53A15 Affine differential geometry
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