##
**Periodic Schur process and cylindric partitions.**
*(English)*
Zbl 1131.22003

This substantial and technical paper is about the so-called periodic Schur process. This process depends on a natural number \(N\), a parameter \(t\) with \(| t| <1\), and \(2N\) specializations of the algebra \(\Lambda\) of symmetric functions \(a[1],a[2],\dots, a[N]\) and \(b[1],b[2],\dots, b[N]\) (where a specialization of \(\Lambda\) is an algebra homomorphism from \(\Lambda\) to \(\mathbb C\). The process lives on periodic sequences of \(N\) partitions, whose period divides \(N\), namely \(\lambda^{(N)}=\lambda^{(0)}\), \(\lambda^{(1)}\), \(\lambda^{(2)},\dots, \lambda^{(N)}\) etc., and assigns a weight proportional to

\[ \sum_{\mu^{(1)},\mu^{(2)},\dots, \mu^{(N)}} t^{| \lambda^{(0)}|} s_{\lambda^{(0)}} \mu^{(1)}(a[1])s_{\lambda^{(1)}}/ \mu^{(1)}(b[1])\dots s_{\lambda^{(N-1)}}/ \mu^{(N)}(a[N]) s_{\lambda^{(N)}}/ \mu^{(N)}(b[N]) \]

to it. Here the \(s_{\lambda/ \mu}\) are the so-called skew Schur functions

(see, e.g., http://www.facstaff.bucknell.edu/pm040/FPSAC07/slides.pdf for some background on skew Schur functions). We emphasise that these weights are complex numbers or formal series in \(\Lambda^{\otimes 2N}\): in particular, though we shall use the language of probability in what follows, some of the probabilities may be negative or just formal series of symmetric functions. Note that the constant of proportionality is obtained explicitly in the article under review.

It is in fact often convenient to look at the so-called shift-mixed periodic Schur process which is the product measure of the periodic Schur process with a measure on \({\mathbb Z}\) which gives to \(S\subseteq {\mathbb Z}\) the weight proportional to \(z^{S}t^{S^{2}/2}\) for some \(z\in {\mathbb C}\), mapped to the space of point configurations in \(\{1,2,\dots, N\}\times ({\mathbb Z}+1/2)\) via the map

\[ (\lambda,S)\rightarrow \{S+\lambda_{i}^{(1)}-i+\tfrac{1}{2}\}_{i\geq 1} \sqcup \dots \sqcup \{S+\lambda_{i}^{(N)}-i+\tfrac{1}{2}\}_{i\geq 1}. \]

In other words, all points of the random point configuration of the periodic Schur process are shifted by an independent integral-valued random variable distributed as \(S\).

Correlation functions (for the periodic Schur process) are defined to be probabilities (with the above caveat!) that the random point configuration contains a fixed finite set of points: \[ \rho_{n}(\tau_{1}, x_{1},\tau_{2},x_{2},\dots, \tau_{n},x_{n})=P\{x_{j}\in \{\lambda_{i}^{(\tau_{j})}-i+\tfrac{1}{2}\}_{i\geq 1}\mid j=1,\dots, n\}. \]

They are defined similarly for the shift-mixed periodic Schur process.

The first result of the paper under review is that the correlation functions of the shift-mixed periodic Schur process are determinants of suitable matrices depending on a so-called correlation kernel \(K\) which depends on the quantities \(a_{m}[k]:=(1/m)p_{m}(a[k])\) and \(b_{m}[k]:=(1/m)p_{m}(b[k])\) where the \(p_{m}\)s are Newton power sums. This statement can be viewed both as a statement about formal identity of series in \(\Lambda^{\otimes 2N}\) or (under suitable convergence conditions) about numeric identities. The author notes that his proof was inspired by formalisms in the so-called Fock space, a point he hopes to explore further later.

The second main result concerns the ‘bulk-scaling’ limit of the correlation functions of the periodic Schur process, and its shift-mixed version, as \(t\rightarrow 1\) with \(N\) and the specializations \(a[k]\) and \(b[k]\) stay fixed. Again the limit is a determinant, this time defined in terms of a contour integral, under some technical assumptions (see the original paper). Various corollaries of the result are noted.

The author then considers the implications of his results for cylindric partitions – that is, ways of filling the boxes of the square grid in the quarter-plane with nonnegative integers so that the numbers do not increase as we move to infinity in the directions of the axes, and the total number of non-zero entries is also finite. (An alternative perspective on the results in this paper is that they initiate the study of random cylindric partitions.) We do not even attempt to describe these results in detail, referring to Theorem C in the paper, and the discussion before it.

\[ \sum_{\mu^{(1)},\mu^{(2)},\dots, \mu^{(N)}} t^{| \lambda^{(0)}|} s_{\lambda^{(0)}} \mu^{(1)}(a[1])s_{\lambda^{(1)}}/ \mu^{(1)}(b[1])\dots s_{\lambda^{(N-1)}}/ \mu^{(N)}(a[N]) s_{\lambda^{(N)}}/ \mu^{(N)}(b[N]) \]

to it. Here the \(s_{\lambda/ \mu}\) are the so-called skew Schur functions

(see, e.g., http://www.facstaff.bucknell.edu/pm040/FPSAC07/slides.pdf for some background on skew Schur functions). We emphasise that these weights are complex numbers or formal series in \(\Lambda^{\otimes 2N}\): in particular, though we shall use the language of probability in what follows, some of the probabilities may be negative or just formal series of symmetric functions. Note that the constant of proportionality is obtained explicitly in the article under review.

It is in fact often convenient to look at the so-called shift-mixed periodic Schur process which is the product measure of the periodic Schur process with a measure on \({\mathbb Z}\) which gives to \(S\subseteq {\mathbb Z}\) the weight proportional to \(z^{S}t^{S^{2}/2}\) for some \(z\in {\mathbb C}\), mapped to the space of point configurations in \(\{1,2,\dots, N\}\times ({\mathbb Z}+1/2)\) via the map

\[ (\lambda,S)\rightarrow \{S+\lambda_{i}^{(1)}-i+\tfrac{1}{2}\}_{i\geq 1} \sqcup \dots \sqcup \{S+\lambda_{i}^{(N)}-i+\tfrac{1}{2}\}_{i\geq 1}. \]

In other words, all points of the random point configuration of the periodic Schur process are shifted by an independent integral-valued random variable distributed as \(S\).

Correlation functions (for the periodic Schur process) are defined to be probabilities (with the above caveat!) that the random point configuration contains a fixed finite set of points: \[ \rho_{n}(\tau_{1}, x_{1},\tau_{2},x_{2},\dots, \tau_{n},x_{n})=P\{x_{j}\in \{\lambda_{i}^{(\tau_{j})}-i+\tfrac{1}{2}\}_{i\geq 1}\mid j=1,\dots, n\}. \]

They are defined similarly for the shift-mixed periodic Schur process.

The first result of the paper under review is that the correlation functions of the shift-mixed periodic Schur process are determinants of suitable matrices depending on a so-called correlation kernel \(K\) which depends on the quantities \(a_{m}[k]:=(1/m)p_{m}(a[k])\) and \(b_{m}[k]:=(1/m)p_{m}(b[k])\) where the \(p_{m}\)s are Newton power sums. This statement can be viewed both as a statement about formal identity of series in \(\Lambda^{\otimes 2N}\) or (under suitable convergence conditions) about numeric identities. The author notes that his proof was inspired by formalisms in the so-called Fock space, a point he hopes to explore further later.

The second main result concerns the ‘bulk-scaling’ limit of the correlation functions of the periodic Schur process, and its shift-mixed version, as \(t\rightarrow 1\) with \(N\) and the specializations \(a[k]\) and \(b[k]\) stay fixed. Again the limit is a determinant, this time defined in terms of a contour integral, under some technical assumptions (see the original paper). Various corollaries of the result are noted.

The author then considers the implications of his results for cylindric partitions – that is, ways of filling the boxes of the square grid in the quarter-plane with nonnegative integers so that the numbers do not increase as we move to infinity in the directions of the axes, and the total number of non-zero entries is also finite. (An alternative perspective on the results in this paper is that they initiate the study of random cylindric partitions.) We do not even attempt to describe these results in detail, referring to Theorem C in the paper, and the discussion before it.

Reviewer: David B. Penman (Colchester)

### MSC:

22D15 | Group algebras of locally compact groups |

22E30 | Analysis on real and complex Lie groups |

43A15 | \(L^p\)-spaces and other function spaces on groups, semigroups, etc. |

46L05 | General theory of \(C^*\)-algebras |

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