Entry Distributions in Random Contingency Tables

Definitions and mathematical statement

For positive integers $m,n$, let $R=(r_1,\ldots,r_m)\in{\mathbb{Z}}_{\geq0}^{m}$ and $C=(c_1,\ldots,c_n)\in{\mathbb{Z}}_{\geq0}^{n}$ be prescribed margins with equal total sum. Define \[ \mathcal T(R,C)=\left\{(d_{ij})\in{\mathbb{Z}}_{\geq0}^{m\times n}:\ \sum_{j=1}^{n}d_{ij}=r_i\ (1\leq i\leq m),\quad \sum_{i=1}^{m}d_{ij}=c_j\ (1\leq j\leq n)\right\}. \] Let $D$ be uniform on this finite, nonempty set; thus $\Pr(D=d)=|\mathcal T(R,C)|^{-1}$ for every $d\in\mathcal T(R,C)$. A geometric distribution on ${\mathbb{Z}}_{\geq0}$ with parameter $p\in(0,1)$ has probabilities $\Pr(G=k)=p(1-p)^k$, $k\geq0$; the boundary case $p=1$ denotes the point mass at zero.

The general source question asks whether an individual entry of $D$ has an asymptotically geometric distribution as the dimensions grow. To formulate this as a definite asymptotic assertion, one must specify a sequence $(m_t,n_t,R^{(t)},C^{(t)})$, the selected entries $(i_t,j_t)$, admissible growth conditions on the margins and dimensions, the geometric parameters, and the meaning of asymptotic approximation. These are not specified in the supplied record, so no universal convergence assertion is imposed here.

The distinguished special case is precise at each finite size: for every integer $n\geq2$, let \[ R^{(n)}=C^{(n)}=(3n,\underbrace{n,\ldots,n}_{n-1\text{ entries}}),\qquad D^{(n)}\sim\operatorname{Unif}\mathcal T(R^{(n)},C^{(n)}). \] The source asks whether ${\mathbb{E}}[D^{(n)}_{11}]$ grows linearly in $n$. Under the order-of-magnitude interpretation of this phrase, the question is whether there exist constants $a,b>0$ and an integer $n_0\geq2$ such that \[ a n\leq {\mathbb{E}}[D^{(n)}_{11}]\leq b n\qquad\text{for every integer }n\geq n_0. \] The source does not specify whether it instead intends the stronger assertion that ${\mathbb{E}}[D^{(n)}_{11}]/n$ converges to a finite positive constant. No constant or limiting formula is supplied.

Short English statement

Choose uniformly from all nonnegative integer matrices with fixed row and column sums. The general question asks when a selected entry is approximately geometric as the dimensions increase, but the asymptotic regime and approximation criterion are unspecified. For square tables with margins $(3n,n,\ldots,n)$, the concrete question concerns linear growth of the expected upper-left entry. The supplied status is partially solved; this is a statement-only draft requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001782$, title: Entry Distributions in Random Contingency Tables; exact source problem code: catalogue entry. The record gives proposed year 2011 and no named proposer. Its literature assessment attributes the general question to Barvinok.
• Supplied source citation and literature source "original OWR report": Random Matrices, Geometric Functional Analysis and Algorithms (2011), Oberwolfach Reports, pp. 1287--1360, DOI 10.4171/owr/2011/24. https://doi.org/10.4171/owr/2011/24.
• Supplied literature source "Barvinok typical tables": https://arxiv.org/abs/0806.3910.
• Supplied literature source "Dittmer--Lyu--Pak phase transition": https://arxiv.org/abs/1903.08743.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved, states that Barvinok's general question remains open, and attributes geometric limits and phase transitions for important non-uniform margin families to Dittmer--Lyu--Pak. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its status and cited claims require independent checking.
• The registry labels the statement corrected_verified and reports that the official report was checked to clarify the probability space and the distinguished entry. This is a supplied verification claim, not verification performed for this draft.