Concentration of Largest Triangular Submatrices in Random Matrices

Definitions and mathematical statement

Let $M_n=(X_{ij}^{(n)})_{1\leq i,j\leq n}$ be an $n\times n$ random matrix whose entries are independent and identically distributed. The supplied record does not specify their common distribution, their value space, or whether the distribution is fixed as $n$ varies. A definition of lower triangularity requires a distinguished zero in the value space: a square matrix $B=(b_{ij})_{1\leq i,j\leq k}$ is lower triangular in the usual sense if $b_{ij}=0$ whenever $i<j$. Its order is $k$.

The source denotes by $T_n$ the order of the largest lower-triangular submatrix of $M_n$. This quantity requires further conventions that the supplied record does not resolve: whether selected rows and columns retain their original order or may be permuted, and whether diagonal entries must be nonzero. Under the usual definition of a submatrix, the selected rows and columns retain their original order; under the usual definition of lower triangularity, diagonal entries are unrestricted. These conventions cannot be confirmed as the intended source conventions from the supplied record. If no positive-order admissible submatrix exists, an additional convention, such as $T_n=0$, is needed.

Once the entry law and these conventions are specified, a precise asymptotic interpretation of the question is: \[ \text{Do there exist deterministic sets }A_n\subseteq\{0,1,\ldots,n\} \text{ with }|A_n|\leq 2 \text{ such that } \lim_{n\to\infty}\mathbb{P}(T_n\in A_n)=1? \] The sets $A_n$ may depend on $n$ and on the specified model, but not on the realized matrix. The supplied wording does not state whether the two values must be consecutive, whether this asymptotic interpretation is intended, or for which entry distributions the assertion is proposed. Consequently, the display is a conditional formulation of the concentration target, not a fully specified conjecture for a determined random-matrix model.

Short English statement

The question asks whether the maximum order of a lower-triangular submatrix lies in a deterministic set of at most two integers with probability tending to one as the matrix size grows. The entry distribution and submatrix conventions are missing, so the intended conjecture requires clarification. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, permanent ID $30002318$, source problem code catalogue entry, title: "Concentration of Largest Triangular Submatrices in Random Matrices." The record belongs to the Oberwolfach Reports Open Problems collection. It records proposed year $2013$ and supplies no proposer attribution.
• Supplied source citation: Combinatorics and Probability (2014), Oberwolfach Reports, DOI 10.4171/owr/2013/18, pp. 1087--1152. https://doi.org/10.4171/owr/2013/18. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The citation's year $2014$ is retained alongside the record's proposed year $2013$; their relationship is not resolved here.
• Supplied literature reference: Izhakian--Janson--Rhodes, "triangular submatrices." https://doi.org/10.1090/S0002-9939-2014-12301-X. No fuller bibliographic citation is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, states that the report poses concentration on at most two values as open, and states that the cited asymptotic result does not settle this finer concentration question. This is supplied dated curation derived from the supplied OWR material, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification description is supplied provenance and has not been independently checked for this draft.