Bulk Large Deviations in Stationary Point Fields

Definitions and mathematical statement

A stationary random point field on ${\mathbb{R}}^d$ can be represented by a random locally finite counting measure $\Xi$ whose law is invariant under translations. For a bounded Borel set $A$, write $N(A)=\Xi(A)$ for its number of points. When the intensity $\rho$ is finite, stationarity gives the usual centering $N(A)-\rho|A|$; finite intensity is not explicitly assumed in the supplied record.

The report-level question is: what large-deviation behavior governs bulk fluctuations of stationary random point fields, as distinguished from the boundary-effect fluctuations described in the source for the number of points in a large disk? A large-deviation formulation requires specifying the point-field law, the bulk observable or event, its scaling, a speed tending to infinity, and a rate function or an asymptotic probability. None of these is fixed by the extraction. In particular, stationarity alone is not a specification of one universal bulk law, and the disk-count comparison does not identify a particular process or precise boundary-fluctuation theorem. This remains an exploratory question; no particular hole-probability asymptotic, counting-function principle, or beta-ensemble model is silently substituted.

Short English statement

Describe large deviations of bulk fluctuations in the intended stationary point fields, beyond the boundary-driven disk-count fluctuations mentioned by the source. The process, bulk event, and asymptotic scale need to be recovered before a single precise large-deviation theorem can be stated.

Sources

• Local UnsolvedMath registry, permanent ID $30000966$, source code catalogue entry, "Bulk Large Deviations in Stationary Point Fields." Registry curation: Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying sources retain their supplied provenance.
• Stochastic Analysis (2009), Oberwolfach Reports, pp. 1361--1424, DOI https://doi.org/10.4171/owr/2008/25.
• Supplied "Valko--Virag large gaps theorem": https://arxiv.org/abs/0811.0007.
• The supplied literature triage checked 2026-08-21 labels the problem partially solved: bulk large-gap and related large-deviation asymptotics for beta-ensemble limits have been proved, while the broader report-level bulk question remains. This is a dated source assessment, not an independent status check.