Hexagon

math.NT — Number Theory

Rare events in multiplicative sequences: Poisson laws, clusters, and observation

Contributed by Brice Pouly

A multiplicative identity can persist at every scale while disappearing from a rare-event count. Another identity can survive as a cluster whose points lie far apart. Which distinction matters depends on what is observed. We explain this through one completely calculated model: a uniform germ can trigger two occurrences in disjoint spatial windows. Its two marginal counts become Poisson, but their sum need not do so. A coupling identifies the correct same-grid target and separates the mean number of points from the mean number of active germs. We then isolate the additional work required by multiplicative sources: private prime coordinates, weighted relations between windows, corrections to prime-site intensities, and relations visible at the target scale. A final passage from marked configurations to records explains the roles of ties, recognition and observation clocks. Selected results are stated with their hypotheses; a short concordance leads to their proofs. The elementary calculations are proved here, while the arithmetic and functional limit theorems are taken from the four cited research articles.

Primarily AI-generated textHuman understanding: no partsPoisson approximationarithmetic clustersexpository articlemarked point processesrandom multiplicative functionsrare events

Oct 18, 2026PDF ↗View work →

hexagon:2610.00336v1

math.NT — Number Theory

Sparse prime surgery in completely multiplicative sequences: Normality, rare-event realization and optimal repair

Contributed by Brice Pouly

We study how changing prime signs affects fixed-word frequencies, rare-event laws and extremes of completely multiplicative sequences. The cost is $d_\sigma(f,g)=\sum_{f(p)\ne g(p)}p^{-\sigma}$ . A known theorem of Frantzikinakis-Lemańczyk-de la Rue preserves normality at finite $d_1$ ; we give its elementary specialization. Using explicit arithmetic inputs from Long runs, we then establish three families of results. For each fixed $0<\sigma\le1$ and almost every base, reversible surgery realizes every stationary finite-intensity marked point-process law along selected scales, with signs, exact excess lengths and converging actual intensity, at arbitrarily small $d_\sigma$ cost and with a prescribed prefix unchanged. Almost surely over all integer scales, the minimum number of prime changes needed for a run of length $\lceil c\log_2N\rceil$ is asymptotic to $c\alpha_c\log_2N$ for $1<c<3/2$ , where $\alpha_c\in(0,1/2)$ satisfies $H_2(\alpha_c)=1-1/c$ and $H_2(u)=-u\log_2u-(1-u)\log_2(1-u)$ . At each scale, with probability tending to one, symbolic and private large-prime repair maxima coincide on the same random base for $k_N=o(\log N)$ ; unrestricted prime repair agrees when $k_N=o(\log\log N)$ . This comparison transfers the classical lattice laws of runs with errors to adaptive prime repair. Elementary programming from arbitrary bases and an autonomous finite-range-dependent process construction complement these results.

Primarily AI-generated textHuman understanding: no partscompletely multiplicative functionslong runsnormalityoptimal repairpoint-process realizationprime-sign perturbationsrare events

Oct 18, 2026PDF ↗View work →

hexagon:2610.00333v1

math.PR — Probability

Records, scale flows, and Gaussian limits from lattice Poisson fields

Contributed by Brice Pouly

How does a rare-event field determine the history of its extremes? Starting from the complete signed run-field comparison in Long runs and rare patterns, we develop records, censoring, occupation and Gaussian limits for the resulting lattice Poisson target. The arithmetic input retains every site, full exact length and actual sign, at growing intensity under a one-factor information-intensity budget. Strict integer records, their signed imbalance and logarithmic leadership time have a joint phase-uniform Brownian limit with explicit covariance, independent of the terminal critical field and its extremal paths. Right censoring gives scalar $M_1$ convergence for the visible maximum and $J_1$ convergence for the first-holder sign. Physical occupation instead has a nondegenerate phase-indexed limit. Canonical interpolation connects threshold thinning and a stationary scale flow without identifying their record rules or their Gaussian constants. The signed spatial theory includes weighted exact levels, fixed-window Ornstein-Uhlenbeck paths, two correlated threshold-compression noises, and rectangle fields with explicit grid centering. Direct partition and quadratic-reconstruction proofs provide alternative routes. For each specified observation, the target calculations identify its limit and the dependencies within the joint constructions.

Primarily AI-generated textHuman understanding: no partsGaussian limitsextremal processesfunctional limit theoremslattice Poisson fieldsoccupation timesrandom multiplicative functionsrecords

Oct 18, 2026PDF ↗View work →

hexagon:2610.00335v1

math.PR — Probability

Moving prime environments and arithmetic cluster limits

Contributed by Brice Pouly

We study shrinking-target occurrences of completely multiplicative functions with arbitrarily moving phases on a fixed finite set of exceptional primes and independent Haar values at the remaining primes. After exact removal of the environment-only points, effective integer-subgroup profiles classify all subsequential marked cluster limits, while target-scale pair overlaps characterize Poisson convergence. The quantitative input is a direct Haar comparison with the moving germ kernel, uniform in the exceptional phases and with stretched-exponential error. For deterministic profiles and interval targets, the complete spatial covariance measure reconstructs the unmarked law through primitive multiplicative rays and finite interval geometry. A positive forest formula makes this reconstruction valid even with unbounded cluster sizes. For a fixed known bounded Borel target, retaining positions and angular marks yields the corresponding identification from the complete marked covariance, using signed overlap channels. Restricting the observation to an interior spatial window imposes both angular and logarithmic-spatial constraints. The maximal number of descendants of one germ becomes a rectangle-occupancy number for a subgroup of $\mathbb{R}^2$ , finite exactly when that subgroup is discrete. Explicit rank-one laws illustrate what counts forget, while environmental mixtures show why deterministic identification does not extend by averaging covariances.

Primarily AI-generated textHuman understanding: no partsPoisson approximationarithmetic clusterscovariance reconstructionmoving environmentsrandom multiplicative functionsrelation latticesshrinking targets

Oct 18, 2026PDF ↗View work →

hexagon:2610.00342v1

math-ph — Mathematical Physics

Ghosts in Poisson Brane Gases

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

A Poisson gas of branes with mass intensity $\nu$ dresses the graviton propagator by the random entire function $\prod_k(1+p^2/t_k)$ . We prove that the gas average of the propagator equals $p^{-2}\exp[-\int p^2\nu(\dd t)/(p^2+t)]$, so that the massive states carry total weight $e^{-\nu(0,\infty)}-1<0$ . For $\nu=\lambda(t)\,\dd t/t$ the averaged spectrum consists of ghosts alone exactly when $\lambda\le1$ , and every average with a finite four-dimensional tadpole has states of both signs. Frozen gases keep alternating residues.

A mix of human-written and AI-generated textHuman understanding: some partsKallen-Lehmann representationLaguerre-Polya classPoisson point processbrane gascomplete Bernstein functionghost

Oct 18, 2026PDF ↗View work →

Matching works

Rare events in multiplicative sequences: Poisson laws, clusters, and observation

Sparse prime surgery in completely multiplicative sequences: Normality, rare-event realization and optimal repair

Records, scale flows, and Gaussian limits from lattice Poisson fields

Moving prime environments and arithmetic cluster limits

Ghosts in Poisson Brane Gases

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