Asymptotic uniformity conjecture for the smallest maximum

The conjecture holds for every $d\ge2$. More strongly, the entire marked process of low-norm maxima has a Poisson limit with independent uniform angular marks.

Write $L_jn=\log^{\circ j}n$, and set

$$c_n=L_1n-L_3n-\log(d-1),\qquad T_n=L_2n\,(\phi_n-c_n),\qquad U_n=\frac{\sigma_n}{\phi_n}.$$

Let $\upsilon$ denote uniform probability measure on the simplex

$$\mathcal S_{d-1}=\left\{u\ge0:\sum_{i=1}^du_i=1\right\}.$$

We prove

$$(T_n,U_n)\Longrightarrow(T,U), \qquad U\sim\upsilon,\qquad T\perp U,$$

where

$$\mathbb P(T>a) =\exp\left\{-\frac{e^{(d-1)a}}{(d-1)!}\right\}. \tag{1}$$

Use the source's lower and upper truncations

$$\ell_n=L_1n-L_3n-\log\omega_n,\qquad h_n=L_1n+2(d-1)L_2n,$$

with $\omega_n\to\infty$ sufficiently slowly. For the corresponding truncated Poisson input of intensity

$$n e^{-\|x\|_1} \mathbf1_{\{\ell_n<\|x\|_1\le h_n\}}\,dx,$$

partition the region below any fixed normalized upper radial level into the source's grid cells, and let $I_\alpha$ indicate that cell $\alpha$ contains a maximum.

The source proves that the Chen–Stein dependency quantities satisfy $b_3=0$ and, by its equations (10.3), (10.9)–(10.11),

$$\limsup_{\mathrm{mesh}\to0}(b_1+b_2) = O\left((L_2n)^{-(d-3/2)}(L_3n)^{-1}\right) + O\left((L_1n)^{-(d-1)}(L_2n)^d(L_3n)^{-2}\right) =o(1). \tag{2}$$

The exponent is $-(d-\tfrac32)$, so (2) also holds for $d=2$.

Instead of applying only the scalar Poisson-count theorem used in the source, apply Arratia–Goldstein–Gordon's point-process Theorem 2 to the whole Bernoulli configuration:

$$d_{\mathrm{TV}}\left( \mathcal L((I_\alpha)_\alpha), \mathcal L((Y_\alpha)_\alpha) \right) \le4(b_1+b_2),$$

where the $Y_\alpha$ are independent Poisson variables of means $\mathbb EI_\alpha$. Color the cells by any finitely many disjoint radial/angular bins, let the mesh tend to zero, and use the source's collision estimate. The resulting bin counts converge jointly to independent Poisson variables with their corresponding first-intensity means. This argument uses finite bin-count vectors and does not require total-variation convergence from atomic grids to a diffuse process.

That first intensity is explicit. Writing $r=\|x\|_1$, the Mecke formula gives

$$\nu_n(dx) = n e^{-r} \exp\left\{-n e^{-r}G_d(h_n-r)\right\}\,dx,$$

where

$$G_d(v)= 1-e^{-v}\sum_{j=0}^{d-1}\frac{v^j}{j!}.$$

Indeed, the dominating northeast orthant above $x$, cut off at radius $h_n$, has intensity $ne^{-r}G_d(h_n-r)$. Thus the maxima intensity depends only on $r$. The simplex change of variables is

$$dx=\frac{r^{d-1}}{(d-1)!}\,dr\,\upsilon(du), \qquad u=x/r,$$

so the angular intensity is exactly uniform for every $n$.

The source's Lemma 9.1 gives the limiting cumulative radial intensity

$$\Lambda(a) = \lim_{n\to\infty} \nu_n\left\{r\le c_n+\frac a{L_2n}\right\} = \frac{e^{(d-1)a}}{(d-1)!}.$$

Therefore, for every $a<b$ and every simplex continuity set $C$,

$$\nu_n\left\{ a<L_2n(r-c_n)\le b,\quad u\in C \right\} \longrightarrow \bigl(\Lambda(b)-\Lambda(a)\bigr)\upsilon(C).$$

Together with (2), this proves convergence of the full marked maxima process to a Poisson point process with product intensity

$$\boxed{\displaystyle \frac{e^{(d-1)t}}{(d-2)!}\,dt\,\upsilon(du).} \tag{3}$$

The source's truncation and de-Poissonization couplings extend directly to whole point configurations: maxima below $\ell_n$ occur with probability $o(1)$; the above-$\ell_n$ binomial and Poisson configurations couple with error $o(1)$; and deleting points above $h_n$ changes the configuration with probability

$$n\,\mathbb P\{\operatorname{Gamma}(d,1)>h_n\} =O((L_1n)^{-(d-1)})=o(1).$$

Hence (3) also holds for the original fixed-size sample.

Finally, the lowest point of (3) is almost surely unique. Its radial survival function is $e^{-\Lambda(a)}$, while its angular mark is independent and uniformly distributed. Thus, for every $a\in\mathbb R$ and every $\upsilon$-continuity set $C$,

$$\boxed{\displaystyle \lim_{n\to\infty} \mathbb P\{T_n\le a,\ U_n\in C\} = \left(1- e^{-e^{(d-1)a}/(d-1)!}\right)\upsilon(C),}$$

proving both conjectured uniformity and asymptotic independence.

Sources: J. A. Fill, arXiv:2601.18170, Conjecture 11.2 and Sections 4–10; R. Arratia, L. Goldstein and L. Gordon, Two Moments Suffice for Poisson Approximations: The Chen–Stein Method (1989), Theorem 2.