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arxiv: 2606.04837 · v1 · pith:NGX2IUF5new · submitted 2026-06-03 · 🧮 math-ph · cond-mat.stat-mech· math.MP· math.PR

Maximal Minimal Spacing for Random Points

Fabio Deelan Cunden , Noemi Cuppone , Giovanni Gramegna , Pierpaolo Vivo This is my paper

Pith reviewed 2026-06-28 04:04 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.stat-mechmath.MPmath.PR
keywords maximal minimal spacingrandom points on a linegap distributionsresetting random walkfirst-passage functionalsoptimal block sumsthreshold crossing
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The pith

The maximal minimal spacing from grouping random gaps equals the number of threshold resets completed by a random walk.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that choosing M+1 points out of N+1 random points on a line to maximize the smallest gap between them is the same as partitioning the N gaps into M consecutive blocks to maximize the smallest block sum. For any distribution of the initial gaps, it derives exact formulas for the distribution of this optimal spacing that hold for every M up to N. The formulas arise by mapping the block-sum problem onto the cycle counts of a random walk that resets to zero each time its position exceeds a fixed threshold. The probability that the optimal spacing is larger than a given value is exactly the probability that the walk finishes at least M full reset cycles inside N steps, which in turn supplies an expression through first-passage quantities of the walk.

Core claim

For arbitrary i.i.d. gap distributions and every M ≤ N, the distribution of the maximal minimal spacing is given exactly by the distribution of the number of threshold-reset cycles completed by an associated random walk; in particular, the survival function of the optimal spacing equals the probability that the walk completes at least M resets within N increments.

What carries the argument

The threshold-resetting random walk, which takes successive random steps and returns to the origin whenever its position crosses a fixed threshold; the number of such completed cycles directly encodes the largest achievable minimal block sum.

If this is right

  • The cumulative distribution function of the optimal spacing is identical to the cumulative distribution of the number of resets in the walk.
  • Asymptotics of the optimal spacing for large N follow from the large-deviation or renewal properties of the resetting walk.
  • The same mapping supplies a Monte-Carlo scheme that samples reset cycles rather than enumerating partitions, accurate for regimes where direct search over groupings becomes infeasible.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The walk representation may allow transfer of known limit theorems for first-passage processes directly to spacing problems without re-deriving them.
  • If the gap distribution belongs to a domain of attraction, the resetting walk may inherit a scaling limit that yields a closed-form asymptotic law for the maximal minimal spacing.
  • The equivalence could be tested numerically for non-i.i.d. gaps to see how far the cycle-count picture extends beyond the stated assumption.

Load-bearing premise

The gaps between the original N+1 points are independent and identically distributed.

What would settle it

For small fixed N and M, generate many realizations of N i.i.d. gaps, compute the exact maximal minimal block sum by enumeration, and check whether its empirical distribution matches the cycle-count distribution obtained from the corresponding resetting walk.

Figures

Figures reproduced from arXiv: 2606.04837 by Fabio Deelan Cunden, Giovanni Gramegna, Noemi Cuppone, Pierpaolo Vivo.

Figure 1
Figure 1. Figure 1: Choose 𝑀 +1 points out of 𝑁 +1 so as to maximize their minimal spacing. Here 𝑁 = 7, and 𝑀 = 4. The top line contains the initial configuration (black circles). The 6 3  = 20 possible selections of points are in the lines below. The optimal ones (the max-min spacing in this case is (𝑃5 − 𝑃3)) are in black squares, while the others are in white circles. 2. Definition of the model The problem is translation … view at source ↗
Figure 2
Figure 2. Figure 2: Sketch of the threshold-resetting random walk. In this example, there are 𝐾𝑁 (𝑠) = 3 complete cycles of length 𝐿1(𝑠) = 4, 𝐿2(𝑠) = 3, 𝐿3(𝑠) = 2 in 𝑁 = 12 steps. 3. Main results 3.1. Distribution-free exact formula. Our main result is an exact formula for the tail distribution of 𝑆 ∗ , valid for all distributions of the i.i.d. gaps 𝑇𝑗’s. A central role is played by first-passage time at level 𝑠 > 0, that we … view at source ↗
Figure 3
Figure 3. Figure 3: Max-min spacing 𝑆 ∗ for gaps (𝑇𝑗)𝑗≥1 uniformly distributed in the interval [0, 1]. Numerical simulations (histograms) compared to the explicit formula (3.3). Here 𝑁 = 6, and the sample size is 105 . Equation (3.10) selects the tilted law under which the typical value of 𝜏1(𝑠,𝑧) matches the constraint 𝑀/𝑁 = 𝛼. The theorem identifies not only the rate function 𝜓(𝑠) governing the probability of atypical fluct… view at source ↗
Figure 4
Figure 4. Figure 4: Relative max-min spacing 𝑀𝑆efor exponential gaps. Numerical simulations (histograms) compared to the explicit formula (4.4). Here 𝑁 = 6, and the sample size is 105 . Remark 4.2. The relative max-min spacing 𝑆efor the model above coincides with the max-min spacing for 𝑁 − 1 uniform points on the unit interval. More precisely, let 𝑈1, . . . ,𝑈𝑁 −1 be i.i.d. random variables uniformly distributed in the inter… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of −(1/𝑁) log 𝑓𝑀𝑆e(𝑠) (black dots) for exponential gaps with 𝑁 = 100 and 𝛼 = 0.215, against the rate function 𝜓𝑆e(𝑠) (red dashed curve) in (4.24) and the asymptotics (4.22) including the first subleading correction (blue solid curve). The inset is a zoom around the typical value ˜𝑠 = 0.785.... 4.2.2. Large deviations. Using Stirling approximation on (4.5)-(4.6), one gets the following large devi… view at source ↗
Figure 6
Figure 6. Figure 6: Max-min spacing 𝑆 ∗ for points with random i.i.d. exponential gaps. Nu￾merical simulations (histograms) over a sample size of 104 obtained using the iterative interval refined method described in Sec. 5. In black the explicit formula (4.5). The dotted-red curve is the Gaussian approximation (4.16). Here 𝑁 = 60, and 𝑀 = 20. (Total number of selections: |𝐼𝑀,𝑁 | = [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Contours of integration appearing in the calculation of P(𝑆 ∗ ≥ 𝑠) Therefore, if 𝑠 > 𝑠∗ , the contour 𝐶𝑧 (𝑠) is contained in the unit disk where the integrand 𝑒 𝑁𝑔(𝑧) (1−𝑧) 𝑧 has a single pole at 𝑧 = 0 (see [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
read the original abstract

From $N+1$ random points on a line we wish to select $M+1$ points so as to maximize the minimal spacing between them. We consider an initial configuration with independent and identically distributed spacings. The problem is equivalent to optimally grouping consecutive gaps into $M$ blocks and maximizing the smallest block sum. For general gap distributions, and for all $M\leq N$, we derive exact distributional identities for the optimal spacing and obtain its asymptotic behavior. The problem admits a reformulation in terms of a threshold-resetting random walk. The walk advances by successive random increments and is reset to the origin upon exceeding a fixed threshold. The probability that the optimal spacing exceeds a given value coincides with the probability that the walk completes at least $M$ reset cycles within $N$ steps. This yields an exact representation in terms of first-passage functionals of the walk. The same mapping suggests a numerical scheme for the max-min spacing problem in the regime of large $N$ and $M$, whose accuracy is tested against the exact results obtained here.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper investigates selecting M+1 points out of N+1 random points on a line to maximize the minimal spacing. Assuming i.i.d. spacings, it equates this to maximizing the smallest sum when grouping consecutive gaps into M blocks. For general gap distributions, exact distributional identities for the optimal spacing are derived for all M ≤ N, along with asymptotic behavior. The problem is mapped to a threshold-resetting random walk, where the probability that the optimal spacing exceeds a threshold equals the probability of completing at least M reset cycles in N steps. This gives an exact representation using first-passage functionals and a numerical scheme for large N, M tested against the exact identities.

Significance. The equivalence between the max-min spacing problem and the number of reset cycles in a random walk is a significant contribution if the identities hold, as it provides exact results for general distributions and links to established random walk theory. The ability to obtain exact distributional identities and test numerical methods against them strengthens the work. This could have implications for optimization in random media and stochastic processes.

minor comments (1)
  1. [Abstract] The abstract is quite dense; expanding slightly on how the random walk equivalence is constructed from the gap grouping would improve accessibility for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, the recognition of the equivalence to reset cycles in a random walk as a significant contribution, and the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives an exact equivalence between the maximal minimal spacing and the number of reset cycles in a threshold-resetting random walk directly from the i.i.d. gap assumption via consecutive block sums and first-passage functionals. No fitted parameters are renamed as predictions, no self-citations serve as load-bearing uniqueness theorems, and no ansatz is smuggled in. The identities are obtained by explicit grouping of sums, which is a definitional mapping rather than a reduction to the input by construction. This is the standard case of an internally consistent exact result with no circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivations rest on the domain assumption of i.i.d. gap lengths and on standard properties of random walks and first-passage times; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption Spacings between the initial N+1 points are independent and identically distributed.
    Explicitly stated as the initial configuration in the abstract.

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