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arxiv: 2604.18427 · v2 · submitted 2026-04-20 · 🧮 math.PR · cond-mat.stat-mech· math.MG

Expected perimeter of the convex hull of planar Brownian motion stopped upon exiting the unit disk

Hugo Panzo , Stjepan \v{S}ebek This is my paper

Pith reviewed 2026-05-10 03:44 UTC · model grok-4.3

classification 🧮 math.PR cond-mat.stat-mechmath.MG
keywords Brownian motionconvex hullexpected perimeterharmonic measuretruncated diskexit timeunit diskplanar processes
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The pith

The expected perimeter of the convex hull of planar Brownian motion exiting the unit disk is given by an exact closed-form expression.

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

This paper establishes an exact formula for the expected perimeter of the convex hull of a planar Brownian motion run until it leaves the unit disk. The key step is reducing the perimeter to twice the expected maximum horizontal displacement at the exit time. This quantity is then computed exactly by solving a harmonic measure problem in a truncated version of the disk. The work complements results on reflecting Brownian motion and shows why the expected area is more difficult to compute exactly. Readers interested in stochastic geometry would find value in having a precise geometric descriptor for confined Brownian paths.

Core claim

We reduce the expected perimeter to the expected maximum horizontal displacement of the Brownian motion at the exit time from the unit disk and recast this as a harmonic measure problem in the truncated disk, thereby obtaining an exact expression for the expected perimeter.

What carries the argument

Reduction of the perimeter to the expected maximum horizontal displacement at exit time, solved via harmonic measure in the truncated disk

If this is right

  • An exact expression is obtained for the expected perimeter.
  • Nontrivial bounds and a Monte Carlo estimate are given for the expected area of the convex hull.
  • Further results are provided on the expected areas of the star hull and topological hull.
  • Computing the exact expected area is shown to be considerably harder than the perimeter.

Where Pith is reading between the lines

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

  • Similar harmonic measure reductions might yield exact expressions for other moments or functionals of the convex hull.
  • The hardness of the area problem likely stems from it requiring integration over the entire path rather than a boundary value problem.
  • These techniques could be tested on Brownian motion exiting other domains with rotational symmetry.
  • Verification via simulation of the perimeter formula would provide a direct falsification test.

Load-bearing premise

The perimeter of the convex hull equals twice the expected maximum horizontal displacement of the Brownian motion at the exit time from the unit disk.

What would settle it

Simulate many independent planar Brownian paths until exit from the unit disk, compute the average perimeter of their convex hulls, and compare it to the value of the derived closed-form expression.

Figures

Figures reproduced from arXiv: 2604.18427 by Hugo Panzo, Stjepan \v{S}ebek.

Figure 1
Figure 1. Figure 1: Convex hull (depicted in green) of a standard planar Brownian motion path starting at 0 and run until it exits the unit disk at WτD . maximum of the trajectory along a fixed direction, as well as the growth of the convex hull. In the planar case d = 2, they obtained a precise large-time asymptotic for the mean perimeter E[Pt ] of the convex hull of a Brownian particle reflected at the boundary of a disk of… view at source ↗
Figure 2
Figure 2. Figure 2: The truncated disk domain Da with a = 1 2 . Define the exit time of Da analogously to that of D and denote it by τDa , that is, τDa = inf{t ≥ 0 : Wt ∈ D/ a} [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The wedge Wa = la(Da) with a = 1 2 and β(a) = 2π 3 . 3.2. Opening the wedge to the upper half-plane. As we already observed, the wedge Wa has opening angle β(a) = π − arccos a. The power map pa(w) = w π/β(a) (with an appropriate branch) maps Wa conformally onto the upper half-plane H, fixing the ray θ = 0 and sending the ray θ = β(a) to the negative real axis. For a similar example of a power map being use… view at source ↗
Figure 4
Figure 4. Figure 4: Illustrations of the same planar Brownian motion trajectory run until first exiting the unit disk, together with the three associated hulls. Brownian path is approximated by a polygonal trajectory w0, w1, . . . , wN obtained from a time discretization with step ∆t = 10−6 , and taking Gaussian steps with expectation zero, and variance equal to ∆t. The interval [0, 2π) is divided into m = 2000 equally spaced… view at source ↗
read the original abstract

We study the convex hull of planar Brownian motion run until the exit time from the unit disk. Our primary objective is to compute the expected perimeter of this convex hull, thereby complementing recent results on the convex hull of reflecting Brownian motion in confined geometries. We reduce the problem to computing the expected value of the Brownian motion's maximum horizontal displacement at the exit time, and then recast this maximum in terms of harmonic measure in a domain we call the truncated disk. In particular, we obtain an exact expression for the expected perimeter. We also obtain nontrivial bounds and a Monte Carlo estimate for the expected area of this convex hull and comment on why computing the exact expected area is a much harder problem. We conclude with further results on the expected areas of two related hulls of the path run until exiting the disk, namely, the star hull and topological hull.

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 / 3 minor

Summary. The manuscript computes the expected perimeter of the convex hull of planar Brownian motion stopped at the exit time from the unit disk. It reduces the perimeter exactly to 2π times the expected maximum horizontal displacement at exit via the support-function representation and rotational invariance, then expresses this expectation as an integral over a of the value at the origin of the harmonic function u_a solving the Dirichlet problem in the truncated disk D_a with boundary data 1 on the vertical chord at Re z = a and 0 elsewhere. An exact expression is thereby obtained. The paper also supplies nontrivial bounds and a Monte Carlo estimate for the expected area of the same hull and gives results for the expected areas of the star hull and topological hull of the same path.

Significance. The central result supplies an exact, parameter-free formula for a geometrically natural functional of planar Brownian motion in a bounded domain, obtained by a direct and standard application of the hitting-probability interpretation of harmonic measure. This is a clear strength and complements the recent literature on convex hulls of reflecting Brownian motion. The clean reduction and the explicit identification of why the area functional is substantially harder (non-harmonic) are also valuable. The Monte Carlo and comparison results for related hulls provide useful context.

minor comments (3)
  1. [Section 3] The final exact expression for the expected perimeter (the integral involving u_a(0)) should be displayed as a numbered equation and referenced explicitly in the abstract and introduction.
  2. [Section 4] The Monte Carlo section should state the number of independent paths, the time-step size, and the method used to compute the convex hull (e.g., Graham scan) so that the reported estimate and error bars are reproducible.
  3. [Section 4] The explanation that the area is harder because it is not a harmonic functional could be sharpened by exhibiting the specific non-harmonic integrand that appears in the area formula.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, including the clear summary of our main results on the expected perimeter and the recognition of the value of our bounds, Monte Carlo estimates, and comparisons for related hulls. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central derivation begins with the support-function formula for the perimeter of a convex set (∫ h(θ) dθ from 0 to 2π) and uses rotational invariance of planar Brownian motion to reduce the expected perimeter exactly to 2π E[max horizontal displacement at exit time τ]. This quantity is then expressed as ∫_0^1 u_a(0) da where each u_a solves a standard Dirichlet problem in the truncated disk with boundary values 1 on the chord Re z = a and 0 elsewhere; this is the classical hitting-probability representation for Brownian motion and requires no fitted parameters or ansatz. No self-definitional steps, no renaming of known results as new predictions, and no load-bearing self-citations appear in the perimeter chain. The area results are explicitly noted as harder and only bounded or estimated, confirming the perimeter derivation stands independently on classical potential theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the derivation relies on standard tools from probability theory. No free parameters, new entities, or ad-hoc axioms are mentioned.

axioms (1)
  • standard math Harmonic measure is well-defined and yields computable expectations for Brownian exit distributions in the truncated disk.
    Harmonic measure is a classical object in potential theory for Markov processes; the abstract invokes it as the recasting mechanism for the maximum displacement.

pith-pipeline@v0.9.0 · 5455 in / 1477 out tokens · 47841 ms · 2026-05-10T03:44:58.476093+00:00 · methodology

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Reference graph

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