On the first hitting time of a high-dimensional orthant

Emmanuel Humbert; Kilian Raschel

REVIEW 2 major objections 3 minor 19 references

On the first hitting time of a high-dimensional orthant

T0 review · 2 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that the principal Dirichlet eigenvalue of the complement of a high-dimensional orthant satisfies $\lambda_1(d) \in [c\,d/2^d,\, C\,d^3/2^d]$, so the survival exponent $p_d$ goes to zero as $d\to\infty$.

desk verdict The high-dimensional orthant eigenvalue limit is a good question and the lower bound is clean, but the upper-bound test function does not vanish on the boundary, so Theorem 1 is unproved as written. read the letter →

arxiv 2411.17023 v1 pith:SG4RNDZ2 submitted 2024-11-26 math.PR math.COmath.SP

Emmanuel Humbert , Kilian Raschel This is my paper · ORCID

classification math.PRmath.COmath.SP MSC 60J6560G4058C40

keywords survivalprobabilityBrownianmotioninconesorthantprincipaleigenvalueLaplace–Beltramioperatorhigh-dimensionalsphereYamabefunctionalfirsthittingtime

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies how long a collection of $d$ independent Brownian particles, started with at least one particle positive, needs before all $d$ particles become negative. Exact distributions are known only for $d=1,2$, so the paper attacks the asymptotic regime $d\to\infty$ through the tail exponent $p_d$ of the survival probability $P_x(\tau_d>t)\sim V_d(x)/t^{p_d/2}$. Its main claim, Theorem 1, is that the principal Dirichlet eigenvalue $\lambda_1(d)$ of the Laplace–Beltrami operator on the sphere cut by the complement of the negative orthant is trapped between $c\,d/2^d$ and $C\,d^3/2^d$, and therefore $\log\lambda_1(d)/d\to\log(1/2)$. Corollary 2 then gives $p_d\to0$ with $p_d\sim\lambda_1(d)/d$, meaning the survival probability decays more and more slowly as the dimension grows. The paper's techniques are spectral-geometric, and the eigenvalue estimates are offered as a result of independent interest.

What carries the argument

The load-bearing object is $\lambda_1(d)$, the principal Dirichlet eigenvalue of the Laplace–Beltrami operator on the spherical domain $U_d$, the intersection of the sphere $\mathbb{S}^{d-1}$ with the complement of the negative orthant. The lower bound follows from the Yamabe functional's Sobolev inequality: any function on the sphere satisfies an $L^{2d/(d-2)}$ estimate, and applying it to an eigenfunction of $U_{d+1}$ extended by zero yields $\lambda_1(d+1) \gtrsim d/2^d$. The upper bound is produced by an explicit cut-off test function $\eta_d(x)=\max_i \theta_d(x_i)$, where $\theta_d$ is 1 below $-a_d$ and 0 above 0, inserted into the variational formula $\lambda_1(d) = \inf_{u \in H^1_0(U_d)} \int_{U_d}|\nabla u|^2 \big/ \int_{U_d} u^2$; the paper bounds the volume of the transition layer $\Sigma_d(a) = [-a,1]^d \cap \mathbb{S}^{d-1}$ by $C\omega_{d-1}/2^d$ to control the gradient term.

What would settle it

Evaluate the proposed test function on the spherical boundary point with one coordinate equal to 0 and another equal to $-1$ (renormalized to the sphere): $\eta_d$ equals 1 there, not 0, so the variational upper bound in Section 2.3 uses a function outside $H^1_0(U_d)$; replacing it with an admissible function, or computing $\lambda_1(d)$ numerically for $d=4,\dots,20$, would decide whether the upper bound $C d^3/2^d$ still holds.

Watch

Extended reading notes

Core claim

The paper's central claim is that the small positive eigenvalue $\lambda_1(d)$ of $-\Delta_{\mathbb{S}^{d-1}}$ with Dirichlet conditions on the boundary of $U_d = (\mathbb{R}^d \setminus \mathbb{R}^d_-) \cap \mathbb{S}^{d-1}$ is controlled, for every $d \ge 1$, by two constants $c, C>0$: $\lambda_1(d) \in [c\,d/2^d,\, C\,d^3/2^d]$. In particular $\log \lambda_1(d)/d \to \log(1/2)$, so the eigenvalue is exponentially small in dimension, matching the fraction of the sphere occupied by the positive orthant. Through the identity $p_d = \sqrt{\lambda_1(d) + (d/2-1)^2} - (d/2-1)$, this forces the survival exponent $p_d$ in $P_x(\tau_d>t) \sim V_d(x)/t^{p_d/2}$ to go to zero, with $p_d \sim \lambda_1(d)/d$.

Load-bearing premise

The upper-bound half of the proof relies on the cut-off function $\eta_d(x)=\max_i\theta_d(x_i)$ belonging to the space of test functions that vanish on the boundary of $U_d$; on boundary points where one coordinate is 0 and another coordinate is $\le -a_d$, however, $\eta_d$ equals 1, so this admissibility condition fails and the upper bound is not supported as written.

Editorial extensions

If this is right

Corollary 2: as $d\to\infty$, $p_d\to0$ and $p_d\sim\lambda_1(d)/d$, so the survival probability $P_x(\tau_d>t)\sim V_d(x)/t^{p_d/2}$ decays more slowly than any fixed power of $t$ once the dimension is large enough.
The uniform bounds $c\,d/2^d \le \lambda_1(d) \le C\,d^3/2^d$ hold for every $d\ge1$, so the exponential rate $\log(1/2)$ is not merely asymptotic but sandwiched by polynomial factors in $d$.
Through the relation in [18, Thm 2], Theorem 1 converts into a quantitative approximation for the short-time behaviour of the occupation time of a large-dimensional orthant by Brownian motion.
The eigenfunction expansion of the survival probability from [1, Thm 1] inherits these bounds, giving quantitative control of the full survival probability rather than only its leading exponent.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the claimed rates are correct, the natural joint $d,t\to\infty$ scaling for the survival probability is likely set by the ratio $t/2^d$ rather than by $t$ alone; this joint scaling is not derived in the paper and could be tested numerically.
The same spectral strategy should apply to linear images of orthants, the unequal-variance case the paper lists as an open problem: one would expect $\lambda_1$ to decay like the volume fraction of the corresponding transformed spherical cap.
Because the claimed rate depends only on the fraction $1/2^d$ of the sphere occupied by the positive orthant, the result suggests that spherical-cap approximations used heuristically in the physics literature capture the correct exponential order for the all-negative first-passage problem.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: the main theorem is not established. The lower bound via the Yamabe inequality is a nice piece of work, and the question — the exponential decay rate of the principal Dirichlet eigenvalue for the complement of a high-dimensional orthant — is natural and worth asking. The paper is clearly written and honest about the state of the art, including that p_3 is unknown. But the upper bound in Theorem 1 rests on a trial function that is not admissible for the Dirichlet problem.

The bug is in §2.3. The domain U_d is the complement of the negative orthant on the sphere, so a point is outside U_d when all coordinates are non-positive. The paper says the opposite: it claims x outside U_d implies every coordinate is positive. That is reversed. Consequently the function η_d, defined as max_i θ_d(x_i), takes the value 1 on the boundary whenever at least one coordinate is ≤ −a_d, and it does not vanish. It is not in H^1_0(U_d), so the Rayleigh quotient bound λ_1 ≤ Y(η_d) is invalid. This is not a minor typo; it removes the only mechanism for the upper bound.

There is also a secondary algebraic gap in the induction for Lemma 3. The paper states that the condition on a immediately gives a/√(1−a²) ≤ ε_D√(d − (d−k)ε_D²). That does not follow from the displayed inequality in (13). If Lemma 3 collapses, the volume estimate in Step 1 lacks support, though the lemma itself is plausible and may be provable by a different argument.

On the positive side, the lower bound argument is coherent and likely correct: the Yamabe functional approach gives λ_1(d) ≥ c d / 2^d without any fitted parameter. The asymptotic log λ_1 / d → log(1/2) is genuinely new and would be a meaningful result if the upper bound can be fixed. Nothing in the paper suggests the result is false; the flaws are in the proof, not in the conclusion.

For a referee: I would not accept the paper in this form. A serious referee should see it, because the question is good and the lower bound is worth preserving. The authors need to supply a valid test function — a product-type cutoff that actually vanishes on ∂U_d would be the natural direction — and repair the Lemma 3 induction. With that, the paper could be publishable. My recommendation is to send it to review, but with the expectation of substantial revision.

Referee Report

2 major / 3 minor

Summary. The paper studies the first time all independent standard Brownian particles become negative, equivalently the first exit time from the complement of the negative orthant in R^d. The key quantity is the principal Dirichlet eigenvalue λ1(d) of the spherical domain U_d = (R^d \ R^d_-) ∩ S^{d-1}. Theorem 1 claims c d / 2^d ≤ λ1(d) ≤ C d^3 / 2^d for all d, and hence lim_d log(λ1)/d = log(1/2); Corollary 2 translates this into the asymptotic vanishing of the survival exponent p_d. The lower bound is proved via the Yamabe inequality, and the upper bound via an explicit trial function together with a volume estimate for spherical slabs.

Significance. If Theorem 1 is established, the paper provides the first high-dimensional asymptotic for the principal eigenvalue of the complement of an orthant and identifies the exponential rate log(1/2) for the survival exponent. The lower-bound argument is coherent, parameter-free, and not circular: it uses only the variational characterization of λ1, the Yamabe inequality, Hölder's inequality, and the known volume of U_{d+1}. These are genuine virtues. The upper-bound proof, however, rests on an inadmissible test function, and the volume lemma used in that proof is not established because of an algebraic error in the induction. The main claim is therefore unproved as written, although the approach appears repairable.

major comments (2)

[§2.3, properties of η_d] The trial function η_d(x) = max_i θ_d(x_i) is not admissible in the Rayleigh quotient for λ1(d). The paper states that η_d is supported in U_d because "when x ∉ U_d then x_i > 0 for all i"; this reverses the definition of U_d. Since U_d = {x : x_i > 0 for some i}, its complement is the closed negative orthant {x : x_i ≤ 0 for all i}. On ∂U_d, e.g. at the point (0, -1, 0, ..., 0), one coordinate is 0 and another is ≤ -a_d, so η_d equals 1 there. Hence η_d does not vanish on ∂U_d and is not in H^1_0(U_d). The inequality λ1(d) ≤ Y(η_d), and with it the entire upper-bound half of Theorem 1, is therefore unjustified.
[§2.4, induction step of Lemma 3] The verification that a/√(1-a^2) satisfies the induction hypothesis is algebraically incorrect. Starting from a^2 ≤ ε_D^2(d+1 - (d-k)ε_D^2), the displayed chain claims a/√(1-a^2) ≤ ε_D√(d - (d-k)ε_D^2 + (1-ε_D^2)) ≤ ε_D√(d - (d-k)ε_D^2). The second inequality would require 1 - ε_D^2 ≤ 0, which is false for ε_D < 1. A numerical check already contradicts the claim: for d = 10, k = 0, and ε_D = 0.3, the allowed a can be as large as about 0.953, but then a/√(1-a^2) is about 3.15, whereas ε_D√(d - (d-k)ε_D^2) is about 0.905. Thus the induction does not close and Lemma 3, which supplies the volume estimate used in Step 1 of the upper bound, is not proved.

minor comments (3)

[§2.2, after Eq. (7)] The integrand in the Hölder step is written as |u_d|^{2d/(d-2)}, but it should be |u_{d+1}|^{2d/(d-2)}, since u_{d+1} is the eigenfunction under consideration.
[§2.3, Step 2] The expression "|Σ1|d1" appears to be a typo; it should read |Σ1|_{d-1}.
[§2.3, support bullet] The sentence "when x ∉ U_d then x_i > 0 for all i" should be corrected to "x_i ≤ 0 for all i"; the sign error is substantive, not merely typographical, because it is the source of the inadmissible trial function.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the eigenvalue bounds are derived from spectral inequalities and volume estimates, not from the survival asymptotics they feed.

full rationale

I walked the derivation chain of Theorem 1. The lower bound (Section 2.2) starts from the Yamabe inequality (7), applies it to an eigenfunction of U_{d+1}, and uses the explicit volume Vol(U_{d+1})=(1-2^{-d})omega_d; no parameter is fitted to reproduce the target asymptotic. The upper bound (Section 2.3) is a Rayleigh-quotient argument with the trial function eta_d and the volume estimate of Lemma 3; the factor 2^{-d} enters through the quantitative volume bound (10), not through the statement being proved. Corollary 2 merely rewrites the known relation (2)-(3) after Theorem 1 is established, and (2) from [11] and [1] motivates p_d but is not used as an input in the spectral proof. The self-citations present in the bibliography (e.g. [4] and [14], which include one of the authors) are contextual and not load-bearing for any step of the proof. I therefore find no step that reduces by construction to its own inputs. For clarity, this verdict concerns circularity only: there is a separate mathematical correctness issue in Section 2.3, where the claim that eta_d is supported in U_d because 'when x not in U_d then x_i > 0 for all i' is reversed, since U_d is the complement of the negative orthant; that is a potential flaw in the upper-bound proof, but it is not a self-referential or fitted derivation.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard spectral-geometry results (Yamabe inequality, variational eigenvalue characterization) and on the prior survival asymptotics (2). No free parameters are fitted; no new entities are postulated. The main burden is the validity of the test-function construction, which is not captured here as an axiom but is flagged under weakest_assumption and red flags.

assumptions (3)

standard math Yamabe inequality on the sphere: the constant function minimizes the Yamabe functional
Used in Section 2.2 to derive the lower bound on lambda_1(d+1).
standard math Variational characterization of the principal Dirichlet eigenvalue of a spherical domain
Used in Section 2.3 to upper-bound lambda_1(d) by the Rayleigh quotient Y(eta_d).
domain assumption Asymptotic survival probability P_x(tau_d > t) ~ V_d(x) / t^{p_d/2} (DeBlassie; Banuelos-Smits)
Cites equation (2) to connect the eigenvalue to the first-passage exponent; not needed for the eigenvalue theorem itself.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On the first hitting time of a high-dimensional orthant." pith.science (2026). https://pith.science/paper/SG4RNDZ2

@misc{pith2026241117023,
  author       = {Pith},
  title        = {Pith review of: On the first hitting time of a high-dimensional orthant},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SG4RNDZ2}},
  note         = {Machine review of arXiv:2411.17023}
}

read the original abstract

We consider a collection of independent standard Brownian particles (or random walks), starting from a configuration where at least one particle is positive, and study the first time they all become negative. This is clearly equivalent to studying the first hitting time from the negative orthant or the first exit time from the complement of the negative orthant. While it turns out to be possible to compute the distribution of these hitting times for one and two particles, the distribution (and even its tail asymptotics) is not known in closed form for three or more particles. In this paper we study the tail asymptotics of the distribution as the number of particles tends to infinity. Our main techniques come from spectral geometry: we prove new asymptotic estimates for the principal eigenvalue of the complement of a high-dimensional orthant, which we believe are of independent interest.

Figures

Figures reproduced from arXiv: 2411.17023 by the authors.

**Figure 1.** Random walks staying in the complement of the d-dimensional orthant, for d = 1, 2, 3 where Vd(x) is the unique (up to multiplicative scaling) harmonic function in the cone C = R d \ R d − (the complement of the negative orthant) and the exponent pd is given by pd = q λ1 + [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 18 canonical work pages

[1]

Bañuelos and R

R. Bañuelos and R. G. Smits , Brownian motion in cones, Probab. Theory Relat. Fields, 108 (1997), pp. 299– 319

work page 1997
[2]

Ben-Naim and P

E. Ben-Naim and P. L. Krapivsky , First-passage exponents of multiple random walks, J. Phys. A, Math. Theor., 43 (2010), p. 16. Id/No 495008

work page 2010
[3]

, Kinetics of first passage in a cone, J. Phys. A, Math. Theor., 43 (2010), p. 14. Id/No 495007

work page 2010
[4]

Bogosel, V

B. Bogosel, V. Perrollaz, K. Raschel, and A. Trotignon , 3d positive lattice walks and spherical trian- gles, J. Comb. Theory, Ser. A, 172 (2020), p. 47. Id/No 105189

2020
[5]

Bousquet-Mélou, Square lattice walks avoiding a quadrant, J

M. Bousquet-Mélou, Square lattice walks avoiding a quadrant, J. Comb. Theory, Ser. A, 144 (2016), pp. 37–79

work page 2016
[6]

, Enumeration of three-quadrant walks via invariants: some diagonally symmetric models, Can. J. Math., 75 (2023), pp. 1566–1632

work page 2023
[7]

Bousquet-Mélou and M

M. Bousquet-Mélou and M. W allner , Walks avoiding a quadrant and the reflection principle, Eur. J. Comb., 119 (2024), p. 43. Id/No 103803

work page 2024
[8]

Bramson and D

M. Bramson and D. Griffeath , Capture Problems For Coupled Random Walks, Birkhäuser Boston, Boston, MA, 1991, pp. 153–188

work page 1991

Show all 19 references

[9]

A. J. Bray, S. N. Majumdar, and G. Schehr , Persistence and first-passage properties in nonequilibrium systems, Advances in Physics, 62 (2013), pp. 225–361

2013
[10]

Dahne and B

J. Dahne and B. Sal vy , Computation of tight enclosures for Laplacian eigenvalues, SIAM J. Sci. Comput., 42 (2020), pp. a3210–a3232

2020
[11]

R. D. DeBlassie , Exit times from cones inRn of Brownian motion, Probab. Theory Relat. Fields, 74 (1986), pp. 1–29

1986
[12]

Denisov and V

D. Denisov and V. W achtel , Random walks in cones, Ann. Probab., 43 (2015), pp. 992–1044

2015
[13]

Duraj, Random walks in cones: the case of nonzero drift, Stochastic Processes Appl., 124 (2014), pp

J. Duraj, Random walks in cones: the case of nonzero drift, Stochastic Processes Appl., 124 (2014), pp. 1503– 1518

2014
[14]

Garbit and K

R. Garbit and K. Raschel , On the exit time from a cone for random walks with drift, Rev. Mat. Iberoam., 32 (2016), pp. 511–532

2016
[15]

Kabluchko and A

Z. Kabluchko and A. Marynych , Random Walks in the High-Dimensional Limit I: The Wiener Spiral. Preprint, arXiv:2211.08538 [math.PR] (2022), 2022

2022 arXiv
[16]

Kesten, An absorption problem for several Brownian motions, in Seminar on stochastic processes, held at the University of California, Los Angeles, CA, USA, from March 23-25, 1991

H. Kesten, An absorption problem for several Brownian motions, in Seminar on stochastic processes, held at the University of California, Los Angeles, CA, USA, from March 23-25, 1991. Dedicated to the memory of Steven Orey, Boston, MA etc.: Birkhäuser, 1992, pp. 59–72

1991
[17]

J. M. Lee and T. H. Parker , The Yamabe problem, Bull. Am. Math. Soc., New Ser., 17 (1987), pp. 37–91

1987
[18]

Meyre and W

T. Meyre and W. Werner , On the occupation times of cones by Brownian motion, Probab. Theory Related Fields, 101 (1995), pp. 409–419

1995
[19]

Mustapha, Non-D-finite walks in a three-quadrant cone, Ann

S. Mustapha, Non-D-finite walks in a three-quadrant cone, Ann. Comb., 23 (2019), pp. 143–158. Université de Tours, Institut Denis Poisson, Tours, France. Email: emmanuel.humbert@univ-tours.fr CNRS, International Research Laboratory France-Vietnam in mathematics and its applica...

2019

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Bañuelos and R

R. Bañuelos and R. G. Smits , Brownian motion in cones, Probab. Theory Relat. Fields, 108 (1997), pp. 299– 319

work page 1997

[2] [2]

Ben-Naim and P

E. Ben-Naim and P. L. Krapivsky , First-passage exponents of multiple random walks, J. Phys. A, Math. Theor., 43 (2010), p. 16. Id/No 495008

work page 2010

[3] [3]

, Kinetics of first passage in a cone, J. Phys. A, Math. Theor., 43 (2010), p. 14. Id/No 495007

work page 2010

[4] [4]

Bogosel, V

B. Bogosel, V. Perrollaz, K. Raschel, and A. Trotignon , 3d positive lattice walks and spherical trian- gles, J. Comb. Theory, Ser. A, 172 (2020), p. 47. Id/No 105189

2020

[5] [5]

Bousquet-Mélou, Square lattice walks avoiding a quadrant, J

M. Bousquet-Mélou, Square lattice walks avoiding a quadrant, J. Comb. Theory, Ser. A, 144 (2016), pp. 37–79

work page 2016

[6] [6]

, Enumeration of three-quadrant walks via invariants: some diagonally symmetric models, Can. J. Math., 75 (2023), pp. 1566–1632

work page 2023

[7] [7]

Bousquet-Mélou and M

M. Bousquet-Mélou and M. W allner , Walks avoiding a quadrant and the reflection principle, Eur. J. Comb., 119 (2024), p. 43. Id/No 103803

work page 2024

[8] [8]

Bramson and D

M. Bramson and D. Griffeath , Capture Problems For Coupled Random Walks, Birkhäuser Boston, Boston, MA, 1991, pp. 153–188

work page 1991