{"id":"3ba63479-13a4-46d7-bbfa-d7df5ad4e469","arxiv_id":"2506.02982","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A corrected and extended proof that bounded discrete bridges obey a Rayleigh height distribution, including the periodic case.","lead":"This paper repairs a flawed 2010 proof that the height of a random discrete bridge follows a Rayleigh limit law, and extends the result to periodic walks. It also derives higher-order asymptotic expansions for Lukasiewicz bridges in terms of Hermite polynomials.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2's small-circle estimate is invalid for non-integer h: the Γr integral of z^{-n+h} does not vanish as the gap shrinks, so Theorem 3 is not proved for the stated range.","rationale":"The reader's verdict is CONDITIONAL, and our analysis also supports CONDITIONAL: the Rayleigh limit itself is known to be true and the manuscript contains the ingredients for a repair, but the proof as written has a load-bearing gap. The reader identified Lemma 4 and branch handling as the weakest assumptions; we find a more specific and more central flaw in Lemma 2 of the aperiodic section. Lemma 2 is the step that justifies replacing Bh by its dominant-root asymptotic expression on the contour, and its claimed o(r^n) bound for Γr relies on an analytic continuation that fails for non-integer h because of the branch point at 0. The integer case, which is what is needed for actual bridges, is deferred to Section 4.5.1 and may be salvageable, but Theorem 3 is stated for all x∈(0,∞) without the integer restriction and Section 4.5.2 explicitly omits the non-integer periodic proof. A concrete numerical or analytic check of the model integral z^{-n+h} settles the issue. Because the central claim is likely true and the gap is fixable by an integer-height argument, we do not recommend REJECT; the verdict remains CONDITIONAL, pending repair of the non-integer small-circle estimate.","tokens_in":29584,"tokens_out":29135,"duration_ms":316425,"concrete_test":"Evaluate analytically J(s) = ∫_{s/r}^{2π-s/r} (r e^{iν})^{h-n} i r e^{iν} dν for a non-integer h, e.g. n=64, h=8.5, r=0.01, and take s/r→0. The limit is (e^{2π i h}-1)/(i(h-n)) · r^{h-n+1}, which is not o(r^n). If this integral is computed numerically for s/r=10^{-250}, it will match this nonzero O(r^{h-n+1}) value, contradicting the estimate in Section 3.4 that the Γr contribution is o(r^n).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The core of the aperiodic proof is Lemma 2, which discards the integral of Bh(z)/z^{n+1} on the almost-closed small circle Γr. In Section 3.4, Equations (37)-(42), the proof reduces each term to Jjk = ∫_{Γr} z^{-n+h} z^α dz and claims this is o(r^n) by replacing (1-t/n)^{-n} with e^t and integrating an entire function. This replacement is not legitimate for the non-integer case that Section 3.4 explicitly treats. For h = xσ√n, z^h has a branch point at 0; the contour Γr is the circle |z|=r with an arc of half-angle s/r removed around the positive real axis. As s/r→0, the values of z^h at the two endpoints differ by the phase e^{2π i h}, which is not 1 for generic non-integer h. The integral over the almost-closed circle therefore tends to (e^{2π i h}-1)/(i(-n+h+α+1)) · r^{1-n+h+α}, which is O(r^{1-n+h+α}) — enormously larger than o(r^n), not smaller. The proof in Equations (38)-(41) computes ∫ e^t t^k dt as if the integrand were entire; this changes the analytic type and drops the branch phase. Thus Lemma 2 is false as stated for non-integer heights. The later sentence in Section 3.4 referring the integer case to Section 4.5.1 does not repair the gap in the theorem as stated for all x∈(0,∞), because Section 4.5.1 only treats integer h and the non-integer periodic case is explicitly omitted in Section 4.5.2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to give a rigorous proof, for aperiodic and for periodic directed lattice walks whose characteristic polynomial has no repeated factor, that the probability that a bridge of length n exceeds height h = xσ√n converges to the Rayleigh tail e^{-2x^2ρ/τ^2} with an explicit O(1/√n) error. The proof strategy is to avoid the invalid disk-wide root-domination estimate used by Banderier and Nicodème by integrating on a thin contour around the positive real segment, then applying singularity analysis and Hankel integrals. The paper also gives higher-order expansions for Łukasiewicz bridges involving Hermite polynomials, a numerical check against the André reflection principle for Dyck bridges, and a conjecture on tightness of arch extrema.","tokens_in":29914,"tokens_out":13701,"duration_ms":134060,"significance":"If the main theorems were established, the paper would correct a genuine gap in the 2010 proof of Banderier and Nicodème, extend the result to periodic walks, and provide a useful, checkable expansion machinery for Łukasiewicz bridges. The numerical verification against the exact André formula for Dyck bridges is a definite strength, as is the careful identification of the Wallner counterexample to disk-wide domination. However, the central proof as written fails for non-integer heights: the key estimate Lemma 2 is false in the stated generality, and the periodic non-integer case is explicitly left unfinished. The Rayleigh law itself is a known result, so the paper's contribution is the analytic proof and its refinements, not the discovery of the limit law.","major_comments":[{"comment":"For non-integer h, the estimate Jjk = o(r^n) is false. Since h = xσ√n is not an integer for generic x, the integrand z^{-n+h+α} has a branch point at z = 0. As the gap in Γr closes, ∫_{Γr} z^{-n+h+α} dz tends to (e^{2π i h} - 1)/(i(-n+h+α+1)) r^{1-n+h+α}, which is not o(r^n); because h ∼ √n, the right-hand side is much larger than r^n as r → 0. The manipulations in Eqs (38)–(41) integrate the truncated expansion (1 - t/n)^h as if it were a single-valued entire function on the whole contour, which is only legitimate when h is an integer. Consequently Lemma 2 and Eq (42) do not hold in the stated generality, and Theorem 3 is not proved for all x ∈ (0,∞).","section":"§3.4, Lemma 2 and Eqs (37)–(42)"},{"comment":"The non-integer periodic case is not proved: the text states \"We omit the end of the proof that follows the same steps as in the aperiodic case.\" Because the aperiodic steps fail for non-integer h (see previous comment), this omission cannot be repaired by copying them. The same branch-point obstruction applies to each arc Γr,ℓ in Lemma 6. Hence Theorem 6 is not established for x ∈ (0,∞).","section":"§4.5.2"},{"comment":"Equation (54) is missing a factor 1/τ. Theorem 1 gives V_n = ρ^{-n}/(τ σ √(2πρ n)) (1 + O(1/n)), not ρ^{-n}/(σ √(2πρ n)). With the displayed formula, the ratio I0 / b_n^{<∞} in Eqs (56)–(58) would acquire an extra factor τ and would not equal e^{-2x^2ρ/τ^2}. The authors should correct Eq (54) and recheck the subsequent ratio computation.","section":"§3.5, Eq (54)"},{"comment":"The proof of Lemma 4 is too compressed to be verifiable. In the rational case y = g + x/p, the assertions \"There exists an integer k and m = kx ≤ δ such that P(χ_m) = τ\" and \"This implies that q is a period of P(u)\" are not justified, and the step from a finite set K to a smallest argument 2π/q with q > p needs a clearer argument. Since Lemma 4 underpins the periodic saddle-point analysis and the claim that no other singularities lie in |z| ≤ ρ, the periodic theorem requires a complete proof of this lemma.","section":"§4.1, Lemma 4"}],"minor_comments":[{"comment":"The inequality in Theorem 5 is reversed: the KMT theorem bounds the probability that the maximum deviation exceeds C log n + x, so the left-hand side should be Pr(max |S_k - B_k| > C log n + x).","section":"§3.6, Eq (61)"},{"comment":"The error term O(Â^n) should be O(Â^h) (or O(e^{-c h})) since h = Θ(√n); as written, the claim is stronger than what the domination bound actually gives.","section":"§3.4, Eq (24)"},{"comment":"The caption writes \"s = o(r^4 r)\", which appears to be a typo; Lemma 2 uses s = o(r^{3n}) and Eq (44) uses s = o(r^{4n}).","section":"Figure 3 caption"},{"comment":"The notation alternates between B(z) and B_h(z) without consistency (e.g., Eq (45) defines B_h(z) while Lemma 6 integrates B(z)); please unify the notation.","section":"§4.5.2 and Lemma 6"},{"comment":"The notation He'1_k is undefined and appears to be a typo for He1_k; the recurrence and the projection-to-1 notation should be clarified.","section":"§5.2.2, Eq (107)"},{"comment":"The Maple worksheet output is raw and hard to read; consider presenting the numerical comparison in a table and moving the worksheet to supplementary material.","section":"Figure 5"},{"comment":"Reference [13] is a personal communication; since the Wallner counterexample motivates the paper, it would help to describe it in the text with explicit numerical parameters.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript addresses a real gap in the literature and contains useful material, but the main theorems are not proved as stated. I recommend major revision rather than rejection because the integer-height case appears sound, and a standard monotonicity/floor argument could plausibly extend it to all x ∈ (0,∞); however, the authors must remove or correct the false non-integer Lemma 2, complete the omitted periodic non-integer proof, and fix the factor 1/τ in Eq (54) before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper is worth taking seriously. It openly corrects the known flaw in Banderier-Nicodeme 2010, replaces the invalid disk-wide domination argument with a thin contour around the real segment, and adds a periodic-case Rayleigh theorem and higher-order Lukasiewicz expansions with Hermite polynomials. Those are real contributions, and the numerical check against the Andre reflection principle is a legitimate sanity check. The author is honest about what is inherited and what is new.\n\nThe soft spots are concentrated in the proof of Lemma 2 for non-integer h. The claim that the Gamma_r integral of z^{-n+h} z^alpha is o(r^n) does not survive the almost-closed contour: for generic non-integer h, the two endpoints of the gap differ by the phase e^{2pi i h}, and the integral is of order r^{1-n+h+alpha}, which is not smaller than r^n. The proof's replacement of (1-t/n)^{-n} by e^t and integration as if the integrand were entire hides exactly this branch issue. Since the theorem states x in (0,infty), non-integer h is the generic case; pointing to the integer-h proof in Section 4.5.1 does not cover it. The periodic non-integer case is explicitly left unfinished, so Lemma 6 inherits the same problem.\n\nSmaller issues are easy to list: Equation (54) is missing the 1/tau factor, the KMT inequality in Theorem 5 has the direction reversed, and Lemma 4 is compressed enough that the maximum claim on the circle |u|=tau needs a fuller proof. None of these are fatal by themselves, but together with the Lemma 2 gap they mean the manuscript is not yet a complete rigorous proof of the stated theorems.\n\nFor whom is this useful? Specialists in lattice path asymptotics will want to know about the periodic treatment and the Hermite structure, and the paper is a good map of where the 2010 proof went wrong. I would not cite the current version for the main theorem until the non-integer h argument is fixed or the theorem is narrowed to integer h. I would still send it to a serious referee: the question is important, the author is engaging the right difficulties, and the periodic and expansion results deserve a careful check.","headline":"A genuine repair attempt with a new periodic case and useful expanded asymptotics, but the non-integer height proof has a branch-cut gap that blocks the main theorem as stated.","tokens_in":30445,"tokens_out":11066,"would_cite":false,"duration_ms":124155,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F05","60G50","05A15","05A16"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves the Rayleigh limit law for the height of discrete bridges, correcting a flawed 2010 proof and extending the result to periodic walks.","keywords":["discrete bridges","Rayleigh limit law","periodic walks","kernel method","Hankel integrals","semi-large powers","Hermite polynomials","Lukasiewicz bridges"],"falsifier":"One concrete check: for a reduced periodic walk such as $P(u)=u^9+u^3+u^{-3}$, evaluate $|1/P(u)|$ numerically on the circle $|u|=\\tau$; any maximum point other than $\\kappa_\\ell\\tau=\\tau e^{2\\pi i\\ell/p}$ refutes Lemma 4 and with it Theorem 6. A second check targets the aperiodic contour: for the paper's own counter-example $P(u)=u+3/u+1/u^2$, compute the integral of $B_h(z)/z^{n+1}$ along the small arc $\\Gamma_r$; if it is not $o(r^n)$ as $r\\to0$ with $s=o(r^{3n})$, the key asymptotic simplification does not hold.","tokens_in":29341,"feed_emoji":"🌉","tokens_out":12308,"duration_ms":97853,"temperature":0.7,"pith_summary":"This paper proves a limit law for the upper tail of the height of discrete bridges: for a directed lattice walk whose characteristic Laurent polynomial has no repeated factor, the probability that a bridge of length $n$ ever reaches above $x\\sigma\\sqrt{n}$ tends to $e^{-2x^2\\rho/\\tau^2}$ as $n\\to\\infty$, with error term $O(1/\\sqrt{n})$. The target is to make rigorous a result announced in 2010, whose published proof used a root-domination property on a disk where the property can fail. The new argument applies the domination property only on a thin lens contour around the positive real axis, and for periodic walks around the $p$ rotated rays, then passes to Hankel contours at the dominant singularity. If the proof is right, the Rayleigh law is settled for both aperiodic and periodic walks, and the higher-order expansions for Lukasiewicz bridges acquire a systematic Hermite-polynomial structure.","feed_headline":"Bridge heights obey Rayleigh limit law, now proven","feed_subtitle":"A corrected analytic-combinatorics proof covers aperiodic and periodic walks, with error rate one over the square root of n.","key_machinery":"The machine is the kernel equation $1-zP(u)=0$ with $d$ large roots $v_k(z)$ and $c$ small roots $u_j(z)$. The generating function of bridges above height $h$ is expressed as\n$$B_h(z)=z\\sum_{j=1}^{c}\\sum_{k=1}^{d}\\left(\\frac{u_j(z)}{v_k(z)}\\right)^h\\frac{Q_k(u_j(z))}{Q_k(v_k(z))}\\frac{u'_j(z)}{v_k(z)},$$\nand the load is to extract $[z^n]B_h(z)$. The device that repairs the earlier proof is a contour shaped like a thin lens around the interval $]0,\\rho[$ (and, in the periodic case, around the $p$ rotated copies), so that the domination inequalities $|u_j(z)|<u_1(z)<v_1(z)<|v_k(z)|$ hold on the integration paths by continuity even though they fail on the full disk $|z|<\\rho$. The contour is then shrunk, the contributions away from the real axis are shown to be $o(r^n)$, and the dominant piece is evaluated by a Hankel contour at $z=\\rho$ with the semi-large-powers scale $h=x\\sigma\\sqrt n$; the resulting Gamma integral evaluates to $e^{-2x^2\\rho/\\tau^2}$.","core_discovery":"The paper's central claim, stated as Theorems 3 and 6, is that under Assumption 1 the bridge-height tail satisfies\n$$\\$beta_n^{{>x\\sigma\\sqrt{n}}$} = \\frac{$b_n^{{>h}}$}{$b_n^{{<\\infty}}$} = $e^{{-2x^2\\rho/\\tau^2}}$\\left(1+O\\left(\\frac1{\\sqrt n}\\right)\\right), \\qquad x>0,$$\nwhere $\\tau$ is the unique positive solution of $P'(\\tau)=0$, $\\rho=1/P(\\tau)$, and $\\sigma=\\sqrt{P''(\\tau)}$. For walks with fundamental period $p$, bridges of length $n$ exist only when $p$ divides $n$, and the same ratio holds: the $p$ symmetric saddle-point contributions multiply both numerator and denominator by $p$ and cancel. The law is independent of the walk's drift and of $\\sigma$, so in the zero-drift probabilistic case $P(1)=1$, $P'(1)=0$ it reduces to the familiar $e^{-2x^2}$ Rayleigh tail.","pith_inferences":["The square-free Assumption 1 looks like a proof artefact rather than a property of the limit: powers such as $P(u)=((u+1/u)/2)^k$ have repeated factors but correspond to reducible periodic walks, and a natural stress test is whether a modified saddle argument yields the same $e^{-2x^2\\rho/\\tau^2}$ tail for them.","Since the limit depends only on $\\rho/\\tau^2$, walks with identical $\\tau$ and $\\rho$ but different jump sets are predicted to be indistinguishable at leading order; the Hermite correction terms give a concrete $1/\\sqrt n$ observable that would separate them, which could be checked by simulation.","Lemma 4 is the compressed point of the periodic proof; replacing it by a direct trigonometric argument based on the period-$p$ structure would make the periodic theorem checkable without continuity heuristics and could extend the method to repeated-factor characteristic polynomials.","The tightness conjecture for arch extrema could be probed on Duchon-style walks $P(u)=u^d+u^{-c}$ by simulating bridges and comparing the distribution of the maximum of positive arches with the Brownian-excursion extreme, expecting the same $O(1/\\sqrt n)$ coupling error."],"forward_implications":["The Rayleigh limit $e^{-2x^2\\rho/\\tau^2}$ is established for the height of aperiodic directed-lattice bridges, with explicit error $O(1/\\sqrt n)$, closing the gap left by the 2010 proof.","Periodic walks, not treated in 2010, obey the same law: only lengths $n=mp$ contribute, the $p$ saddle points multiply both numerator and denominator, and the ratio still converges to $e^{-2x^2\\rho/\\tau^2}$.","Because the limit is independent of drift and of $\\sigma$, all aperiodic walks with the same ratio $\\rho/\\tau^2$ share the same Brownian tail; zero-drift walks with $P(1)=1$ have limit simply $e^{-2x^2}$.","For Lukasiewicz bridges the method yields explicit higher-order corrections whose coefficients are probabilists' Hermite polynomials $\\mathrm{He}_i(4x)$; the numerical test against the reflection principle for the $\\pm1$ walk agrees to about $2\\times10^{-8}$ at $n=64$, $h=9$.","The $O(1/\\sqrt n)$ proximity between discrete and Brownian first-passage points supports the paper's conjecture that the highest and lowest points of long positive and negative arches of discrete bridges are tight to the Brownian limit."],"supporting_citations":[{"why":"Supplies the kernel method, the root-domination lemma for the real segment, and the saddle-point expansion of unbounded bridges on which the Rayleigh computation is built.","marker":"[1]"},{"why":"The 2010 proof whose disk-domination simplification this paper identifies as erroneous and corrects; also the source of the Lukasiewicz Hermite-expansion approach.","marker":"[3]"},{"why":"Provides singularity analysis, Hankel contours, the semi-large powers theorem, and the Gamma-function integral representation used to evaluate the dominant term.","marker":"[7]"},{"why":"Supplies the coalescing-saddle and semi-large-powers argument followed for the dominant Hankel integral $I_0$.","marker":"[2]"},{"why":"Provides the counter-example showing root domination fails inside the full disk $|z|<\\rho$, prompting the lens-contour design.","marker":"[13]"},{"why":"Supplies the strong embedding theorem used in Section 3.6 to position the discrete bridge's first-passage point tight to the Brownian limit.","marker":"[9]"}],"fun_headline_variants":["Rayleigh law for bridge heights, proof corrected","Bridge height tail proven Rayleigh for all periods","Corrected analytic proof: bridge heights follow Rayleigh","Bridge heights converge to Rayleigh distribution","Periodic and aperiodic bridges share Rayleigh limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the root-domination inequalities survive on the thin contour around the positive real axis and, in the periodic case, that $|1/P(u)|$ peaks on the circle $|u|=\\tau$ only at the $p$ points $\\kappa_\\ell\\tau$; if this fails, the asymptotic simplifications collapse and the Rayleigh theorem does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Rayleigh law for bridge heights, proof corrected","Bridge height tail proven Rayleigh for all periods","Corrected analytic proof: bridge heights follow Rayleigh","Bridge heights converge to Rayleigh distribution","Periodic and aperiodic bridges share Rayleigh limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000229,"raw_usage":{"total_tokens":1461,"prompt_tokens":911,"completion_tokens":550,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":482}},"tokens_in":527,"tokens_out":550,"duration_ms":5539,"temperature":1.0,"reasoning_tokens":482,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:16:39.757509+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete check: for a reduced periodic walk such as $P(u)=u^9+u^3+u^{-3}$, evaluate $|1/P(u)|$ numerically on the circle $|u|=\\tau$; any maximum point other than $\\kappa_\\ell\\tau=\\tau e^{2\\pi i\\ell/p}$ refutes Lemma 4 and with it Theorem 6. A second check targets the aperiodic contour: for the paper's own counter-example $P(u)=u+3/u+1/u^2$, compute the integral of $B_h(z)/z^{n+1}$ along the small arc $\\Gamma_r$; if it is not $o(r^n)$ as $r\\to0$ with $s=o(r^{3n})$, the key asymptotic simplification does not hold.","supporting_citations":[{"cited_title":"Basic analytic combinatorics of directed lattice paths","cited_arxiv_id":null,"evidence_quote":"Supplies the kernel method, the root-domination lemma for the real segment, and the saddle-point expansion of unbounded bridges on which the Rayleigh computation is built."},{"cited_title":"Boundeddiscretewalks","cited_arxiv_id":null,"evidence_quote":"The 2010 proof whose disk-domination simplification this paper identifies as erroneous and corrects; also the source of the Lukasiewicz Hermite-expansion approach."},{"cited_title":"Analytic combinatorics","cited_arxiv_id":null,"evidence_quote":"Provides singularity analysis, Hankel contours, the semi-large powers theorem, and the Gamma-function integral representation used to evaluate the dominant term."},{"cited_title":"Random maps, coalescing saddles, singularity analysis, and Airy phenomena.Random Struct","cited_arxiv_id":null,"evidence_quote":"Supplies the coalescing-saddle and semi-large-powers argument followed for the dominant Hankel integral $I_0$."},{"cited_title":"https://lipn.univ-paris13.fr/~nicodeme/Publications/heightofbridge.mpl","cited_arxiv_id":null,"evidence_quote":"Provides the counter-example showing root domination fails inside the full disk $|z|<\\rho$, prompting the lens-contour design."},{"cited_title":"An approximation of partial sums of independant RV’s and the sample DF","cited_arxiv_id":null,"evidence_quote":"Supplies the strong embedding theorem used in Section 3.6 to position the discrete bridge's first-passage point tight to the Brownian limit."}],"review_version":1}