{"id":"23779c67-22bd-49cf-93c2-132a8ec6ddc1","arxiv_id":"2501.11095","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For random subwords of b^K in an irreducible affine Weyl group, the normalized alcove position converges to a central spherical Gaussian, with an explicit variance formula.","lead":"Randomly deleting letters from a repeated word in an infinite symmetry group yields a random location that becomes bell-curve shaped as the word grows. This paper proves the spread of that bell curve and gives an exact formula for it, linking random words, hyperplane arrangements, and billiard walks.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.2's proof of zero mean for regeneration displacements relies on an almost-sure separation that does not follow from the CLT; since the centered CLT in Theorem 1.4 depends on it, this is the main soft spot.","rationale":"Reading in good faith, the main theorem is a nontrivial and plausible result. The Markov-chain regeneration structure on the finite quotient is the right framework, and the variance formulas in Sections 4-5 are internally consistent. The notational blurring between the affine group W and its finite quotient W, flagged by the reader, is real but not logically load-bearing because the intended state space is explicitly finite and the transition kernel (1) only uses quotient elements. The more substantive issue is the proof of Lemma 3.2, which is the linchpin for the centered CLT. The argument needs to show that the expected displacement between consecutive returns to a quotient state is zero; otherwise the return-time sums have drift and u_K^•/√K diverges or limits to a non-centered law. The proof's contradiction step infers an almost-sure linear separation from the CLT, which is not a valid inference in general. The gap is repairable because the increments have exponential tails (Lemma 3.1) and the renewal index ratio t'/t is asymptotically 1, so an SLLN/LIL version of the argument should work. Because the fix is straightforward and the conclusion is very likely correct, the reader's CONDITIONAL verdict need not change. The lack of code for exceptional-type values is a reproducibility concern but does not threaten Theorem 1.4 itself.","tokens_in":24802,"tokens_out":45316,"duration_ms":464521,"concrete_test":"Independently re-derive the missing step in Lemma 3.2 by writing S_t = Σ_{i≤t} D^1_i and S'_{t'} = Σ_{i≤t'} D^x_i, and prove the claimed almost-sure separation using the strong law of large numbers: S_t/t → E[D^1_1] a.s., S'_{t'}/t' → E[D^x_x] a.s., and t' = min{i : L^x_i ≥ L_t} satisfies t'/t → 1 a.s. by renewal theory. If this re-derivation cannot be completed without new assumptions, the proof of Theorem 1.4 is incomplete; if it goes through, the concern is an exposition gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3's proof of Theorem 1.4 is built on regeneration at returns of the finite-quotient Markov chain. The key step is Lemma 3.2, which must show E[D^1_1]=0 so that the return-time sums have zero drift and the CLT yields a centered normal limit. The proof by contradiction claims that if E[D^1_1] != E[D^x_x] for some x, then 'it follows' from the CLT that almost surely |ξ(u_{L_t}) − ξ(u_{L^x_{t'}})| > υ_x |tE[D^1_1] − t'E[D^x_x]| for all large t,t'. This inference is not valid: the CLT is a distributional statement and does not give simultaneous almost-sure linear separation for all large indices. The missing piece can probably be supplied by the SLLN/LIL using the exponential tails from Lemma 3.1 and the renewal fact that t' ∼ t, but as written the proof has a genuine gap. If E[D^1_1] were nonzero, the regeneration sums would drift linearly and v_p(b^K)^•/√K would not converge to N(0, σ_b^2 I_r); thus Theorem 1.4 depends on this step. This is a more substantive concern than the reader's notation point: the finite-quotient chain itself is clearly intended, but the proof that the cycle increments are centered is not fully justified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the random element v_p(b^K) of an irreducible affine Weyl group W obtained by keeping each letter of the word b^K independently with probability p. The main theorem (Theorem 1.4) asserts that the alcove centroids of v_p(b^K), normalized by sqrt(K), converge in distribution to a centered spherical multivariate normal distribution N(0, sigma_b^2 I_r). The proof uses a Markov chain on the finite quotient W of W by the coroot translation lattice, with regeneration at returns to a fixed quotient state, and proves a CLT for the regenerative increments. The paper then derives closed-form formulas for sigma_b^2 (Theorems 1.7 and 1.8), an asymptotic formula for the expected Coxeter length (Corollary 1.6), a recurrence/transience criterion for the associated random billiard walk (Corollary 1.5), and an explicit table of sigma_b^2 for Coxeter words in all irreducible affine types (Theorem 1.10).","tokens_in":25086,"tokens_out":18652,"duration_ms":199343,"significance":"If the proof gaps identified below are repaired, the paper is a solid contribution to the probabilistic study of Coxeter groups and combinatorial billiards. The regenerative approach is natural, and Theorem 1.7 gives a genuinely computable closed form for the variance. The paper is self-contained and does not fit parameters to data; the spherical form of the limiting covariance is a striking and nontrivial consequence of Schur's lemma. The explicit values for the infinite families in Section 5 are derived by hand and are convincing. However, the proof of the central CLT contains a real gap in Lemma 3.2, the state space of the Markov chain is described ambiguously, and two corollaries rely on unproved uniform-integrability or almost-sure transfer statements. These issues are local and repairable, but they are load-bearing for Theorem 1.4 and its corollaries.","major_comments":[{"comment":"The proof that E[D_1^1] = E[D_x^x] is not valid as written. The sentence beginning 'It follows' asserts that the CLT gives almost sure linear separation of the two regenerative sequences for all large t and t'. A CLT is only a distributional statement and does not imply such simultaneous almost sure separation. The conclusion is recoverable: one should use the SLLN for the i.i.d. regenerative increments, together with the exponential tails from Lemma 3.1 and the renewal fact that t' ~ t when t' is chosen as the index after L_t. Since the zero mean of D_1^1 is exactly what ensures that the regenerative sums have zero drift, Theorem 1.4 depends on this step, so the proof must be repaired.","section":"Section 3, Lemma 3.2"},{"comment":"The chain M is described as having finite state space W, but the random variables u_K are elements of the infinite affine group W. If the state space is meant literally, Lemma 3.1's exponential tail bound for T_w^{w'} is false and the assertion that each L_w is infinite fails for r >= 3, where the walk is transient by Corollary 1.5. The argument only works for the quotient chain \\bar u_K = u_K mod Lambda with state space the finite Weyl group W, with L_w and D_w^{w'} defined for returns of \\bar u_K to w in W. Please make this quotient explicit, redefine the notation accordingly, and restate Lemma 3.1 and relation (3) in that setting.","section":"Section 3, Markov chain setup and Lemma 3.1"},{"comment":"The sentence 'It follows from Theorem 1.4 that lim_{K->0} ||u_K^bullet||/K = 0 almost surely' is incorrect as stated: convergence in distribution of u_K^bullet/sqrt(K) to a normal law does not imply almost sure convergence of u_K^bullet/K to zero, and the limit index should be K -> infinity. The almost sure zero-drift statement can be obtained from the regenerative structure and the SLLN after Lemma 3.2 is repaired, but it needs to be proved before applying [Ale02, Corollary 1.17].","section":"Section 3, proof of Corollary 1.5"},{"comment":"The passage from convergence in distribution of <v_p(b^K)^bullet/sqrt(K), beta> to the asserted limit of E[|<v_p(b^K)^bullet/sqrt(K), beta>|] requires uniform integrability, and none is supplied. Convergence in distribution alone does not justify the interchange of limit and expectation of an unbounded function like |x|. Please add a uniform-integrability or tail argument, using the exponential tails available from the regenerative structure, or the proof of the expected length asymptotic is incomplete.","section":"Section 3, proof of Corollary 1.6"},{"comment":"The exceptional-type values in Theorem 1.10 are asserted with the phrase 'Using Theorem 1.8 and a computer', but no code, explicit intermediate quantities, or verifiable algebraic derivation is provided for E6, E7, E8, and F4. Since these four rows of Table 1 are part of a stated theorem, please include the computer code or enough detailed computation to make the results reproducible, or explicitly mark these entries as computer-assisted with a reproducibility statement.","section":"Sections 5.5-5.8 and Table 1"}],"minor_comments":[{"comment":"The indexing in equation (1) uses [0,m-1] while [m] was defined as {1,...,m}; please align the notation.","section":"Section 3, equation (1)"},{"comment":"The displayed limit 'lim_{K->0}' should read 'lim_{K->\\infty}'.","section":"Section 3, proof of Corollary 1.5"},{"comment":"The observation P[|L_{h(K)} - K| > (log K)^2] = o(1) is stated without proof; a one-line renewal-theoretic justification or a citation would help the reader.","section":"Section 3, Lemma 3.3"},{"comment":"The first estimate, P[||u_K^bullet - xi(u_{L_{h(K)}})|| > (log K)^2] = o(1), is asserted without derivation; it follows from a residual renewal time bound and the bounded step displacements, but this should be spelled out.","section":"Section 3, Lemma 3.4"},{"comment":"In the printed table, the rows for \\tilde B_r and \\tilde C_r appear to contain an extra factor r inside or outside the square root when compared with Corollary 1.6; please verify these entries against the formula in Corollary 1.6.","section":"Table 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is well organized and the results are likely correct after the regenerative CLT proof is fixed. The main issues are the ambiguous state space in Section 3, the unjustified almost-sure and uniform-integrability transfer steps, and the lack of reproducibility for the exceptional-type computer computations. None of these appears to be fatal, but they do require careful revision before the paper can be accepted. The self-citations to Def24A and Def24B are contextual and do not affect the central proofs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plain take: this is a real result, and the main theorem is probably right, but the written proof of Lemma 3.2 has a genuine gap that an editor should not wave through.\n\nThe paper's new contribution is a CLT for random subwords in irreducible affine Weyl groups, where the product is the ordinary group product rather than the Demazure or pipe-dream settings. The regeneration method on the finite quotient Weyl group is the right tool, and the explicit variance formulas in Theorems 1.7 and 1.8 are genuinely useful. The Coxeter-word table in Theorem 1.10 is impressive: the infinite families come with complete derivations, and the A_r case includes a nontrivial cyclic-rotation argument. The expected-length corollary is clean, and the billiard interpretation is a nice way to package the model. Overall this is a substantive advance, not a restatement.\n\nNow the soft spots, in order of importance. First, Lemma 3.2: the proof that E[D^1_1]=E[D^x_x] relies on the claim that the CLT implies, almost surely, that the two regeneration sequences are linearly separated for all large indices. That inference does not follow from the CLT; it is a distributional statement and says nothing about almost-sure separation. The gap is repairable: with the exponential tails from Lemma 3.1, an SLLN/LIL argument plus the renewal fact t' ≈ t should give the needed separation. But as written, this is a real hole in a step the centered CLT depends on. I would not call the theorem false—I suspect it is true—but the proof needs revision.\n\nSecond, the E6/E7/E8/F4 entries in Theorem 1.10 are stated as computer results with no code or detailed derivation. That is a reproducibility gap, though minor given the one-line check from Theorem 1.8.\n\nThird, the notation in Section 3 calls the quotient chain's state space W, the same symbol as the infinite affine group. This is confusing but not a mathematical flaw; the finite quotient is clearly intended and the uniform stationary distribution is fine.\n\nAudience: probabilistic combinatorialists, Coxeter theorists, and anyone working on random walks on groups. This paper deserves a serious referee. My recommendation: send it out, and have the referees insist that Lemma 3.2 be fixed and the exceptional-type computations be made reproducible. If those land, this is a solid accept.","headline":"Strong, genuinely new CLT for random subwords in affine Weyl groups, but Lemma 3.2 has a repairable gap and the exceptional-type table needs code.","tokens_in":25622,"tokens_out":3950,"would_cite":true,"duration_ms":43676,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F05","20F55","60B15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Random subwords of repeated words in affine Weyl groups converge to a spherical Gaussian law.","keywords":["random subwords","affine Weyl groups","billiard walks","Coxeter length","multivariate normal distribution","central limit theorem","random walks on groups","alcoves"],"falsifier":"For $W=\\widetilde A_2$, $\\mathsf{b}=s_1s_2s_0$, and $p=1/2$, simulate many independent copies of $v_p(\\mathsf{b}^K)$ for, say, $K=10^4$, and compute the sample covariance of $v_p(\\mathsf{b}^K)^\\bullet/\\sqrt{K}$. The theorem predicts convergence to $\\frac{2}{r(r+1)}\\frac{p}{1-p}I_2=\\frac{1}{3}I_2$; a significantly anisotropic covariance or a different scale would falsify it. A direct check, feasible for small rank, is to enumerate all paths in the finite quotient $\\overline W$ to compute $\\operatorname{Tr}(\\operatorname{Cov}(D^1_1))$ and compare with the paper's formula.","tokens_in":24599,"feed_emoji":"🎱","tokens_out":11780,"duration_ms":113104,"temperature":0.7,"pith_summary":"Take a fixed finite word $\\mathsf{b}$ made from the simple reflections of an irreducible affine Weyl group, and form $\\mathsf{b}^K$ by repeating it $K$ times. Delete each letter independently with probability $1-p$; the surviving letters represent a random group element $v_p(\\mathsf{b}^K)$, which can be viewed geometrically as a random alcove in the Coxeter arrangement. The paper proves that as $K\\to\\infty$, the position of this alcove divided by $\\sqrt{K}$ converges in distribution to a multivariate normal law with mean zero and covariance $\\sigma_{\\mathsf{b}}^2 I_r$, so the limit is a central spherical Gaussian. The positive constant $\\sigma_{\\mathsf{b}}$ depends only on $\\mathsf{b}$ and $p$, and the paper gives an explicit formula for it, with a much simpler formula when $\\mathsf{b}$ uses the affine simple reflection $s_0$ exactly once. As a corollary, the expected Coxeter length grows like an explicit constant times $\\sqrt{K}$; in type $\\widetilde A_r$ with $\\mathsf{b}$ a word using each simple reflection once, the constant is $\\sqrt{\\frac{2}{\\pi}r(r+1)\\frac{p}{1-p}}$.","feed_headline":"Random subwords make affine alcoves Gaussian","feed_subtitle":"Each repetition scales to a spherical normal law; the paper gives the exact variance and the expected Coxeter length.","key_machinery":"The carrying object is the finite-quotient Markov chain on $\\overline W=W/\\Lambda$, where $\\Lambda$ is the normal subgroup of translations by coroot vectors; the state is the image of $u_K$ in $\\overline W$. Between successive visits to a fixed state $w$, the walker accumulates a displacement $D^{w'}_w$ in the coroot lattice, and these displacements are i.i.d. across regeneration times, with exponential tails. The variance $\\sigma_{\\mathsf{b}}^2$ equals $\\frac{1}{r|\\overline W|}\\operatorname{Tr}(\\operatorname{Cov}(D^1_1))$, and the paper evaluates this by a subword summation: for each subset $J\\subseteq[m]$ with weight $p^{|J|}(1-p)^{m-|J|}$, one forms the vector $\\nu_J$ and the operator $R_{\\mathsf{b}}=\\prod R_{s_{i_j}}$ with $R_s=(1-p)I_V+pP_s$, and Theorem 1.7 expresses $\\sigma_{\\mathsf{b}}^2$ as a rational function in $p$ built from $\\nu_J$, $P_{s_J}\\kappa_{\\mathsf{b}}$, and $\\chi_{\\mathsf{b}}$. The invertibility of $I_V-R_{\\mathsf{b}}$ (Lemma 4.2) is what makes the formula well-defined.","core_discovery":"The central claim is Theorem 1.4: if $\\mathsf{b}$ is a finite word over the simple reflections of an irreducible affine Weyl group $W$ that contains every simple reflection at least once, then $v_p(\\mathsf{b}^K)^\\bullet/\\sqrt{K}$ converges in distribution to $N(0,\\sigma_{\\mathsf{b}}^2 I_r)$ as $K\\to\\infty$. The proof works with the Markov chain $u_K=v(K)\\cdots v(1)$ formed from independent copies of $v_p(\\mathsf{b})$, but projected to the finite quotient $\\overline W=W/\\Lambda$ by the coroot translation lattice; on this finite quotient the chain is irreducible and aperiodic, with uniform stationary distribution. Regenerating at returns to the identity gives i.i.d. displacement blocks $D^{w'}_w$ with exponential tails, so the classical central limit theorem applies; the limiting covariance is shown to commute with the full finite Weyl group action, forcing it to be $\\sigma_{\\mathsf{b}}^2 I_r$ by Schur's lemma. The same setup yields an explicit summation formula for $\\sigma_{\\mathsf{b}}^2$ (Theorem 1.7), a simplified version when $s_0$ appears once (Theorem 1.8), and explicit constants for every Coxeter word in every irreducible affine type (Theorem 1.10).","pith_inferences":["The regeneration-block argument is strong enough to yield a functional central limit theorem for the entire trajectory $K\\mapsto v_p(\\mathsf{b}^{\\lfloor K\\rfloor})^\\bullet/\\sqrt{K}$, though the paper does not state one; this would be a natural next step.","The simplification in Theorem 1.8 highlights a phenomenon the paper notes explicitly: with a single occurrence of $s_0$, $\\sigma_{\\mathsf{b}}^2$ is a rational function of $p$ of a special form, while with multiple occurrences the dependence can change; one could test whether a structural dichotomy (e.g., whether $s_0$ appears once) marks exactly where simple formulas exist.","For initial directions that are not scalar multiples of coroots, the infinite word is non-periodic and the paper's method stops; a plausible conjecture is that a spherical Gaussian limit persists for generic directions, with a variance that may depend on the direction through some ergodic invariant.","The Markov chain of type $\\widetilde A_{n-1}$ with $\\mathsf{b}=s_{n-1}\\cdots s_1s_0$ is a variant of the cyclic adjacent transposition shuffle; its mixing time is an open question the paper raises implicitly in its concluding remarks."],"forward_implications":["The limiting distribution of $v_p(\\mathsf{b}^K)^\\bullet/\\sqrt{K}$ is $N(0,\\sigma_{\\mathsf{b}}^2 I_r)$, so after normalization the random alcove position is asymptotically isotropic: no Euclidean direction is preferred.","The expected Coxeter length obeys $\\mathbb{E}[\\ell(v_p(\\mathsf{b}^K))]=\\sqrt{2/\\pi}\\,\\sigma_{\\mathsf{b}}\\sum_{\\beta\\in\\Phi^+}\\|\\beta\\|\\,\\sqrt{K}+o(\\sqrt{K})$; for $\\widetilde A_r$ with $\\mathsf{b}$ a Coxeter word this is $\\sqrt{\\frac{2}{\\pi}r(r+1)\\frac{p}{1-p}}\\,\\sqrt{K}$.","For Coxeter words, $\\sigma_{\\mathsf{b}}^2=\\kappa_W\\frac{p}{1-p}$, where $\\kappa_W$ depends only on the affine type and is tabulated for $\\widetilde A_r$, $\\widetilde B_r$, $\\widetilde C_r$, $\\widetilde D_r$, $\\widetilde E_6$, $\\widetilde E_7$, $\\widetilde E_8$, $\\widetilde F_4$, and $\\widetilde G_2$.","The associated random billiard walk is recurrent if and only if the rank $r\\le 2$, and transient otherwise.","Words that are cyclically commutation equivalent have the same $\\sigma_{\\mathsf{b}}^2$, so the variance is an invariant of the flip-equivalence class of the word's acyclic orientation."],"supporting_citations":[{"why":"Establishes that every power of a Coxeter word is reduced, which underlies the billiard and pseudolight interpretation of the random subword.","marker":"[Spe09]"},{"why":"Supplies the flip-equivalence classification of cyclic commutation equivalence used to reduce Theorem 1.10 to checking one Coxeter word per affine type.","marker":"[DMR16]"},{"why":"Gives the polynomial-growth random walk estimate used to prove the transience part of Corollary 1.5.","marker":"[Ale02]"},{"why":"Provides the classical recurrence criterion for centered lattice random walks used to prove the recurrent case $r\\le 2$ of Corollary 1.5.","marker":"[Spi76]"}],"fun_headline_variants":["Random subwords turn affine alcoves Gaussian","Spherical Gaussian limit for random subword walks","Billiard walks in affine Weyl groups hit normality","Exact variance for Gaussian subword limits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the walk modulo the coroot translation lattice living on a finite set of states, so returns to each state occur infinitely often; without that finiteness there would be no uniform regenerative blocks to which a central limit theorem could apply.","fun_headline_variants_meta":{"raw":{"variants":["Random subwords turn affine alcoves Gaussian","Spherical Gaussian limit for random subword walks","Billiard walks in affine Weyl groups hit normality","Exact variance for Gaussian subword limits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000264,"raw_usage":{"total_tokens":1727,"prompt_tokens":1192,"completion_tokens":535,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":808,"completion_tokens_details":{"reasoning_tokens":476}},"tokens_in":808,"tokens_out":535,"duration_ms":5379,"temperature":1.0,"reasoning_tokens":476,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T18:38:58.307939+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $W=\\widetilde A_2$, $\\mathsf{b}=s_1s_2s_0$, and $p=1/2$, simulate many independent copies of $v_p(\\mathsf{b}^K)$ for, say, $K=10^4$, and compute the sample covariance of $v_p(\\mathsf{b}^K)^\\bullet/\\sqrt{K}$. The theorem predicts convergence to $\\frac{2}{r(r+1)}\\frac{p}{1-p}I_2=\\frac{1}{3}I_2$; a significantly anisotropic covariance or a different scale would falsify it. A direct check, feasible for small rank, is to enumerate all paths in the finite quotient $\\overline W$ to compute $\\operatorname{Tr}(\\operatorname{Cov}(D^1_1))$ and compare with the paper's formula.","supporting_citations":[],"review_version":1}