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Random Subwords and Billiard Walks in Affine Weyl Groups

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Pith's one-line read Random subwords of repeated words in affine Weyl groups converge to a spherical Gaussian law.

desk verdict 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. read the letter →

arxiv 2501.11095 v1 pith:5M6F6KVH submitted 2025-01-19 math.PR math.CO

Colin Defant , Pakawut Jiradilok , Elchanan Mossel This is my paper · ORCID
classification math.PRmath.CO MSC 60F0520F5560B15
keywords randomsubwordsaffineWeylgroupsbilliardwalksCoxeterlengthmultivariatenormaldistributioncentrallimittheoremonalcoves
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

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 $\sqrt{\frac{2}{\pi}r(r+1)\frac{p}{1-p}}$.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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).

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

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).

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 (5)
  1. [Section 3, Lemma 3.2] 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.
  2. [Section 3, Markov chain setup and Lemma 3.1] 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.
  3. [Section 3, proof of Corollary 1.5] 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].
  4. [Section 3, proof of Corollary 1.6] 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.
  5. [Sections 5.5-5.8 and Table 1] 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.
minor comments (5)
  1. [Section 3, equation (1)] The indexing in equation (1) uses [0,m-1] while [m] was defined as {1,...,m}; please align the notation.
  2. [Section 3, proof of Corollary 1.5] The displayed limit 'lim_{K->0}' should read 'lim_{K->\infty}'.
  3. [Section 3, Lemma 3.3] 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.
  4. [Section 3, Lemma 3.4] 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.
  5. [Table 1] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central CLT and variance formula are derived from Markov-chain regeneration with no fitted input; self-citations are contextual and non-load-bearing.

full rationale

The paper's central claim (Theorem 1.4) is derived from a regeneration argument on the finite quotient Markov chain M on W, not from any fitted parameter. The random element v_p(b^K) is represented as Y_{Km}...Y_1 with i.i.d. factors Y_i, and the proof analyzes return displacements D_w^{w'} of the quotient chain; their covariance is shown finite via exponential tails, and Lemma 3.2 establishes zero mean by comparing return displacements. The only soft spot is that Lemma 3.2 says an almost-sure separation 'follows' from the Central Limit Theorem; the CLT alone would not give that separation, but the needed statement follows from the SLLN for the i.i.d. return increments, so this is a proof-wording issue rather than circularity. Theorem 1.7 then evaluates Tr(Cov(D_1^1)) through explicit finite sums over subwords involving the quantities nu_J and kappa_b; this is an algebraic computation, not a renaming of the target result. Section 5 verifies the formula on each Cartan-Killing type with direct linear algebra; the E6/E7/E8/F4 entries are computer-assisted exact calculations, not fits to simulated data. Self-citations to Def24A and Def24B appear only in the introduction and related-work discussion, e.g., 'this problem was considered in [Def24A]' and 'one can view sub_p(w_stair) as a random pipe dream (see [Def24B,MPPY24])'; these citations carry no load in the proofs of Theorems 1.4, 1.7, 1.8, or 1.10. Auxiliary facts are supported by external references (Speyer's reducedness result, Alexopoulos's local limit theorem, Schur's lemma, standard random-walk recurrences). No equation in the paper reduces to its own input by construction, and no fitted quantity is relabeled as a prediction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted. The only assumptions are standard background from Coxeter theory and probability, plus the standing assumption that b covers all simple reflections. No new physical or mathematical entities are introduced.

assumptions (4)
  • domain assumption The word b contains each simple reflection in S at least once.
    Imposed before Theorem 1.4; if not, the walk is confined to a finite subgroup and σ_b^2 = 0. It also guarantees invertibility of IV - R_b in Lemma 4.2.
  • domain assumption W is an irreducible affine Weyl group and the finite Weyl group W acts irreducibly on V.
    Irreducibility is used in the Schur's Lemma argument in Theorem 1.4 and in the proof that E[D_1^1] = 0 in Lemma 3.2.
  • standard math The Markov chain on the finite quotient Weyl group has a uniform stationary distribution.
    This follows from right-invariance of the transition probabilities; the proof is given in Section 3, but the finite quotient structure is not always made explicit.
  • standard math Classical results in random walk theory: Spitzer's recurrence criterion, Pólya's theorem, and Alexopoulos's local limit theorem.
    Used in the proof of Corollary 1.5 to characterize recurrence versus transience.

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Pith. "Pith review of Random Subwords and Billiard Walks in Affine Weyl Groups." pith.science (2026). https://pith.science/paper/5M6F6KVH

@misc{pith2026250111095,
  author       = {Pith},
  title        = {Pith review of: Random Subwords and Billiard Walks in Affine Weyl Groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5M6F6KVH}},
  note         = {Machine review of arXiv:2501.11095}
}
abstract

Let $W$ be an irreducible affine Weyl group, and let $\mathsf{b}$ be a finite word over the alphabet of simple reflections of $W$. Fix a probability $p\in(0,1)$. For each integer $K\geq 0$, let $\mathsf{sub}_p(\mathsf{b}^K)$ be the random subword of $\mathsf{b}^K$ obtained by deleting each letter independently with probability $1-p$. Let $v_p(\mathsf{b}^K)$ be the element of $W$ represented by $\mathsf{sub}_p(\mathsf{b}^K)$. One can view $v_p(\mathsf{b}^K)$ geometrically as a random alcove; in many cases, this alcove can be seen as the location after a certain amount of time of a random billiard trajectory that, upon hitting a hyperplane in the Coxeter arrangement of $W$, reflects off of the hyperplane with probability $1-p$. We show that the asymptotic distribution of $v_p(\mathsf{b}^K)$ is a central spherical multivariate normal distribution with some variance $\sigma_{\mathsf{b}}^2$ depending on $\mathsf{b}$ and $p$. We provide a formula to compute $\sigma_{\mathsf{b}}^2$ that is remarkably simple when $\mathsf{b}$ contains only one occurrence of the simple reflection that is not in the associated finite Weyl group. As a corollary, we provide an asymptotic formula for $\mathbb{E}[\ell(v_p(\mathsf{b}^K))]$, the expected Coxeter length of $v_p(\mathsf{b}^K)$. For example, when $W=\widetilde A_{r}$ and $\mathsf{b}$ contains each simple reflection exactly once, we find that \[\lim_{K\to\infty}\frac{1}{\sqrt{K}}\mathbb{E}[\ell(v_p(\mathsf{b}^K))]=\sqrt{\frac{2}{\pi}r(r+1)\frac{p}{1-p}}.\]

Figures

Figures reproduced from arXiv: 2501.11095 by the authors.

Figure 1
Figure 1. The trajectory of a random billiard walk corresponding to a random subword sub4/5 ((s2s1s0) 200) in type Ae2. Line segments traversed more frequently are drawn darker. The red dot indicates the starting point z0. of alcoves, so the map u 7→ Au is a bijection from W to the set of alcoves. The alcoves adjacent to Au are those of the form Asu for s ∈ S. We denote by u • the centroid of the alcove Au. Fix a point z0 in … view at source ↗
Figure 2
Figure 2. The trajectory of a random billiard walk corresponding to a random subword sub4/5 ((s1s2s1s0) 150) in type Ce2. Line segments traversed more frequently are drawn darker. The red dot indicates the starting point z0. We now introduce a random process that we call a random billiard walk. As before, start by shining a beam of light from z0 in the direction of η. But now, whenever the beam of light hits a hyperplane H ∈ … view at source ↗
Figure 3
Figure 3. The trajectory of the billiard walk corresponding to the subword s2s1s0s2 s2s1 s2 s0s2s1s0 of (s2s1s0) 5 in type Ae2. The red dot indicates the starting point z0. As mentioned above, we will primarily focus on random subwords of the form subp(b K), where b is a fixed finite word. Such words arise from random billiard walks with “rational” initial directions. More precisely, the word wz0 (η) is periodic if and only i… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Homology in Combinatorial Refraction Billiards

    math.CO 2025-02 accept novelty 8.0 of 10

    A graph is expelling, sending every billiard loop around the torus non-contractibly, exactly when it is bipartite; ensnaring graphs have a partial structural theory.

Reference graph

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