{"id":"de4add3b-e332-4f43-81cc-227678eae308","arxiv_id":"2501.15483","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A weighted 'pure snake' model on a torus is shown to be exactly solvable, and its scaling limit yields a new representation of ASEP on a ring as a change of measure from non-colliding walkers.","lead":"Pure snake configurations generalize lozenge tilings by allowing paths to go down as well as up or right. The paper builds an exactly solvable version of this model and uses its scaling limit to derive a traffic representation of the asymmetric exclusion process on a ring.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The torus-link winding property in Lemma 3.4/Proposition 3.2 is the least secure input to Theorem 2.1: the lemma as stated is false for null-homotopic components, and the sketch does not prove that pure snake cycles are essential or have equal windings; if cycles could wind differently, the…","rationale":"The reader identifies the torus-link winding property as the weakest assumption, and I agree. The whole Kasteleyn derivation is a sequence of determinant manipulations that is individually checkable; the only step that imports external topology is Proposition 3.2, used in Lemma 3.5 to reach the cancellation identity (3.18). If the winding pairs of cycles were not all equal, or if (q1,q2) were not coprime, the parity calculation in equations (3.8)-(3.13) would break and the four-sector signed sum would not reduce to the positive weight w(sigma). The paper's Lemma 3.4 is overbroad: disjoint knots on a torus need not have equal winding numbers when a null-homotopic component is present. The manuscript does not explicitly rule out such components for pure snake configurations, nor does it give the standard algebraic-intersection proof for essential curves. This is a proof gap rather than a demonstrated falsehood: the no-left constraint makes null-homotopic cycles very likely impossible, since q1 = q2 = 0 would force a path with no net horizontal displacement and no net vertical displacement, which on a circle cannot be a simple cycle without repeating a vertex. Thus I do not think the central claim should be rejected; it should remain conditional until the topological input is proved or replaced by a direct combinatorial proof. A small-torus enumeration would settle whether the proposition is actually violated and whether the determinant identity holds numerically, giving a concrete way to test whether the concern lands. The reader's conditional verdict is exactly the right level of caution, so no verdict change is needed.","tokens_in":48058,"tokens_out":12766,"duration_ms":130763,"concrete_test":"Enumerate all pure snake configurations on small tori, e.g. m1,m2 in {(3,3),(4,3),(3,4)}, and for each configuration compute the winding pair of every nontrivial cycle from the counts of right, up, and down moves. If any configuration has cycles with different winding pairs, Proposition 3.2 is false. Independently, for several generic parameter choices such as (alpha,beta,gamma,delta) = (2,1,1,0.5) and (3,2,1,0.25), compute Zm from the definition (2.5) by enumeration and compare with the determinant sum (2.7). A mismatch on any small torus would be a concrete counterexample to Theorem 2.1; agreement would show the identity survives in tested cases but would still leave the proof gap in Lemma 3.4 needing a written argument that snake cycles are essential and homology-equal.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, Theorem 2.1, is obtained by summing the pointwise identity (3.18), and (3.18) rests on Lemma 3.5, whose proof uses Proposition 3.2: every long cycle of a pure snake configuration has the same winding pair (q1,q2), with q1,q2 coprime. This is where the topology of the torus enters the Kasteleyn sign cancellation, and it is the weakest link. The supporting Lemma 3.4 states that any disjoint knots on the torus have equal winding numbers. As written, that statement is false: a small contractible circle on the torus and an essential circle can be disjoint, with winding numbers (0,0) and (1,0). The proof sketch replaces each knot by a linear representative and asserts that disjointness forces all q^i to coincide; this is only plausible for essential simple closed curves, and even then requires the algebraic-intersection argument (for primitive classes (a,b) and (c,d), disjointness forces ad-bc = 0 and hence equal slopes). The paper never proves that a pure snake cycle cannot be null-homotopic, nor that two disjoint cycles of different essential homology classes cannot occur. For the no-left step set, a contractible cycle would have q1 = q2 = 0 and would force a repeated vertex, so Proposition 3.2 may well be true; but the present proof is a genuine gap at the exact point where the four determinants are assembled into the partition function. Because every later result, including the ASEP applications, depends on Theorem 2.1, this is the most load-bearing concern.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a finite-torus model of pure snake configurations, bijections sigma on Tm = Zm1 x Zm2 with moves right, up, or down (no left moves), equipped with the Fibonacci weighting J(sigma). The two foundational claims are Theorem 2.1, expressing the partition function Zm as a signed sum of four determinants of operators K_{theta,m} whose determinants factor into explicit products over roots of unity, and Theorem 2.2, giving determinantal correlation functions for right moves in the probabilistic regime alpha^2 - 4 gamma delta >= 0; Theorem 2.5 extends this to general moves. The remainder of the paper derives scaling limits (Theorems 2.6 and 2.7), identifies a continuous-time scaling limit with non-colliding asymmetric Poisson walkers on the ring (Theorem 2.8), and proves an ASEP traffic representation via an exponential martingale change of measure (Theorem 2.9). The proofs develop a Kasteleyn theory for snake configurations, using a shape projection that deletes two-cycles and a topological statement about torus links.","tokens_in":48406,"tokens_out":6459,"duration_ms":65999,"significance":"If the main results are correct, the paper provides a genuinely new exactly solvable model that generalizes lozenge tilings and the bead model, with explicit determinant and product formulas that are not fitted to the conclusions. The derived ASEP traffic representation extends the first author's prior TASEP result and gives a concrete probabilistic application. Strengths of the manuscript include the explicit nature of the eigenvalue computation for det(K_{theta,m}), the self-contained treatment of the Fibonacci weighting via the shape projection, and the fact that the correlation formulas are stated in a form suitable for the subsequent scaling limits. The main caveat is that one load-bearing topological step in the proof of Theorem 2.1 is only sketched and, as stated, is not correct in the generality used; several later statements are also asserted with omitted or very compressed proofs.","major_comments":[{"comment":"Lemma 3.4 is false as a general statement about disjoint torus knots: a small contractible loop and an essential longitude on the torus can be disjoint and have winding numbers (0,0) and (1,0). The proof sketch simply asserts that disjointness forces all q^i_1 and q^i_2 to coincide, without justification. This is load-bearing because Lemma 3.5, equation (3.6), and the key identity (3.18) used to prove Theorem 2.1 all depend on Proposition 3.2. The manuscript needs either a direct proof that every long cycle of a pure snake configuration is an essential simple closed curve with primitive winding, together with the standard fact that two disjoint essential simple closed curves on the torus have the same winding up to sign, or an alternative proof of Proposition 3.2 that avoids the false lemma.","section":"3.4, Lemma 3.4 and Proposition 3.2"},{"comment":"The statement preceding Lemma 4.4 that the measure P^{gamma,delta}_{ell,n} is supported on configurations with occupation number ell almost surely is explicitly said to have an omitted proof, and Lemma 4.4 (Markov property of (U_s)) is also stated without proof. The proof of Theorem 2.7 is a two-sentence sketch relying on a contour-integral approximation and a density argument. These results are used downstream in the scaling limits leading to Theorems 2.8 and 2.9, so the omissions are not merely cosmetic. The authors should either supply the deferred proofs or give precise cross-references showing that the finite-cylinder analogues are proved in Section 5 and that the relevant properties are inherited in the m1 -> infinity and n -> infinity limits.","section":"4.1 and 4.2, support fact and Theorem 2.7"},{"comment":"The cyclic Karlin-McGregor formula with Kasteleyn weightings is stated as a theorem, but the paper says 'For the proof of this result see [21] or [36]' after having announced that it will 'state and prove a version' of the formula. Since this formula is central to the identification of P^{ell,n} with the non-colliding walkers in Theorem 6.1, the statement should either be proved in the paper or explicitly and precisely imported as an external result, including the conditions under which the winding-number weighting is valid for even ell.","section":"6.1, Theorem 6.2"}],"minor_comments":[{"comment":"The contour integral in (1.3) is written with both endpoints equal to w_{alpha,beta,gamma}; as printed this is not a valid contour. Presumably one endpoint should be the conjugate, and this should be corrected.","section":"1, equation (1.3)"},{"comment":"The denominators in the displayed formulas for G_{theta,m} and H_{theta,m} appear to be missing the factor w^{-1} on the delta term; the eigenvalue formula (2.8) and Lemma 3.7 use delta w^{-1}, so the kernel formulas should be checked and aligned.","section":"2.1, equations (2.12) and (2.26)"},{"comment":"The sentence 'In the case m2 = 2, e2 = -e2. So WLOG, we will consider down jumps not to exist' is unclear, since the move set already identifies up and down moves and the definition of pureness excludes two-cycles; please clarify the convention.","section":"3.1, paragraph on m2 = 2"},{"comment":"The proof refers to 'Theorem 5.1' when it appears to mean Proposition 5.1; the reference should be corrected.","section":"5.2, proof of Lemma 5.7"},{"comment":"The sentence 'The latter equation, (2.8), follows from diagonalising the operator K_{theta,m}' appears in a discussion of Theorem 2.2, but equation (2.8) is part of Theorem 2.1; rephrase to avoid confusion.","section":"2.2, paragraph after Theorem 2.2"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle to acceptance is the torus-link step in the proof of Theorem 2.1: the cited Lemma 3.4 is false in general and the manuscript does not prove the restricted version needed for snake cycles. This is a fixable gap rather than a demonstrated counterexample to the theorem, but it must be addressed before the paper can be considered rigorous. The later sections also contain several omitted or compressed arguments, so a careful revision is warranted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on the integrable snake model paper.\n\nThe pure snake model is a genuine new object—it widens lozenge tilings to paths that can go down, and the Fibonacci weighting that makes it solvable is a neat trick. The determinant formulas in Theorems 2.1 and 2.2 are explicit and concrete, and the scaling limit that ends in a traffic representation of ASEP (Theorem 2.9) is a real generalization of the earlier TASEP result. If the core formula is correct, this is a substantial contribution to integrable probability.\n\nBut there is a spot I want you to look at before taking anything on trust. Theorem 2.1 is obtained by summing the pointwise identity (3.18), and that identity rests on Proposition 3.2: every long cycle of a pure snake configuration on the torus has the same coprime winding pair (q1,q2). The proof imports Lemma 3.4 from knot theory, and as stated that lemma is false. A small contractible circle on the torus and an essential curve can be disjoint and have different windings. The sketch never proves that a pure snake cycle cannot be null-homotopic or that two disjoint cycles can't lie in different homology classes. This is not cosmetic. If cycles could wind differently, the sign cancellation in (3.18) breaks and Theorem 2.1 falls. My guess is the missing statement is actually true for these specific paths, because a null-homotopic cycle would force a repeated vertex in the no-left-step setting, but the paper doesn't prove it. Until that is fixed, the foundation is shaky.\n\nThe other soft spots are less serious: Theorem 2.7 is only sketched, Lemma 4.4 and the support claim are omitted, and the cyclic Karlin-McGregor formula is imported without a derivation. These are the kind of gaps a referee can ask to fill without throwing the whole paper away.\n\nI'd send this to a serious referee, specifically asking them to check Proposition 3.2 in the discrete torus setting. If it survives, the paper is a solid addition to the exactly solvable models literature. My own verdict would be conditional, not reject.","headline":"Clever new exactly solvable model, but the torus-link winding lemma underpinning Theorem 2.1 has a real proof gap; the paper deserves peer review but is not yet established.","tokens_in":48927,"tokens_out":3244,"would_cite":false,"duration_ms":29700,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B20","82B21","60K35","60J27"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a Fibonacci-weighted snake model on a torus is exactly solvable: its partition function is a signed sum of four determinants and its correlation functions are determinantal, yielding a traffic representation of ASEP…","keywords":["snake model","Kasteleyn matrix","determinantal point process","lozenge tiling","ASEP","non-colliding walks","torus link","Fibonacci weighting"],"falsifier":"Enumerate all pure snake configurations on a small torus, say $T_{3,3}$ or $T_{3,4}$, compute $\\sum_\\sigma w(\\sigma)J(\\sigma)$ directly for generic parameters $\\alpha,\\beta,\\gamma,\\delta$, and compare with the four-determinant formula (2.7); a mismatch would disprove Theorem 2.1. More directly, search the same configurations for one whose cycles have differing winding numbers — the existence of such a configuration would violate Proposition 3.2 and break identity (3.18).","tokens_in":47863,"feed_emoji":"🐍","tokens_out":8627,"duration_ms":70281,"temperature":0.7,"pith_summary":"The paper introduces a new exactly solvable statistical-mechanics model: 'pure snake configurations', bijections of a torus (or cylinder/plane) where every point maps to itself, one step right, one step up, or one step down, with no two-cycles. These generalize lozenge tilings, whose path representation only allows right/up moves. The authors prove that a natural Fibonacci-weighted version of this model is integrable: the partition function is a signed sum of four determinants, and the correlations of random snakes are given by explicit determinants of inverse Kasteleyn operators. Because the model interpolates between dimers and exclusion processes, the exact formulas feed into scaling limits that recover the discrete sine kernel and a representation of ASEP on the ring as a change of measure of non-colliding Poisson walkers.","feed_headline":"Snake tiling model solved by four determinants","feed_subtitle":"A Fibonacci-weighted generalization of lozenge tilings gets exact correlations and maps ASEP to non-colliding walkers.","key_machinery":"The load-bearing object is the family of four Kasteleyn operators $K_{\\theta,m}$ on $T_m$, defined by $K_{\\theta,m}(x,y)=\\alpha\\mathbf 1_{y=x}+\\beta e^{\\pi i\\theta_1/m_1}\\mathbf 1_{y=x+e_1}+\\gamma e^{\\pi i\\theta_2/m_2}\\mathbf 1_{y=x+e_2}+\\delta e^{-\\pi i\\theta_2/m_2}\\mathbf 1_{y=x-e_2}$, whose eigenvalues are $\\alpha+\\beta z_\\theta+\\gamma w_\\theta+\\delta w_\\theta^{-1}$ over $m_1m_2$ root pairs; the determinant product (2.8) is the partition function, and the inverse operator supplies the correlation kernel $G_{\\theta,m}$. The argument also turns on the torus-link property (Proposition 3.2): a pure snake configuration on a torus decomposes into cycles that all have the same coprime winding numbers $(q_1,q_2)$, which makes the Kasteleyn sign identity (3.18) hold term by term. The Fibonacci weighting $J(\\sigma)=\\prod_{g} f_{|g|}(-\\gamma\\delta/\\alpha^2)$ is what converts the signed sum over generalized configurations (including snakelets) back into a positive sum over pure ones: $f_n$ obeys $f_n=f_{n-1}+\\lambda f_{n-2}$ and counts nest placements in vertical gaps by domino tilings. The whole structure then pushes forward: the shape map $sh$ deletes snakelets, and correlations under the genuine measure are sums, with coefficients $\\lambda_{\\theta,m}=C_{\\theta,m}\\det(K_{\\theta,m})/Z_m$, of signed determinantal correlations.","core_discovery":"The central discovery is that the pure snake model on the discrete torus $T_m=\\mathbb{Z}_{m_1}\\times\\mathbb{Z}_{m_2}$, weighted by $w(\\sigma)J(\\sigma)$ with Fibonacci gap weights $J$, is exactly solvable. Theorem 2.1 expresses the partition function as $Z_m=\\sum_{\\theta\\in\\{0,1\\}^2}C_{\\theta,m}\\det(K_{\\theta,m})$, where each $K_{\\theta,m}$ is a four-term Toeplitz-like operator whose determinant factorizes as $\\prod_{z^{m_1}=(-1)^{\\theta_1},w^{m_2}=(-1)^{\\theta_2}}(\\alpha+\\beta z+\\gamma w+\\delta w^{-1})$. In the probabilistic regime $\\alpha^2\\ge 4\\gamma\\delta$, Theorem 2.2 gives a determinantal formula for probabilities of prescribed right moves, with a kernel $G_{\\theta,m}$ written as a contour/root sum; Theorem 2.5 extends this to arbitrary events involving up and down moves through words in the indicators $R^f_x$. The proof route is a Kasteleyn theory: the signed weight of a generalized snake configuration equals a sum over boundary-twisted operators, thanks to a torus-link lemma stating that all cycles share the same coprime winding numbers; the downward/upward two-cycles ('snakelets') are integrated out by the Fibonacci weights, which count domino tilings of vertical gaps.","pith_inferences":["The same machinery should produce exact formulas for descendants of this model — e.g., snake configurations with reflecting or absorbing horizontal boundaries — by mimicking the torus-link Kasteleyn theory on other surfaces, since the only topology-dependent input is the winding-number lemma.","The Fibonacci weights tie the model to domino tilings of gaps, so tuning $\\gamma\\delta/\\alpha^2$ near the critical value $1/4$ (where $f_n(-\\lambda)$ stops being sign-definite) may expose a phase transition in the number of snakelets; the determinant product (2.8) gives a concrete way to probe this numerically.","Because the limiting processes are determinantal, standard CLT/variance bounds for determinantal point processes should transfer to fluctuations of right-move counts in large torus limits, giving fluctuation scales not explicit in the paper.","The ASEP traffic representation suggests that other exclusion-process observables (current, tagged particle speed) can be computed from the conditioned-walker determinantal structure, extending the connection to non-intersecting path theory."],"forward_implications":["The partition function and $k$-point correlations of the Fibonacci-weighted snake model can be evaluated in closed form by taking determinants, so statistical questions about random snakes reduce to spectral data of $K_{\\theta,m}$.","Letting $m_1\\to\\infty$ yields explicit limiting measures $P^{\\gamma,\\delta}_{\\ell,n}$ on the cylinder $\\mathbb{Z}\\times\\{0,\\dots,n-1\\}$, with phase transitions governed by how many $n$-th roots of $\\pm1$ satisfy $|1+\\gamma w+\\delta w^{-1}|<\\beta$; these limits are Markov in the horizontal coordinate.","Letting both dimensions go to infinity recovers an extension of the extended discrete sine kernel (with $1+\\gamma w+\\delta w^{-1}$ in place of $1+\\gamma w$), and setting $\\delta=0$ reproduces the lozenge-tiling correlation kernel (1.3).","In the low-$\\gamma,\\delta$ scaling limit, the snake model is exactly the law of $\\ell$ independent asymmetric Poisson walkers on the ring conditioned never to collide, with a determinantal transition kernel and stationary weights $\\Delta(\\mathbf y)^2/n^\\ell$.","ASEP on the ring is an exponential martingale change of measure of these conditioned walkers, with Radon-Nikodym density involving the traffic of the configuration; this generalizes the TASEP traffic representation of [30] to ASEP."],"supporting_citations":[{"why":"Supplies the determinant-sign technique: expressing weighted permutation sums as determinants of adjacency-like operators.","marker":"[32]"},{"why":"Supplies the torus-link fact that all disjoint knots on a torus have the same coprime winding numbers, used to prove identity (3.18).","marker":"[38]"},{"why":"Supplies Jacobi's identity, which converts signed correlations of generalized configurations into determinants of the inverse operator.","marker":"[26]"},{"why":"Supplies the coordinate change that views lozenge tilings as bijections with $\\sigma(x)\\in\\{x,x+e_1,x+e_2\\}$, the model the snake configurations extend.","marker":"[23]"},{"why":"Supplies the lozenge-tiling extended discrete sine kernel that the snake correlations reproduce in the $\\delta=0$ limit.","marker":"[25]"},{"why":"Supplies the bead-model scaling limit of lozenge tilings whose semidiscrete torus version motivates the scaling limits used here.","marker":"[11]"},{"why":"Supplies the TASEP traffic representation that the paper generalizes to ASEP (Theorem 2.9).","marker":"[30]"},{"why":"Supplies the cyclic Karlin-McGregor formula for nonintersecting paths on the cylinder, a key tool in the walker identification.","marker":"[21]"},{"why":"Supplies the circular nonintersecting transition density that inspired the cyclic Karlin-McGregor formula used in Section 6.","marker":"[36]"}],"fun_headline_variants":["Snake model exactly solved by four determinants","Fibonacci-weighted snake tilings get exact correlations","From snake paths to ASEP via integrable exact solution","Four determinants crack the integrable snake model","Snake model maps to ASEP with exact formulas"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every cycle of a pure snake configuration on the torus winds around the torus the same number of times horizontally and vertically, and that those two winding numbers are coprime; if the cycles could wind differently, the signed cancellations that turn the determinant sum into the partition function would fail.","fun_headline_variants_meta":{"raw":{"variants":["Snake model exactly solved by four determinants","Fibonacci-weighted snake tilings get exact correlations","From snake paths to ASEP via integrable exact solution","Four determinants crack the integrable snake model","Snake model maps to ASEP with exact formulas"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000222,"raw_usage":{"total_tokens":1516,"prompt_tokens":1073,"completion_tokens":443,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":689,"completion_tokens_details":{"reasoning_tokens":372}},"tokens_in":689,"tokens_out":443,"duration_ms":4140,"temperature":1.0,"reasoning_tokens":372,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:15:10.805988+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all pure snake configurations on a small torus, say $T_{3,3}$ or $T_{3,4}$, compute $\\sum_\\sigma w(\\sigma)J(\\sigma)$ directly for generic parameters $\\alpha,\\beta,\\gamma,\\delta$, and compare with the four-determinant formula (2.7); a mismatch would disprove Theorem 2.1. More directly, search the same configurations for one whose cycles have differing winding numbers — the existence of such a configuration would violate Proposition 3.2 and break identity (3.18).","supporting_citations":[{"cited_title":"Lectures on random lozenge tilings (V ol","cited_arxiv_id":null,"evidence_quote":"Supplies the determinant-sign technique: expressing weighted permutation sums as determinants of adjacency-like operators."},{"cited_title":"Non-intersecting paths, random tilings and random matrices","cited_arxiv_id":null,"evidence_quote":"Supplies the torus-link fact that all disjoint knots on a torus have the same coprime winding numbers, used to prove identity (3.18)."},{"cited_title":"H., /a.sc/n.sc/d.sc F/o.sc/x.sc, R","cited_arxiv_id":null,"evidence_quote":"Supplies Jacobi's identity, which converts signed correlations of generalized configurations into determinants of the inverse operator."},{"cited_title":"/a.sc/n.sc/d.sc P/r.sc/o.sc/p.sc/p.sc, J.(1996)","cited_arxiv_id":null,"evidence_quote":"Supplies the coordinate change that views lozenge tilings as bijections with $\\sigma(x)\\in\\{x,x+e_1,x+e_2\\}$, the model the snake configurations extend."},{"cited_title":"Tropical combinatorics and Whittaker func- tions","cited_arxiv_id":null,"evidence_quote":"Supplies the lozenge-tiling extended discrete sine kernel that the snake correlations reproduce in the $\\delta=0$ limit."},{"cited_title":"Nonintersecting lattice paths on the cylinder","cited_arxiv_id":null,"evidence_quote":"Supplies the TASEP traffic representation that the paper generalizes to ASEP (Theorem 2.9)."},{"cited_title":"On the distribution of the length of the longest incr easing subsequence of random permutations","cited_arxiv_id":null,"evidence_quote":"Supplies the cyclic Karlin-McGregor formula for nonintersecting paths on the cylinder, a key tool in the walker identification."},{"cited_title":"B., K/r.sc/i.sc/s.sc/h.sc/n.sc/a.sc/p.sc/u.sc/r.sc, M., P/e.sc/r.sc/e.sc/s.sc, Y ., & V/i.sc/r.sc/aacute.sc/g.sc, B.(2006)","cited_arxiv_id":null,"evidence_quote":"Supplies the circular nonintersecting transition density that inspired the cyclic Karlin-McGregor formula used in Section 6."}],"review_version":1}