The Integrable Snake Model

Rohan Shiatis; Samuel G. G. Johnston

REVIEW 3 major objections 5 minor 56 references

The Integrable Snake Model

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2501.15483 v4 pith:UV2PDBRC submitted 2025-01-26 math.PR math.CO

Samuel G. G. Johnston , Rohan Shiatis This is my paper · ORCID

classification math.PRmath.CO MSC 82B2082B2160K3560J27

keywords snakemodelKasteleynmatrixdeterminantalpointprocesslozengetilingASEPnon-collidingwalkstoruslinkFibonacciweighting

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

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.

What carries the argument

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.

What would settle it

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).

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my take on the integrable snake model paper.

The 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.

But 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.

The 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.

I'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.

Referee Report

3 major / 5 minor

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.

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 (3)

[3.4, Lemma 3.4 and Proposition 3.2] 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.
[4.1 and 4.2, support fact and Theorem 2.7] 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.
[6.1, Theorem 6.2] 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.

minor comments (5)

[1, equation (1.3)] 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.
[2.1, equations (2.12) and (2.26)] 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.
[3.1, paragraph on m2 = 2] 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.
[5.2, proof of Lemma 5.7] The proof refers to 'Theorem 5.1' when it appears to mean Proposition 5.1; the reference should be corrected.
[2.2, paragraph after Theorem 2.2] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the determinant and correlation formulas are derived by direct Kasteleyn computation, and the only self-citation generalizes earlier TASEP work rather than supplying an unverified input.

full rationale

The central derivation is self-contained and does not reduce to its inputs. The Fibonacci weighting J(σ) is defined independently as a sum over nests (2.2), and Lemma 2.3 with Lemma 3.1 proves the identity Σ_{σ∈sh^{-1}(σ)} w(σ) = w(σ)J(σ) by an explicit snakelet-counting/recurrence argument. The Kasteleyn sign identity (3.18) is then proved from the definition of Qθ,m and from the winding-number property of torus links, which is cited to Murasugi rather than to the authors' own work; no self-citation is load-bearing. The determinant product (2.7)-(2.8) follows by diagonalizing Kθ,m, and the correlation formulas (2.11), (2.25) are standard Jacobi-identity consequences of the inverse kernel. The scaling limits and the ASEP result Theorem 2.9 are proven independently: Theorem 2.9 is derived from an explicit generator computation and a martingale change of measure, generalizing rather than importing [30]. The one self-citation, [30], is merely the TASEP result being generalized and does not serve as an unverified premise. The paper does contain correctness-relevant gaps that are not circularity: Proposition 3.2/Lemma 3.4 rest on a sketchy torus-link claim that is not fully proved and may fail for null-homotopic components, and a few supporting proofs are explicitly omitted (e.g., the support statement after Theorem 2.6 and Lemma 4.4). These are gaps in external/auxiliary inputs, not circular reductions, and therefore do not raise the circularity score beyond the minor-self-citation level.

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

The model has four positive parameters alpha, beta, gamma, delta which are inputs, not fitted quantities. The only constructed object is the Fibonacci weighting J(sigma), defined in (2.2); it is a definition chosen to make the model determinantal, and is validated by the internal proofs rather than by external data. The paper relies on standard mathematics: knot theory facts on torus links, the cyclic Karlin-McGregor formula from the literature, and standard determinant identities. In the probabilistic regime alpha^2 - 4 gamma delta >= 0, J(sigma) is non-negative and P_m is a genuine probability measure.

assumptions (4)

standard math All cycles of a pure snake configuration on the discrete torus have identical coprime winding numbers (Proposition 3.2).
Used in Lemma 3.5 to prove the Kasteleyn sign identity (3.7). The proof is sketched and credited to knot theory (Murasugi, Chapter 7).
standard math The cyclic Karlin-McGregor formula with Kasteleyn weightings (Theorem 6.2) is valid.
This formula identifies the determinant of twisted transition probabilities with non-collision probabilities on the ring. It is cited to Fulmek and Liechty-Wang and used in Lemma 6.3 to prove Theorem 6.1.
domain assumption The parameters lie in the probabilistic regime alpha^2 - 4 gamma delta >= 0 so that J(sigma) > 0 and P_m is a probability measure.
This condition is stated in (2.9). The paper notes it is sufficient, and discusses more complicated conditions outside this regime in Section 4.
domain assumption The limiting measures P^{gamma,delta}_tau depend continuously on the parameters tau and beta, allowing extension from generic beta to all tau.
In the proof of Theorem 2.7, the authors state the argument holds on a dense set of beta and extends by continuity, without giving the continuity proof.

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Pith. "Pith review of The Integrable Snake Model." pith.science (2026). https://pith.science/paper/UV2PDBRC

@misc{pith2026250115483,
  author       = {Pith},
  title        = {Pith review of: The Integrable Snake Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UV2PDBRC}},
  note         = {Machine review of arXiv:2501.15483}
}

abstract

A pure snake configuration is a bijection $\sigma:\mathbb{Z}^2 \to \mathbb{Z}^2$ containing no two-cycles and such that for each $x \in \mathbb{Z}^2$ we have $\sigma(x) \in \{ x , x+ \mathbf{e}^1, x+\mathbf{e}^2 , x- \mathbf{e}^2 \}.$ The non-trivial cycles of a pure snake configuration may be regarded as a collection of non-intersecting paths in $\mathbb{Z}^2$ that may travel right, up, or down (but not left) from a given vertex. Pure snake configurations are a generalisation of lozenge tilings, which are in natural correspondence with paths that only travel right or up. We introduce a partition function on a finite version of this model and study the probabilistic properties of random pure snake configurations chosen according to their contribution to this partition function. Under a suitable weighting, the model is integrable in the sense that we have access to explicit formulas for its partition function and correlation function. We utilise the integrable structure of this model in several applications through its various scaling limits, such as to prove a traffic representation of ASEP on the ring, generalising the analogous result for TASEP by the first author.

Figures

Figures reproduced from arXiv: 2501.15483 by the authors.

**Figure 1.** A generalised snake configuration. The snakelets are highlighted in red. If the two-cycles corresponding to snakelets were removed and replaced with fixed points, then we would obtain a pure snake configuration. We begin by constructing a partition function on pure snake configurations on the discrete torus Tm in the vein of (1.2). To this end, given a generalised snake configuration σ : Tm → Tm we define (σ) := #{x… view at source ↗

**Figure 2.** The diagram illustrates one possible nest, N. The underlying snake configuration σ is marked by the blue points and the snakelets are marked with red points [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** The red lines join the points in the same equivalence class, i.e. they represent the vertical gaps. so proving Lemma 2.3 amounts to showing that (2.2) is an equivalent definition. The following lemma tells us that we can recast J(σ) in terms of a product over the vertical gaps of σ – c.f. (2.2) and the preceding discussion. Lemma 3.1. Let Gσ be the set of all vertical gaps for a given snake configuration σ. Then J(σ… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: The blue curve denotes the ellipse Eγ,δ and the dotted red curve is the circle Cβ. The points on the blue curve of the form 1 + γw + δw −1 where w n = (−1) η are denoted in black. The points lying inside the red curve are Lη and outside are Rη. Proof. The statement of …

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Asymptotics of Plancherel measures for symmetric g roups

B/o.sc/r.sc/o.sc/d.sc/i.sc/n.sc, A., O/k.sc/o.sc/u.sc/n.sc/k.sc/o.sc/v.sc , A., & O/l.sc/s.sc/h.sc/a.sc/n.sc/s.sc/k.sc/i.sc, G.(2000). Asymptotics of Plancherel measures for symmetric g roups. J. Amer. Math. Soc., 13(3), 481-515

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The bead model and limit behaviors of dimer models, Ann

B/o.sc/u.sc/t.sc/i.sc/l.sc/l.sc/i.sc/e.sc/r.sc, C.(2009). The bead model and limit behaviors of dimer models, Ann. Probab., 37(1), 107–142

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B/u.sc/f.sc/e.sc/t.sc/o.sc/v.sc , A. /a.sc/n.sc/d.sc G/o.sc/r.sc/i.sc/n.sc, V .(2019). Fourier transform on high-dimensional unitary gro ups with applications to ran- dom tilings Duke Math. J. 168(13), 2559-2649

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Asymptotic domino statistics in the Aztec diamond

C/h.sc/h.sc/i.sc/t.sc/a.sc, S., J/o.sc/h.sc/a.sc/n.sc/s.sc/s.sc/o.sc/n.sc, K., & Y /o.sc/u.sc/n.sc/g.sc, B.(2015). Asymptotic domino statistics in the Aztec diamond. Ann. Appl. Probab., 1232-1278

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C/o.sc/h.sc/n.sc, H., E/l.sc/k.sc/i.sc/e.sc/s.sc, N. /a.sc/n.sc/d.sc P/r.sc/o.sc/p.sc/p.sc, J.(1996). Local statistics for random domino tilings of the Az tec diamond, Duke Math. J., 85(1), 117-166 THE INTEGRABLE SNAKE MODEL 47

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C/o.sc/h.sc/n.sc, H., K/e.sc/n.sc/y.sc/o.sc/n.sc, R. /a.sc/n.sc/d.sc P/r.sc/o.sc/p.sc/p.sc, J.(2001). A variational principle for domino tilings, J. Amer. Math. Soc. , 14(2), 297–346

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Tropical combinatorics and Whittaker func- tions

C/o.sc/r.sc/w.sc/i.sc/n.sc, I., O’C/o.sc/n.sc/n.sc/e.sc/l.sc/l.sc, N., S/e.sc/p.sc/p.sc/adieresis.sc/l.sc/adieresis.sc/i.sc/n.sc/e.sc/n.sc, T., & Z/y.sc/g.sc/o.sc/u.sc/r.sc/a.sc/s.sc, N.(2014). Tropical combinatorics and Whittaker func- tions. Duke Math. J. , 163(3), 513

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H., /a.sc/n.sc/d.sc F/o.sc/x.sc, R

C/r.sc/o.sc/w.sc/e.sc/l.sc/l.sc, R. H., /a.sc/n.sc/d.sc F/o.sc/x.sc, R. H.(2012). Introduction to knot theory (V ol. 57). Springer Science & Business Media

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The connective constant of the honeycomb lattice eq uals √ 2 + √

D/u.sc/m.sc/i.sc/n.sc/i.sc/l.sc-C/o.sc/p.sc/i.sc/n.sc, H., S/m.sc/i.sc/r.sc/n.sc/o.sc/v.sc , S.(2012). The connective constant of the honeycomb lattice eq uals √ 2 + √

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Ann. of Math. (2), 175(3), 1653-1665
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Alternating-sign matrices and domino tilings (Par t I)

E/l.sc/k.sc/i.sc/e.sc/s.sc, N., K/u.sc/p.sc/e.sc/r.sc/b.sc/e.sc/r.sc/g.sc, G., L/a.sc/r.sc/s.sc/e.sc/n.sc, M., & P/r.sc/o.sc/p.sc/p.sc, J.(1992). Alternating-sign matrices and domino tilings (Par t I). J. Algebraic Combin., 1, 111-132

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Nonintersecting lattice paths on the cylinder

F/u.sc/l.sc/m.sc/e.sc/k.sc, M.(2004). Nonintersecting lattice paths on the cylinder. Sém. Lothar. Combin. 52 16 pp. (electronic)

2004
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Limit shapes of large skew Y oung tableaux and a modiﬁ cation of the TASEP process

G/o.sc/r.sc/d.sc/e.sc/n.sc/k.sc/o.sc, A.(2020). Limit shapes of large skew Y oung tableaux and a modiﬁ cation of the TASEP process. arXiv preprint arXiv:2009.10480

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Lectures on random lozenge tilings (V ol

G/o.sc/r.sc/i.sc/n.sc, V .(2021). Lectures on random lozenge tilings (V ol. 193). Cambridge University Press

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From alternating sign matrices to the Gaussian unitary ensemble

G/o.sc/r.sc/i.sc/n.sc, V .(2014). From alternating sign matrices to the Gaussian unitary ensemble. Comm. Math. Phys., 332(1), 437-447

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Universality of local statistics for noncolliding random walks

G/o.sc/r.sc/i.sc/n.sc, V ., & P/e.sc/t.sc/r.sc/o.sc/v.sc , L.(2019). Universality of local statistics for noncolliding random walks. Ann. Probab. , 47(5), 2686-2753

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A., & J/o.sc/h.sc/n.sc/s.sc/o.sc/n.sc, C

H/o.sc/r.sc/n.sc, R. A., & J/o.sc/h.sc/n.sc/s.sc/o.sc/n.sc, C. R.(2012). Matrix analysis. Cambridge university press

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Random domino tilings and the arctic circle theorem

J/o.sc/c.sc/k.sc/u.sc/s.sc/c.sc/h.sc, W ., P/r.sc/o.sc/p.sc/p.sc, J., & S/h.sc/o.sc/r.sc, P .(1998). Random domino tilings and the arctic circle theorem . arXiv preprint math/9801068

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Non-intersecting paths, random tilings and random matrices

J/o.sc/h.sc/a.sc/n.sc/s.sc/s.sc/o.sc/n.sc, K.(2002). Non-intersecting paths, random tilings and random matrices. Probab. Theory Related Fields , 123(2), 225-280

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J/o.sc/h.sc/n.sc/s.sc/t.sc/o.sc/n.sc, S. G. G.(2022). Continuous Kasteleyn theory for the bead model. arX iv preprint arXiv:2207.13538

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Coincidence probabilities

K/a.sc/r.sc/l.sc/i.sc/n.sc, S., & M/c.scG/r.sc/e.sc/g.sc/o.sc/r.sc, J.(1959). Coincidence probabilities. Paciﬁc J. Math. : 1141-1164

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W .(1961)

K/a.sc/s.sc/t.sc/e.sc/l.sc/e.sc/y.sc/n.sc, P . W .(1961). The statistics of dimers on a lattice: I. The number o f dimer arrangements on a quadratic lattice. Physica, 27(12), 1209-1225

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K/e.sc/n.sc/y.sc/o.sc/n.sc, R. & O/k.sc/o.sc/u.sc/n.sc/k.sc/o.sc/v.sc , A.(2005). What is... a dimer?, Notices Amer. Math. Soc., 52(3)

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Dimers and amoebae

K/e.sc/n.sc/y.sc/o.sc/n.sc, R., O/k.sc/o.sc/u.sc/n.sc/k.sc/o.sc/v.sc , A., & S/h.sc/e.sc/f.sc/f.sc/i.sc/e.sc/l.sc/d.sc, S.(2006). Dimers and amoebae. Ann. of Math. (2) , 1019-1056

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E.(2003)

L/y.sc/o.sc/n.sc/s.sc, R., & S/t.sc/e.sc/i.sc/f.sc, J. E.(2003). Stationary determinantal processes: phase multip licity , Bernoullicity , entropy , and domination. Duke. J. Math. , 515-575

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Nonintersecting Brownian motions on the unit circl e

L/i.sc/e.sc/c.sc/h.sc/t.sc/y.sc , K., & W /a.sc/n.sc/g.sc, D.(2016). Nonintersecting Brownian motions on the unit circl e. Ann. Probab., 44(2), 1134-1211

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A.(2001)

M/a.sc/c.scM/a.sc/h.sc/o.sc/n.sc, P . A.(2001). Combinatory analysis, volumes I and II (V ol. 137). American Mathematical Society

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Knot Theory and its Applications

M/u.sc/r.sc/a.sc/s.sc/u.sc/g.sc/i.sc, K.(1996). Knot Theory and its Applications . Birkhäuser

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Geometric RSK correspondence, Whittaker function s and symmetrized random polymers

O’C/o.sc/n.sc/n.sc/e.sc/l.sc/l.sc, N., S/e.sc/p.sc/p.sc/adieresis.sc/l.sc/adieresis.sc/i.sc/n.sc/e.sc/n.sc, T., & Z/y.sc/g.sc/o.sc/u.sc/r.sc/a.sc/s.sc, N.(2014). Geometric RSK correspondence, Whittaker function s and symmetrized random polymers. Invent. Math., 197(2), 361-416

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(2003).Correlation function of Schur process with application to l ocal geometry of a random 3-dimensional Y oung diagram

O/k.sc/o.sc/u.sc/n.sc/k.sc/o.sc/v.sc , A., & R/e.sc/s.sc/h.sc/e.sc/t.sc/i.sc/k.sc/h.sc/i.sc/n.sc, N. (2003).Correlation function of Schur process with application to l ocal geometry of a random 3-dimensional Y oung diagram. J. Amer. Math. Soc., 16(3), 581-603

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Inﬁnite wedge and random partitions

O/k.sc/o.sc/u.sc/n.sc/k.sc/o.sc/v.sc , A.(2001). Inﬁnite wedge and random partitions. Selecta Math. (N.S.) , 7(1), 57

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Choosing a spanning tree for the integer lattice uni formly .Ann

P/e.sc/m.sc/a.sc/n.sc/t.sc/l.sc/e.sc, R.(1991). Choosing a spanning tree for the integer lattice uni formly .Ann. Probab., 1559-1574

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Asymptotics of random lozenge tilings via Gelfand– Tsetlin schemes

P/e.sc/t.sc/r.sc/o.sc/v.sc , L.(2014). Asymptotics of random lozenge tilings via Gelfand– Tsetlin schemes. Probab. Theory Related Fields , 160(3), 429-487

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The surprising mathematics of longest increasing subseque nces (No

R/o.sc/m.sc/i.sc/k.sc, D.(2015). The surprising mathematics of longest increasing subseque nces (No. 4). Cambridge University Press

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Scaling limits of loop-erased random walks and unif orm spanning trees

S/c.sc/h.sc/r.sc/a.sc/m.sc/m.sc, O.(2000). Scaling limits of loop-erased random walks and unif orm spanning trees. Israel J. Math. , 118(1), 221-288

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Determinantal random point ﬁelds

S/o.sc/s.sc/h.sc/n.sc/i.sc/k.sc/o.sc/v.sc , A.(2000). Determinantal random point ﬁelds. Russian Math. Surveys, 55(5), 923

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N., & F/i.sc/s.sc/h.sc/e.sc/r.sc, M

T/e.sc/m.sc/p.sc/e.sc/r.sc/l.sc/e.sc/y.sc , H. N., & F/i.sc/s.sc/h.sc/e.sc/r.sc, M. E.(1961). Dimer problem in statistical mechanics-an exact re sult. Philos. Mag., 6(68), 1061-1063

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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On the distribution of the length of the longest incr easing subsequence of random permutations

B/a.sc/i.sc/k.sc, J., D/e.sc/i.sc/f.sc/t.sc, P ., & J/o.sc/h.sc/a.sc/n.sc/s.sc/s.sc/o.sc/n.sc, K.(1999). On the distribution of the length of the longest incr easing subsequence of random permutations. J. Amer. Math. Soc., 12(4), 1119-1178

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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)

H/o.sc/u.sc/g.sc/h.sc, J. 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). Determinantal processes and independence. Probab. Surv., 206-229

work page 2006

[3] [1]

at xi, the snake σ makes a move in the fi direction

≡ 0 mod 2. Thus q1 + q2 + q1q2 ≡ 1 mod 2 and using this fact in ( 3.12) we obtain (m1 + 1)rq1 + rq2(m2 + 1 + rq1) ≡ r(m1q1 + m2q2 + 1) mod 2. (3.13) Using (3.13) in (3.11) and comparing with ( 3.8), we complete the proof of ( 3.6). The equation (3.7) follows immediately from summing ( 3.6) over θ 2 ∈ { 0, 1}. □ We now begin working towards our proof of Th...

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[4] [2]

satisfying (q′ 1, q′

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[5] [3]

Thus we will always suppose without loss of generality that q1 ≥ 0 and that q2 ≥ 0 if q1 = 0

:= ±(q1, q2). Thus we will always suppose without loss of generality that q1 ≥ 0 and that q2 ≥ 0 if q1 = 0. We give a sketch proof of two well-known lemmas for torus knot s and links. For further informa- tion, the reader might consult Chapter 7 of Murasugi [ 38]. Lemma 3.3. If K is a torus knot, its winding number takes the form (q1, q2) where q1 ≥ 0 and...

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[6] [4]

In order for the images of the distinct functionsφ 1,

for the winding numbers of Ki. In order for the images of the distinct functionsφ 1, . . . , φ r not to intersect, it must be the case, for one, that xi ⁄= ( qj 1s, qj 2s) + xj for all j ⁄= i, s ∈ [0, 1), and for seconds that q1 1 = . . . = qr 1 and qr 2 = . . . = qr 2. In particular, the winding numbers of K1, . . . , Kr coincide, completing the proof. □...

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[7] [5]

triangle inequality

T/h.sc/e.scm1 → ∞ /l.sc/i.sc/m.sc/i.sc/t.sc /o.sc/f.sc /o.sc/u.sc/r.sc /m.sc/o.sc/d.sc/e.sc/l.sc 4.1. The integrable snake model on Z × [n]. Let us take stock of what we have achieved in the last section. In the probabilistic regime α 2 − 4γδ ≥ 0 we showed that we can deﬁne a probability measure on pure snake conﬁgurations of the form Pθ 2,m(σ ) = J(σ )w(...

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[8] [6]

Preliminary results

T/h.sc/e.sc E/x.sc/c.sc/l.sc/u.sc/s.sc/i.sc/o.sc/n.sc P/r.sc/o.sc/c.sc/e.sc/s.sc/s.sc /o.sc/n.sc /t.sc/h.sc/e.sc R/i.sc/n.sc/g.sc 5.1. Preliminary results. In this section we study the asymptotic behaviour of the prob ability mea- sure Pγ ,δ ℓ,n (abbreviated to Pℓ,n) occurring in Theorem 2.6 under the scaling regime γ = ε T and δ = ε T′, as ε ↓ 0. In orde...

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