{"id":"2bfd5a4b-913f-4716-a127-8a4a5b8191ed","arxiv_id":"2505.15630","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Random Demazure products on arbitrary order-convex shapes converge to explicit permuton limits with KPZ-type Tracy-Widom fluctuations, and the density inside the classic bubble-sort curve is now known.","lead":"This paper proves exact large-scale limit shapes for several random sorting processes built from the Demazure product, an algebraic operation used in Schubert calculus. For random pipe dreams and bubble-sort-like operators, it gives explicit formulas and shows the fluctuations follow the same universal law seen in growing interfaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 and 1.2 rest on the claimed pipe-dream/TASEP equivalence in Section 4.2, which is asserted but not proved; this is the load-bearing step and needs a formal bijection.","rationale":"The reader's weakest assumption correctly identifies the pipe-dream-to-TASEP coupling in Section 4.2 as the load-bearing step. Theorems 1.1 and 1.2 are not self-contained without a proof that the random pipe dream update is equidistributed with the geometric-jump TASEP: every subsequent asymptotic statement, including the explicit formulas for h_p^{phi,psi} and the KPZ fluctuation result, is obtained by substituting the pipe dream into the TASEP asymptotics of Theorem 4.1. The text gives a plausible description and a small illustrative example, but no formal bijection. This is a rigor gap rather than a demonstrated error: spot checks for small n, including two-particle examples with holes and walls, confirm the asserted transition law, and the known staircase case of Morales-Panova-Petrov-Yeliussizov supports the general mechanism. Still, because the claim is the junction between the combinatorial object and the integrable probability input, it is the weakest point of the proof. The almost-sure-versus-in-probability issue is real but secondary: it affects the mode of convergence in the statement, not the limiting object or the explicit formulas. The reader's CONDITIONAL verdict remains appropriate: the paper should supply a complete proof of the coupling, and ideally also justify the almost-sure convergence, before the main theorems are accepted in their stated form.","tokens_in":24730,"tokens_out":34871,"duration_ms":314160,"concrete_test":"Prove by induction on column index j that the law of the TASEP state iota(Delta_p(omega_1 ... omega_j)) equals the law of the geometric-jump TASEP after j-T steps, using the recursive definition of the TASEP with independent geometric increments. Alternatively, run an exhaustive enumeration for all n <= 6, all order-convex shapes with at most four columns, and p in {1/4, 1/2, 3/4}: compute the Demazure product by the resolution algorithm for every pipe dream filling and compare the resulting distribution of iota(Delta_p(S)) with the distribution of the geometric-jump TASEP started from step initial data and run for the appropriate number of steps. If any mismatch appears, the coupling in Section 4.2 is incorrect; if no mismatch appears for all small cases and the induction goes through, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central results are derived by reducing the height function H_{u_n}(x,y) to the position of a prescribed particle in the TASEP with geometric jumps. In Section 4.2 the text asserts that processing columns T+1 through T' of an order-convex pipe dream updates the state ι(v_j) exactly as one step of the geometric-jump TASEP, with inequality (13) ensuring that the correct particles are mobile. This is the only bridge from the random pipe dream to the integrable asymptotics used in Theorem 4.1. No formal bijection is written out: the TASEP uses independent geometric jump variables G_i(t) for every particle at every time, while the pipe dream uses independent cross/bump tiles shared by all particles in a column. The text does not prove that the induced joint law of the particle configuration after each column coincides with the TASEP transition kernel, including the wall at n+1-a_j and the right-to-left order of the column's word. If a corner case exists, the explicit formula h_p^{phi,psi} and the Tracy-Widom statement would not follow. A secondary but real gap is that the proof upgrades convergence in probability to the stated almost-sure convergence without a Borel-Cantelli argument; the primary risk, however, is the unexplained coupling.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a Demazure product on permutons, defined through the min-plus tropical formula H_{μ⋆ν}(x,y)=min_γ (H_μ(γ,y)+H_ν(x,γ)), and uses it to study two families of random permutations. The first family comes from Demazure products of random subwords of words associated with arbitrary order-convex shapes, viewed as random pipe dreams. The main results, Theorems 1.1 and 1.2, assert that the height functions of these permutations converge to an explicit limiting permuton height function h_p^{φ,ψ} and that the fluctuations inside a certain region converge to the Tracy–Widom GUE distribution with n^{-2/3} scaling, by reducing the pipe dream evolution to the TASEP with geometric jumps. The second family applies deterministic bubble-sort-type operators to random initial permutations; Theorems 1.4–1.6 give limiting permutons in terms of the new Demazure product, recovering and extending DiFranco's bubble-sort curve. The paper also proves Theorem 3.2, the continuity of the permuton Demazure product under weak convergence, and uses it to show that the inversion count of the Demazure product of two independent uniform permutations is binom(n,2)(1-o(1)).","tokens_in":24924,"tokens_out":6971,"duration_ms":65215,"significance":"If the results are correct, this is a substantial contribution to the interface of algebraic combinatorics and integrable probability. The paper generalizes the Grothendieck permutons of Morales–Panova–Petrov–Yeliussizov from staircase shapes to all order-convex shapes, provides a direct TASEP route that bypasses the stochastic six-vertex model, and produces several explicit new families of permutons (peridot, Polyphemus, Platyhelminthes, pointy peanut). The min-plus formulation of the Demazure product on permutons is elegant and likely to be useful beyond this paper, as evidenced by the clean proof of the inversion-count result. The explicit verification of the limit formulas in the parallelogram and rectangle cases of Sections 5.1 and 5.2 is a genuine strength, as is the demonstrable use of external integrable asymptotics (Theorem 4.1) in a new setting. The main caveat is that the bridge from random pipe dreams to the TASEP, which underpins both the scaling limit and the KPZ fluctuation statement, is only sketched; the paper would be fully convincing once that coupling is supplied rigorously.","major_comments":[{"comment":"The column-by-column evolution of the states ι(v_j) is asserted to be equidistributed with the k-particle TASEP with geometric jumps, but no formal bijection is given. The pipe dream has independent tiles shared by all letters in a column, whereas the TASEP transition uses independent geometric variables G_i(t) for each particle at each time; the text does not prove that the joint law after processing columns T+1 through T′ equals the TASEP kernel, including the right-to-left order inside a column (the word ω_j lists contents b_j, b_j−1, …, a_j) and the reflecting wall at n+1−a_j. This coupling is the sole bridge to Theorem 4.1, so it is load-bearing for Theorems 1.1 and 1.2. A complete inductive construction or a direct bijection between pipe-dream tile configurations and the geometric jump variables should be written out.","section":"§4.2, paragraphs after Eq. (13)"},{"comment":"The proof of Theorem 1.1 concludes from the fixed-h limits in (14) being 0 or 1 that H_{u_n}(x,y) converges to h_p(x,y) with probability 1. As written, the threshold argument gives convergence in probability for each fixed (x,y), not almost-sure convergence; an almost-sure statement would require a Borel–Cantelli estimate or an explicit joint coupling across n. The theorem statement and the following sentence 'Equivalently, (π_{u_n}) converges weakly to ζ_D^p' also do not specify whether the convergence is almost sure or in probability. This should be clarified and the proof adjusted accordingly.","section":"§4.2, Eqs. (14)–(16)"},{"comment":"The assertion that '(x−y+H_w(x,y))n is equal to the number of particles in ι(w) occupying positions at or to the right of n−k′+1' is stated as a straightforward computation, but it is a key translation between permutation height functions and TASEP particle counts. A short derivation of this identity, including the handling of the floors in x=k′/n and y=(n−k)/n, would make the reduction self-contained and easier to verify in corner cases.","section":"§4.2, first paragraph"}],"minor_comments":[{"comment":"'In Theorem 1.7, we noted...' refers to Remark 1.7, not a theorem; the cross-reference should be corrected.","section":"§6.2, first sentence"},{"comment":"The displayed formulas for H_{ρ_{α,β}}(x,y) and H_{κ_β}(x,y) write h_1(x,y) in the integrand, but the variable of minimization is γ and the subsequent computations use h_1(x,γ); these are typos that should be corrected to h_1(x,γ).","section":"§5.1 and §5.2, height function formulas"},{"comment":"'one can in principal compute' should read 'one can in principle compute'.","section":"§4.4, last paragraph"},{"comment":"The rotation operator for permutons is denoted inconsistently: it appears as bµ in Remark 4.3 and as dν or dρ in Section 5.1. Please unify the notation.","section":"Remark 4.3 and §5.1"}],"recommendation":"major_revision","confidential_remarks":"The main results are likely correct and the paper is a strong contribution, but the missing formal pipe-dream/TASEP coupling in §4.2 is a genuine gap that must be closed before the scaling-limit and fluctuation theorems can be considered established. The almost-sure versus in-probability issue is also easy to fix but needs to be addressed. I would support acceptance after these revisions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper is worth a serious look. It generalizes the Grothendieck permuton construction from staircase shapes to arbitrary order-convex shapes, gives the first explicit densities for the standard bubble-sort permutons, and introduces a Demazure product on permutons via the Chan–Pflueger min-plus identity. The Demazure-product-on-permutons part is clean and likely correct; Theorem 3.2 (continuity) is proved in a few lines and is a useful tool. The bubble-sort results in Section 5 are concrete and check out in the special cases.\n\nThe soft spot is exactly the one flagged in the stress test. Section 4.2 asserts that processing columns T+1 through T' of an order-convex pipe dream updates the TASEP state ι(v_j) as one step of the geometric-jump TASEP, with inequality (13) ensuring that the right particles are mobile. This is load-bearing: it is the only bridge from random pipe dreams to the TASEP asymptotics of Theorem 4.1, and therefore to the explicit height formula and the Tracy–Widom statement. But the proof is a sketch. The pipe dream uses independent cross/bump tiles shared by all pipes in a column; the TASEP uses independent geometric jumps for each particle at each time. The text does not write out a bijection proving that the joint law of the particle configuration after each column matches the TASEP transition kernel, including the wall at n+1-a_j and the right-to-left order of the column's word. I don't see an obvious fatal flaw — the example is consistent — but the step needs a formal proof or a reference to a known equivalence.\n\nThere is a second, smaller gap: Theorem 1.1 states almost-sure convergence, but the displayed proof establishes convergence in probability. To get a.s. you need a Borel–Cantelli argument or a rate from Theorem 4.1, which isn't supplied. This is likely fixable.\n\nMinor remarks: Section 4.5 describes a decomposition method but leaves the computation to the reader; that's fine for a remark. The paper cites [22] for Theorem 4.1 without independent verification; acceptable.\n\nWho should read this: anyone working on permuton limits, integrable probability, or KPZ universality. The main ideas are right and the paper is well written. It deserves a serious referee. My recommendation: send it to peer review, but require the author to supply the missing coupling bijection and the a.s. convergence argument. The results are likely correct, but the manuscript as written is not fully rigorous at its core.","headline":"Load-bearing TASEP-coupling gap in Section 4.2 keeps a strong, likely-correct paper from being fully rigorous; worth peer review with a demand for a formal bijection.","tokens_in":25513,"tokens_out":3858,"would_cite":true,"duration_ms":32183,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60F05","05A05","20F55"],"pacs":[],"model":"deepseek-v4-flash","headline":"Demazure-product random permutations converge to explicit permutons, with Tracy–Widom height fluctuations.","keywords":["permutons","Demazure product","random pipe dreams","TASEP","KPZ universality","Tracy-Widom distribution","bubble sort","0-Hecke monoid"],"falsifier":"Take the staircase shape with $p=1/2$ and the point $(x,y)=(0.8,0.55)$, which lies in $K_{\\phi,\\psi}$; simulate the random pipe dream many times. Under the paper's claim the empirical height $H_{u_n}(0.8,0.55)$ must converge to $h_p^{\\phi,\\psi}(0.8,0.55)$ with deviations of order $n^{-2/3}$ whose scaled distribution is Tracy–Widom GUE; a systematic discrepancy in the limit value, or a fluctuation histogram that is not $F_2$ after the stated rescaling, would falsify the central theorem.","tokens_in":24455,"feed_emoji":"📊","tokens_out":19127,"duration_ms":152006,"temperature":0.7,"pith_summary":"Random permutations built from the Demazure product have exact large-scale limits, and this paper writes those limits down. For random pipe dreams filling any order-convex shape, the limiting permuton height is the closed form $h_p^{\\phi,\\psi}(x,y)=\\min(\\max(0,y-x,f_p^{\\phi,\\psi}(x,y)),1-x,y)$, with almost-sure convergence of the finite-$n$ permutations; inside the nontrivial region the height fluctuations are $n^{-2/3}$ and converge to the Tracy–Widom GUE law, so the model belongs to the KPZ universality class. For bubble-sort-type operators applied to a random permutation, the limiting height is a min-plus convolution of the initial permuton with the pipe-dream formula, and for Coxeter words this gives exact densities for the standard bubble-sort permutons whose support had already been computed. The same algebraic tool defines an associative Demazure product on permutons and shows that the Demazure product of two independent uniform random permutations is asymptotically the decreasing permutation, with $\\binom{n}{2}(1-o(1))$ expected inversions.","feed_headline":"One formula gives the limit shapes of two random permutation models","feed_subtitle":"Random pipe dreams and bubble-sort operations both converge to computable permutons with a known fluctuation law.","key_machinery":"The load-bearing object is the Demazure product on $S_n$, whose local generators $\\tau_i$ put entries $i$ and $i+1$ in decreasing order. Chan and Pflueger's formulation of this product as matrix multiplication in the min-plus tropical semiring gives the height-function identity $H_{u\\star v}(x,y)=\\min_\\gamma(H_u(\\gamma,y)+H_v(x,\\gamma))$, which the paper upgrades to a Demazure product on permutons. The second mechanism is a direct coupling: filling an order-convex pipe-dream shape column by column, with crossings resolved, reproduces the dynamics of the totally asymmetric simple exclusion process with geometric jumps under step initial data, so the height function at $(x,y)$ is controlled by the position of a single TASEP particle after $n(\\psi(x)-\\phi(y))$ steps. The explicit limit comes from solving $c_p(x-y+h,\\psi(x)-\\phi(y))=h$ for $h$, with $c_p$ the hydrodynamic speed of that TASEP; the same integrable fluctuations produce the $n^{-2/3}$ Tracy–Widom GUE statement.","core_discovery":"The paper's central discovery is that the Demazure product, the one-step 'sort these two adjacent entries' operation of the 0-Hecke monoid, has a limit theory governed by a single min-plus identity. For permutations $u,v$, the height function satisfies $H_{u\\star v}(x,y)=\\min_{0\\le\\gamma\\le1}(H_u(\\gamma,y)+H_v(x,\\gamma))$, and this identity extends to permutons, making the Demazure product an associative operation on limit shapes. Reading a random pipe dream column by column turns it into the totally asymmetric simple exclusion process with geometric jumps and step initial data; the limiting height $h_p^{\\phi,\\psi}$ is the value $h$ at which the TASEP hydrodynamic speed balances the displacement equation $c_p(x-y+h,\\psi(x)-\\phi(y))=h$, and the same TASEP asymptotics yield Tracy–Widom GUE fluctuations of order $n^{-2/3}$. The paper then uses the permuton product to derive explicit limiting densities for bubble-sort-type operators, recovering and refining DiFranco's support-only description of standard bubble-sort permutons, and to prove that the Demazure product of two independent uniform random permutations converges to the anti-diagonal line segment.","pith_inferences":["The min-plus form of the height function suggests that permuton height functions form a closed algebra under a tropical convolution; one could test whether the Legendre–Fenchel transform of $H_{\\mu\\star\\nu}$ decomposes as a sum, which would give a variational proof of the product rule and a dual method for computing limits.","The Dory memory model is a quantitative corollary of the rectangle-shape limit: the forgotten relevance factors, plotted against forgetting time, should converge to $\\nu_{\\beta(1-\\beta),\\beta}$, so the permuton gives explicit predictions for the empirical distribution of forgotten facts that a simulation of the exact process can check.","The paper's doppelgänger coincidence between rectangle and parallelogram limits hints that the uniform permuton's Demazure product may depend only on a projection of the second factor; checking whether $\\upsilon\\star\\zeta$ is unchanged by certain shape deformations would explain when such coincidences occur.","The visible stripes in the plots likely come from the singular curves in the limiting permuton support; a local-limit analysis near those curves would predict the stripe spacing, and direct sampling at moderate $n$ could confirm the predicted local density profile."],"forward_implications":["The staircase Grothendieck-permuton theorem is the special case $\\phi(z)=0$, $\\psi(z)=z$, and the same theorem produces new explicit families: peridot permutons for rectangles, Polyphemus permutons for trapezoids, and assembled permutons for decomposable non-order-convex shapes.","For every sequence of order-convex shapes whose scaled boundaries converge to $\\phi,\\psi$, the random pipe-dream permutation converges almost surely to the permuton with height $h_p^{\\phi,\\psi}$; inside $K_{\\phi,\\psi}$ the rescaled height fluctuations converge to the Tracy–Widom GUE distribution.","Applying $\\tau_{w(S)}$ to a uniformly random permutation gives the permuton $\\upsilon\\star\\zeta^D_1$, and for Coxeter words with linear boundaries the limit is exactly $\\nu_{\\alpha,\\beta}$, a full density description of the standard bubble-sort permutons.","The Demazure product of two independent uniform random $n$-permutations has expected inversion number $\\binom{n}{2}(1-o(1))$, and every pattern except the decreasing one has vanishing density in the limit; the permuton product of two uniform permutons is the anti-diagonal line segment.","Because the permuton Demazure product is associative and compatible with weak convergence, limits for shapes cut into finitely many order-convex pieces can be assembled by starring the piecewise limits, as illustrated by the Platyhelminthes and pointy-peanut permutons."],"supporting_citations":[{"why":"Supplies the TASEP hydrodynamic limit and Tracy–Widom fluctuation theorem that Theorem 1.2 imports, and is the staircase-shape result this paper generalizes.","marker":"[22]"},{"why":"Gives the min-plus tropical formulation of the Demazure product that yields the height-function identity for $u\\star v$.","marker":"[8]"},{"why":"Provides the full proof of the min-plus formulation used to define and justify the permuton Demazure product.","marker":"[23]"},{"why":"Supplies the equivalence between weak convergence of permutons and pointwise convergence of height functions, together with the pattern-density criterion.","marker":"[17]"},{"why":"Gives the support curve of the standard bubble-sort permutons that Theorem 1.5 refines with exact densities.","marker":"[12]"},{"why":"Introduces the TASEP with geometric jumps whose step-initial-data dynamics are coupled to pipe dreams in Section 4.2.","marker":"[24]"}],"fun_headline_variants":["Permutons meet TASEP: limit shapes via one min-plus formula","Demazure product yields limit laws for random permutations","One identity shapes all: pipe dreams and bubble sort","Random permutations converge to permutons via a single formula","Demazure product: the key to permuton limits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The formulas rest on the claim that reading a random pipe dream column by column reproduces exactly the particle positions of the totally asymmetric simple exclusion process with geometric jumps; if that correspondence fails in any corner case, the explicit height formula and the Tracy–Widom fluctuation law do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Permutons meet TASEP: limit shapes via one min-plus formula","Demazure product yields limit laws for random permutations","One identity shapes all: pipe dreams and bubble sort","Random permutations converge to permutons via a single formula","Demazure product: the key to permuton limits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000765,"raw_usage":{"total_tokens":3451,"prompt_tokens":1063,"completion_tokens":2388,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":679,"completion_tokens_details":{"reasoning_tokens":2306}},"tokens_in":679,"tokens_out":2388,"duration_ms":14237,"temperature":1.0,"reasoning_tokens":2306,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:14:39.184422+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the staircase shape with $p=1/2$ and the point $(x,y)=(0.8,0.55)$, which lies in $K_{\\phi,\\psi}$; simulate the random pipe dream many times. Under the paper's claim the empirical height $H_{u_n}(0.8,0.55)$ must converge to $h_p^{\\phi,\\psi}(0.8,0.55)$ with deviations of order $n^{-2/3}$ whose scaled distribution is Tracy–Widom GUE; a systematic discrepancy in the limit value, or a fluctuation histogram that is not $F_2$ after the stated rescaling, would falsify the central theorem.","supporting_citations":[{"cited_title":"Grothendieck Shenanigans: Permutons from pipe dreams via integrable probability","cited_arxiv_id":"2407.21653","evidence_quote":"Supplies the TASEP hydrodynamic limit and Tracy–Widom fluctuation theorem that Theorem 1.2 imports, and is the staircase-shape result this paper generalizes."},{"cited_title":"Chan and N","cited_arxiv_id":null,"evidence_quote":"Gives the min-plus tropical formulation of the Demazure product that yields the height-function identity for $u\\star v$."},{"cited_title":"Hoppen, Y","cited_arxiv_id":null,"evidence_quote":"Supplies the equivalence between weak convergence of permutons and pointwise convergence of height functions, together with the pattern-density criterion."},{"cited_title":"DiFranco","cited_arxiv_id":null,"evidence_quote":"Gives the support curve of the standard bubble-sort permutons that Theorem 1.5 refines with exact densities."},{"cited_title":"Vershik and S","cited_arxiv_id":null,"evidence_quote":"Introduces the TASEP with geometric jumps whose step-initial-data dynamics are coupled to pipe dreams in Section 4.2."}],"review_version":1}