{"id":"7626b830-f4ea-4c49-86a9-e996515464d9","arxiv_id":"2506.17042","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"On affine buildings, caloric functions with weighted ℓ^1 or radial ℓ^1 initial data converge, after normalization, to the product of a p-mass function and the heat kernel in ℓ^p norm.","lead":"This paper proves that solutions to the discrete heat equation on affine buildings, a non-Archimedean analogue of symmetric spaces, split after long time into a mass function times the heat kernel. The result covers buildings without any transitive group action and gives explicit ℓ^p growth rates and concentration regions for the heat kernel.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem B inherits the pointwise heat-kernel asymptotics (1.25)-(1.26) from [25]; their validity for exotic buildings without a transitive group action is the least secure condition for the central claim.","rationale":"The reader's conditional verdict is appropriate. The central argument is structurally sound: mass functions are defined independently of the asymptotics; the approximation of l^1(w_p) and radial l^1 data by finitely supported functions works via Minkowski's inequality and the uniform bound phi_{s_p} <= 1; and the decoupling on and off the critical regions is correctly reduced to the ratio limits from Section 3. The detected p=2 flaw in Theorem 4.9 is real: the proof sets gamma = 1/(2(|Phi++|+1)), which violates the standing requirement 0 < gamma < 1/(4|Phi++|) whenever |Phi++| >= 2, and for |Phi++| = 1 the required inequality is not strict. However, this only invalidates the stated rate in Theorem 4.9; choosing any admissible gamma, for example gamma = 1/(8|Phi++|), makes both error terms epsilon'_n and epsilon''_n tend to zero, so the l^2 convergence in Theorem B still follows with a slower, unspecified rate. Thus the p=2 issue is a correctness risk in the rate statement, not in the central convergence claim. The load-bearing risk is instead the unverified inheritance of (1.25)-(1.26) for exotic buildings. That condition is not checked in the present paper, and if it fails for buildings without a transitive group action, the main theorem loses its claimed generality. A verification of the scope of [25] would settle this concern. I therefore agree with the reader's weakest assumption and keep the CONDITIONAL verdict unchanged.","tokens_in":41349,"tokens_out":12106,"duration_ms":111592,"concrete_test":"Verify in [25] whether Theorem 4.1 and Corollary 4.10 are stated for all affine buildings satisfying only the standing assumptions of Section 1.3 (regular, thick, locally finite, complete apartment system, no transitive group action), or whether they require a group action or Bruhat-Tits structure. If the required generality is absent, the exotic-building claim in Theorem B is unsupported. As a computational cross-check, run a finite-range isotropic random walk on an explicit exotic affine building (for example, Ronan's example [24]) at moderate n and test whether the ratio asymptotics of Theorem 3.1 match the claimed leading term.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem B (and the supporting Theorems 4.7-4.9) depends on the pointwise heat-kernel asymptotics (1.25) and (1.26), imported from [25, Theorem 4.1 and Corollary 4.10]. The paper applies these expansions to every regular, thick, locally finite affine building with a complete apartment system, explicitly including exotic buildings with no transitive group action, and it inherits their uniform error terms over the critical regions N^p_n and over y in the support of f. If [25]'s expansions hold only under a Bruhat-Tits or transitive-group hypothesis, or fail in the exotic setting, then the proof of Theorem B for exotic buildings is not established. This is the least secure condition for the central claim because it is not verified anywhere in the present text. A secondary internal issue, the p=2 gamma choice in Theorem 4.9, is concrete but does not threaten the convergence claim; it only invalidates the stated rate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the large-time asymptotic behavior of solutions to the discrete heat equation on affine buildings, including buildings without a transitive automorphism group. For each p in [1,∞] it introduces a p-mass function M_p(f) and proves that, under suitable weighted-ℓ^1 or radial assumptions on the initial datum f, the caloric function u(n,·) decouples asymptotically as M_p(f)(·) k_n(o,·) in ℓ^p norm. The paper also establishes three regimes for the ℓ^p growth of the heat kernel (Theorem A) and describes the critical regions where the mass dominates. The proofs rely on pointwise heat-kernel asymptotics imported from [25], on new ratio-limit theorems, and on a Helgason transform and Plancherel formula developed for affine buildings, including exotic ones.","tokens_in":41515,"tokens_out":7754,"duration_ms":72629,"significance":"If the imported pointwise asymptotics hold in the claimed generality, this is a substantial contribution: it provides non-Archimedean analogues of classical Euclidean and symmetric-space decoupling results, extends them to buildings without transitive group actions, and reveals genuinely different behavior for p<2, p=2, and p>2. The Helgason transform and Fourier-inversion theorems for exotic buildings (Theorems 4.3 and 4.6) are of independent interest. The paper is clearly organized and the algebraic/geometric setup is careful. The main risk is that the central theorem inherits unverified pointwise asymptotics from [25] for exotic buildings, and the p=2 rate in Theorem 4.9 contains an internal inconsistency with Theorem 2.5.","major_comments":[{"comment":"The proof of the main decoupling theorem for arbitrary regular, thick, locally finite affine buildings with a complete apartment system, including exotic buildings without a transitive group action, is built on the pointwise heat-kernel asymptotics (1.25)–(1.26) imported from [25, Theorem 4.1 and Corollary 4.10]. The present text neither states the precise hypotheses of those results nor verifies that they hold for buildings without a transitive automorphism group. Since Theorem B is claimed for exactly this class, the authors should either supply the missing verification or restrict the statement to the setting in which (1.25)–(1.26) are actually established.","section":"Section 1.5, Eqs. (1.25)–(1.26); Theorems 4.7–4.10"},{"comment":"In the proof of Theorem 4.9, the case p=2 chooses γ = 1/(2(|Φ++|+1)) so that ε''_n = O(ε'_n). However, Theorem 2.5 requires 0 < γ < 1/(4|Φ++|). For every |Φ++| ≥ 1, the chosen value violates this strict inequality (it equals the upper bound when |Φ++|=1 and exceeds it for larger |Φ++|). Consequently the stated rate ε_n = n^{-|Φ++|/(2|Φ++|+2)} is not justified by the estimates proved earlier; only the convergence assertion (with a possibly slower rate) is supported. The theorem and its proof need to be reconciled, for instance by modifying the critical region N^2_n or the admissible γ range, or by proving the endpoint case separately.","section":"Theorem 4.9 (p=2) vs. Theorem 2.5"}],"minor_comments":[{"comment":"The keyword list contains the typo \"excotic building\"; it should read \"exotic building\".","section":"Abstract / keywords"},{"comment":"The text says \"caloric functions associated with anisotropic random walk\" but the definition that follows describes an isotropic (vectorial-distance-invariant) random walk. The terminology should be made consistent.","section":"Section 1.5, opening paragraph"},{"comment":"The BC_r case in the proof of Lemma 1.1(iii) is dispatched with a short observation about the Weyl group; since BC_r buildings are explicitly covered, a few more details would improve readability.","section":"Lemma 1.1(iii)"},{"comment":"For p=2 two mass functions are introduced and said not to coincide in general; a concrete example or reference illustrating their difference would help the reader gauge the significance of this phenomenon.","section":"Section 4.2, Eqs. (4.6a)–(4.6b)"}],"recommendation":"major_revision","confidential_remarks":"The central convergence result is likely correct, and the paper is well written. The two major issues are: (1) the unverified dependence on [25]'s pointwise asymptotics for exotic buildings, which is load-bearing for the advertised scope; and (2) the p=2 rate in Theorem 4.9, which is internally inconsistent with Theorem 2.5 and needs a genuine fix, not just a cosmetic change. The rate error is local and does not undermine the convergence claim, but it should be corrected before publication. Please ask the authors to clarify the scope of the imported asymptotics and to repair the p=2 rate argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the Papageorgiou–Trojan paper on caloric functions on affine buildings. Worth your time if you work on analysis on buildings. The main result, Theorem B, says that for an admissible random walk, caloric functions with initial data in ℓ¹(w_p) or radial ℓ¹ decouple as M_p(f)·k_n asymptotically in ℓ^p norm. That is the right analogue of the symmetric-space theorems, and doing it for exotic buildings without a transitive group action is a genuine step forward. The Helgason transform they introduce for this setting is new and independently useful.\n\nThe proof strategy is sound: define mass functions a priori from the geometry (Busemann function, harmonic measure, Macdonald functions), prove ratio limits for heat kernels, then combine concentration estimates. The heat kernel norm results (Theorem A) with the three regimes p<2, p=2, p>2 are also well done.\n\nNow the soft spots. The concrete one: Theorem 4.9 claims an error rate n^{-|Φ++|/(2|Φ++|+2)} for p=2. To get it, the proof chooses γ=1/(2(|Φ++|+1)), but Theorem 2.5, which supplies the outside-region estimate, requires γ<1/(4|Φ++|). For every |Φ++|≥1 the chosen γ violates that bound—equality at |Φ++|=1, strict violation for larger. So the stated rate is not derived. This is not fatal: you can pick a smaller valid γ and still get convergence with a slightly worse rate. But the theorem statement overclaims.\n\nBigger caveat: the whole proof imports the pointwise heat kernel asymptotics (1.25)–(1.26) from Trojan's JEMS paper [25], applied to every regular thick locally finite affine building with complete apartment system, including exotic ones. The present text does not verify that those expansions hold in that generality. If [25] is limited to Bruhat–Tits buildings, the exotic-building theorems are not established here. I trust the second author on this, but a referee should check.\n\nThe circularity worry is misplaced—mass functions are defined from building data, not fitted to the limit. Citation pattern looks fine.\n\nVerdict: the paper deserves a serious referee. The main qualitative result is plausible and likely correct; the p=2 rate needs fixing, and the dependence on [25] needs explicit scoping. I'd accept it for review with a request for revision, not reject.","headline":"A credible extension of mass-function asymptotics to affine buildings, with a real but repairable gap in the p=2 rate and heavy dependence on imported heat-kernel estimates.","tokens_in":42090,"tokens_out":8922,"would_cite":true,"duration_ms":78523,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K08","35B40","51E24","58J35","60B15","20E42","20F55","22E35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Discrete heat flow on affine buildings becomes mass times heat kernel","keywords":["affine buildings","caloric functions","discrete heat equation","mass functions","heat kernel asymptotics","exotic buildings","random walks","Macdonald spherical functions"],"falsifier":"On a rank-2 affine building with thickness 2 and an admissible random walk, compute the normalized $\\ell^p$ error $\\|u(n;\\cdot)-M_p(f)k_n(o,\\cdot)\\|_{\\ell^p}/\\|k_n(o,\\cdot)\\|_{\\ell^p}$ for a finitely supported, non-radial $f$ and $p=3/2$; Theorem 4.9 predicts it is $O(n^{-1/2+\\gamma})$. If it fails to tend to 0 for some $f$, the decoupling claim is false for that building; if it tends to 0 but at a different rate, only the error estimate needs revision, while the limit statement may survive.","tokens_in":41084,"feed_emoji":"♨️","tokens_out":18851,"duration_ms":164233,"temperature":0.7,"pith_summary":"This paper establishes a decoupling theorem for the discrete-time heat equation on affine buildings: if the initial datum $f$ is integrable against a $p$-dependent weight, or is radial and in $\\ell^1$, then the caloric function $u(n,\\cdot)$ is asymptotically indistinguishable in $\\ell^p$ norm from the product of a data-dependent mass function $M_p(f)$ and the heat kernel $k_n(o,\\cdot)$, with relative error tending to zero. The mass function is the new ingredient: for $p<2$ it averages the horocycle (Busemann) contribution over the sector boundary, while for $p\\ge2$ it is a convolution with the ground-state spherical function; for radial data both reduce to the single constant obtained by evaluating the Helgason transform at $s_p$. The same machinery yields the sharp $\\ell^p$ growth rates of the heat kernel, which split into three regimes, with $p=2$ behaving discontinuously from its neighbors. The results hold for regular, thick, locally finite affine buildings with complete apartment systems, including exotic buildings on which no group acts transitively, extending classical Euclidean and symmetric-space asymptotics to the non-Archimedean setting.","feed_headline":"Discrete heat flow on affine buildings becomes mass times heat kernel","feed_subtitle":"Extends Euclidean and symmetric-space asymptotics to non-Archimedean buildings, including exotic ones with no group action.","key_machinery":"The load-bearing tool is a pair of sharp pointwise heat-kernel expansions, (1.25) and (1.26): in the Cramér zone at bounded distance from the boundary of $\\mathcal{M}$, $k(n;x_n)\\sim n^{-r/2}\\rho^n e^{-n\\phi(\\delta_n)}\\chi_0(\\sigma(o,x_n))^{-1/2}(\\det B_{s_n})^{-1/2}c(s_n)^{-1}$, while for $\\sigma(o,x_n)/n\\to0$, $k(n;x_n)\\sim n^{-r/2-|\\Phi^{++}|}\\rho^n e^{-n\\phi(\\delta_n)}\\Phi(\\sigma(o,x_n))$. These are converted by Theorem 3.1 and Theorem 3.3 into ratio limits for $k(n;y,x_n)/k(n;o,x_n)$ that are uniform over the support of $f$; it is these ratio limits that force the caloric function to factor as $M_p(f)k_n$. The mass functions themselves are defined through horocycle (Busemann) functions and Macdonald spherical functions, and a Helgason transform on buildings—introduced in this paper, including for exotic buildings—provides the inversion and convolution identities connecting spatial data to the spherical/eigenfunction picture.","core_discovery":"For an admissible random walk on the good vertices of a regular, thick, locally finite affine building with a complete apartment system, the paper proves that the solution $u(n,x)=\\sum_y k_n(y,x)f(y)$ of the discrete heat equation obeys $$\\lim_{n\\to\\infty}\\frac{\\|u(n;\\cdot)-M_p(f)(\\cdot)k_n(o,\\cdot)\\|_{\\ell^p}}{\\|k_n(o,\\cdot)\\|_{\\ell^p}}=0$$ for every $p\\in[1,\\infty]$, whenever $f\\in\\ell^1(w_p)$ or $f$ is radial and in $\\ell^1(V_g)$. The mass function is $M_p(f)(x)=\\sum_y f(y)\\,\\frac{1}{\\nu(\\Omega(o,x))}\\int_{\\Omega(o,x)}\\chi_0(h(o,y;\\omega))^{1/p}\\,d\\nu(\\omega)$ for $p<2$, and $M_p(f)(x)=(1/\\varphi_0(x))(f\\times\\varphi_0)(x)$ for $p\\ge2$; for radial $f\\in\\ell^1(V_g)$ it is the constant $M_p(f)=\\mathcal{H}f(s_p)$ with $s_p=\\eta(2/p-1)$ for $p<2$ and $s_p=0$ for $p\\ge2$. Theorem A supplies the companion heat-kernel norms: $\\|k_n\\|_{\\ell^p}\\approx n^{-r/(2p')}\\rho^n\\kappa(s_p)^n$ for $p\\in[1,2)$, $\\approx n^{-r/4-|\\Phi^{++}|/2}\\rho^n$ for $p=2$, and $\\approx n^{-r/2-|\\Phi^{++}|}\\rho^n$ for $p\\in(2,\\infty]$, together with explicit critical regions $\\mathcal{N}^p_n$ on which the $\\ell^p$ mass concentrates. The result is stated for buildings without any transitive group action, so it covers the so-called exotic affine buildings that have no continuous analogue.","pith_inferences":["The three-regime split in Theorem A suggests a crossover in the mechanism controlling heat decay: for $p<2$ the large-deviation factor $\\kappa(s_p)^n$ dominates, for $p>2$ the volume-growth factor $|\\Phi^{++}|$ dominates, and $p=2$ is the balance point—an interpretation the paper hints at but does not name.","Because the proof for exotic buildings avoids any group action, one could try to transplant the same mass-function construction to other non-classical discrete spaces with similar volume growth, such as products of trees or other graphs with exponential volume growth; the decoupling may survive, but the sharp heat-kernel expansions would need to be re-established.","The Helgason transform developed here, including the inversion formula, is likely to be useful beyond heat equations—for instance in boundary-value problems or in defining Hardy-type spaces on buildings, in analogy with the symmetric-space theory.","For $p<2$, the mass function $M_p(f)(x)$ depends on $x$ through the sector set $\\Omega(o,x)$, so non-radial data produce a spatially varying coefficient; the paper leaves open whether this variation has a geometric interpretation as a boundary integral of the Helgason transform evaluated at the frequency $s_p$."],"forward_implications":["For finitely supported initial data the decoupling carries an explicit rate: the normalized $\\ell^p$ error is $O(n^{-1/2+\\gamma})$ for $p<2$, $O(n^{-|\\Phi^{++}|/(2|\\Phi^{++}|+2)})$ for $p=2$, and $O(n^{-1}r_n)$ for $p>2$, with $r_n$ growing faster than $\\log n$ and slower than $\\sqrt n$ (Theorem 4.9).","The heat kernel's $\\ell^p$ mass concentrates in explicit critical regions: around the drift ray $n\\delta_p$ for $p<2$, in a spherical shell of radius $\\sim\\sqrt n$ away from walls for $p=2$, and in a ball of radius $r_n$ for $p>2$ (Theorems 2.4–2.6).","For radial $\\ell^1$ data, $M_p(f)\\equiv \\mathcal{H}f(s_p)$, so the caloric function is eventually a single heat kernel with that coefficient, in every $\\ell^p$ norm (Theorem 4.10).","The two candidate definitions of the $p=2$ mass function both yield the same $\\ell^2$ limit, although they coincide only for radial data (Remark 1).","For Bruhat–Tits buildings of semisimple groups over local fields, the decoupling extends to radial data in the larger weighted classes $\\ell^1_{s_p}$ (Theorem 4.11)."],"supporting_citations":[{"why":"Supplies the pointwise heat-kernel expansions (1.25) and (1.26) on which Theorems A and B are built.","marker":"[25]"},{"why":"Provides the spherical harmonic analysis, Plancherel formula, and inversion on affine buildings used to define the mass functions and the Helgason transform.","marker":"[22]"},{"why":"The symmetric-space analogue for the heat equation that the paper extends to the non-Archimedean setting.","marker":"[4]"},{"why":"The hyperbolic-space result for radial $L^1$ initial data that motivates and is generalized by the radial $\\ell^1$ case.","marker":"[26]"},{"why":"Introduces mass functions for non-radial data on symmetric spaces, the template for the building mass functions.","marker":"[19]"},{"why":"Constructs affine buildings without transitive group actions, the exotic buildings the paper explicitly aims to cover.","marker":"[24]"}],"fun_headline_variants":["Heat on affine buildings splits into mass and kernel","Caloric functions on buildings: mass times kernel asymptotics","Non-Archimedean heat flow: mass-kernel separation proven","Affine buildings heat asymptotics: mass function and kernel","Discrete heat flow decouples into mass and kernel on buildings"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central proof inherits, without re-proving, the pointwise heat-kernel expansions (1.25) and (1.26) for every regular, thick, locally finite affine building with a complete apartment system; if those expansions fail, or fail to be uniform over the relevant regions and over the support of $f$, the main decoupling theorem is not established for the buildings where they fail, in particular not for exotic buildings.","fun_headline_variants_meta":{"raw":{"variants":["Heat on affine buildings splits into mass and kernel","Caloric functions on buildings: mass times kernel asymptotics","Non-Archimedean heat flow: mass-kernel separation proven","Affine buildings heat asymptotics: mass function and kernel","Discrete heat flow decouples into mass and kernel on buildings"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000359,"raw_usage":{"total_tokens":2046,"prompt_tokens":1148,"completion_tokens":898,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":764,"completion_tokens_details":{"reasoning_tokens":814}},"tokens_in":764,"tokens_out":898,"duration_ms":10104,"temperature":1.0,"reasoning_tokens":814,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:13:43.154994+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a rank-2 affine building with thickness 2 and an admissible random walk, compute the normalized $\\ell^p$ error $\\|u(n;\\cdot)-M_p(f)k_n(o,\\cdot)\\|_{\\ell^p}/\\|k_n(o,\\cdot)\\|_{\\ell^p}$ for a finitely supported, non-radial $f$ and $p=3/2$; Theorem 4.9 predicts it is $O(n^{-1/2+\\gamma})$. If it fails to tend to 0 for some $f$, the decoupling claim is false for that building; if it tends to 0 but at a different rate, only the error estimate needs revision, while the limit statement may survive.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the pointwise heat-kernel expansions (1.25) and (1.26) on which Theorems A and B are built."},{"cited_title":"Parkinson","cited_arxiv_id":null,"evidence_quote":"Provides the spherical harmonic analysis, Plancherel formula, and inversion on affine buildings used to define the mass functions and the Helgason transform."},{"cited_title":"Anker, E","cited_arxiv_id":null,"evidence_quote":"The symmetric-space analogue for the heat equation that the paper extends to the non-Archimedean setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The hyperbolic-space result for radial $L^1$ initial data that motivates and is generalized by the radial $\\ell^1$ case."},{"cited_title":"Papageorgiou.𝐿𝑝 asymptotics for the heat equation on symmetric spaces for non-symmetric solutions.Int","cited_arxiv_id":null,"evidence_quote":"Introduces mass functions for non-radial data on symmetric spaces, the template for the building mass functions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Constructs affine buildings without transitive group actions, the exotic buildings the paper explicitly aims to cover."}],"review_version":2}