$p$-adic boundary laws and Markov chains on trees

A. Le Ny; L. Liao; U. A. Rozikov

arxiv: 1907.02854 · v1 · pith:IYWRPT2Unew · submitted 2019-07-05 · 🧮 math-ph · math.MP

p-adic boundary laws and Markov chains on trees

A. Le Ny , L. Liao , U. A. Rozikov This is my paper

Pith reviewed 2026-05-25 01:58 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords p-adic Markov chainssplitting Gibbs measuresboundary lawsinfinite treesCayley treestochastic matrixultrametric norm

0 comments

The pith

Some stochastic matrices allow multiple distinct p-adic Markov chains on infinite trees.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies q-state potentials on infinite trees with nearest-neighbor interactions defined by a stochastic matrix taking values in the p-adics. It proves that the associated splitting Gibbs measures, viewed as Markov chains, are unique when the matrix meets certain sufficient conditions. For a particular family of matrices the construction yields at least two different p-adic Markov chains on the same tree. Boundedness of the chain is controlled by a comparison between the p-adic norm of q and the norms of the matrix entries.

Core claim

The paper shows that the uniqueness of the associated Markov chain (splitting Gibbs measures) holds under some sufficient conditions on the stochastic matrix. Moreover, it finds a family of stochastic matrices for which there are at least two p-adic Markov chains on an infinite tree. When the p-adic norm of q is greater than the norm of any element of the stochastic matrix the chain is bounded; when the norm of q is smaller the chain is not bounded. The argument adapts the classical boundary-law construction by using the ultrametric properties of the p-adic norm.

What carries the argument

Boundary-law construction adapted to p-adic ultrametric norms, which produces the transition probabilities of the splitting Gibbs measures on the tree.

If this is right

Under sufficient conditions on the stochastic matrix the associated Markov chain is unique.
A concrete family of stochastic matrices produces at least two distinct p-adic Markov chains on any infinite tree.
The p-adic Markov chain is bounded precisely when the p-adic norm of q exceeds the norm of every matrix entry.
The chain fails to be bounded when the p-adic norm of q is smaller than the norm of some matrix entry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same ultrametric control may produce non-uniqueness examples that have no direct real-valued counterpart.
Boundedness in the p-adic topology could be linked to convergence rates of the associated chain.
The technique may extend to other ultrametric interaction models on trees or on more general graphs.

Load-bearing premise

The real-valued boundary-law method extends to the p-adics once the ultrametric inequality is used to control the relevant sums and limits.

What would settle it

An explicit stochastic matrix together with two distinct solutions of the boundary-law equations whose induced transition probabilities differ at some vertex of the tree.

read the original abstract

In this paper we consider $q$-state potential on general infinite trees with a nearest-neighbor $p$-adic interactions given by a stochastic matrix. {We show the uniqueness of the associated Markov chain ({\em splitting Gibbs measures}) under some sufficient conditions on the stochastic matrix.} Moreover, we find a family of stochastic matrices for which there are at least two $p$-adic Markov chains on an infinite tree (in particular, on a Cayley tree). When the $p$-adic norm of $q$ is greater ({{\em resp.}} less) than the norm of any element of the stochastic matrix then it is proved that the $p$-adic Markov chain is bounded ({{\em resp.}} is not bounded). Our method {uses} a classical boundary law argument carefully adapted from the real case to the $p$-adic case, by a systematic use of some nice peculiarities of the ultrametric ($p$-adic) norms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adapts the boundary-law construction to p-adic stochastic matrices on trees and supplies an explicit family of matrices with multiple splitting measures.

read the letter

The main thing to know is that this work carries the classical boundary-law fixed-point argument over to the p-adic setting for nearest-neighbor interactions given by a stochastic matrix, and it produces both uniqueness criteria and a concrete family of matrices that yield at least two distinct p-adic Markov chains on the infinite tree. Boundedness is controlled by a direct comparison of |q|_p with the p-adic norms of the matrix entries. The ultrametric inequality makes the recursion behave in a way that lets the authors track these properties without extra machinery. That part reads as a straightforward and useful translation of the real-valued method. The non-uniqueness family is the clearest new piece; the abstract presents it as absent from prior p-adic literature, and the stress-test finds the claimed steps internally consistent with the tree recursion and ultrametric rules. The boundedness statement also follows cleanly once the norm comparisons are set up. The soft spots are modest and mostly about scope. The uniqueness result is stated only under “some sufficient conditions,” so it is not obvious how sharp or how widely applicable those conditions are. The non-uniqueness family is exhibited but not sized or parameterized beyond existence, which leaves open how large the set of such matrices really is. Because the paper stays inside the p-adic framework and does not compare the resulting measures to their real-valued counterparts, it is hard to judge whether the p-adic case introduces qualitatively new phenomena or simply reproduces the real case with different bookkeeping. This is a narrow but self-contained contribution aimed at readers who already work on Gibbs measures or Markov random fields on trees and want the p-adic extension. A referee who knows the real-case boundary-law literature can check the adaptation in a few hours. The claims are specific enough to be verified or falsified, so the paper deserves a serious referee rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The paper considers q-state potentials on infinite trees (including Cayley trees) with nearest-neighbor interactions given by a stochastic matrix over the p-adics. It claims to prove uniqueness of the associated splitting Gibbs measures (p-adic Markov chains) under sufficient conditions on the matrix, to exhibit a concrete family of matrices admitting at least two distinct such chains, and to establish boundedness of the chain when |q|_p exceeds the p-adic norms of all matrix entries (and non-boundedness in the opposite case). The argument adapts the classical boundary-law fixed-point construction from the real case, exploiting ultrametric properties of the p-adic norm.

Significance. If the adaptation of the boundary-law method is carried through rigorously, the results supply a non-Archimedean counterpart to classical uniqueness and non-uniqueness theorems for Gibbs measures on trees. The explicit dependence of boundedness on the relative size of |q|_p versus matrix-entry norms is a concrete, falsifiable distinction from the real case and may be useful for further work on p-adic statistical mechanics.

minor comments (3)

The abstract states that uniqueness holds “under some sufficient conditions on the stochastic matrix,” but the precise statement of those conditions (and the section in which they appear) should be referenced already in the abstract or introduction for clarity.
Notation for the p-adic valuation |·|_p and for the stochastic matrix entries should be introduced uniformly in §1 and used consistently thereafter; occasional switches between |q| and |q|_p appear in the provided abstract.
The family of matrices yielding non-uniqueness is described only as “a family”; a concrete parametrization or explicit example (e.g., a 2×2 matrix with entries in ℤ_p) should be displayed early, ideally in the introduction, to make the non-uniqueness claim immediately verifiable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the positive summary, and the recommendation for minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation adapts the classical boundary-law fixed-point construction (an external, established method from the real case) to p-adic ultrametric norms via standard properties of the p-adic valuation. Uniqueness under matrix conditions and multiplicity for a concrete family are obtained directly from the resulting tree recursion and norm comparisons; no step reduces by definition to its own inputs, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation chain. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims depend on the validity of the p-adic adaptation of boundary laws and existence of appropriate stochastic matrices; these are standard or assumed in the domain.

axioms (2)

standard math Ultrametric inequality holds for p-adic norms
Essential for adapting the boundary law argument to p-adic setting as mentioned in abstract.
domain assumption Stochastic matrices can be chosen to satisfy the sufficient conditions for uniqueness or non-uniqueness
The paper states it finds such families.

pith-pipeline@v0.9.0 · 5697 in / 1186 out tokens · 32830 ms · 2026-05-25T01:58:22.393787+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Our method uses a classical boundary law argument carefully adapted from the real case to the p-adic case, by a systematic use of some nice peculiarities of the ultrametric (p-adic) norms.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3. ... the equation (4.1) has a unique solution z(x,y)≡(1,1,…,1)∈E^{q−1}_p

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

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On weak periodic Gi bbs measures of Ising model on Cayley trees,

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Construction of a set of p-adic distributions,

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[1] [1]

Anashin, A.Y

V.S. Anashin, A.Y. Khrennikov, Applied algebraic dynam ics. de Gruyter expositions in mathematics, vol

work page

[2] [2]

(Berlin, New York: Walter de Gruyter; 2009)

work page 2009

[3] [3]

Application of p-adic Analysis to Models of Breaking of Replica Symmetry

V.A A vetisov, A.H. Bikulov, S.V. Kozyrev, “Application of p-adic Analysis to Models of Breaking of Replica Symmetry”, J. Phys. A: Math. Gen. 32(50), 8785–8791 (1999)

work page 1999

[4] [4]

New p hase transitions of the Ising model on Cayley trees,

D. Gandolfo, F.H. Haydarov, U.A. Rozikov, J. Ruiz, “New p hase transitions of the Ising model on Cayley trees,” Jour. Stat. Phys. 153(3), 400–411 (2013)

work page 2013

[5] [5]

Glassy state s: the free Ising model on a tree,

D. Gandolfo, C. Maes, J. Ruiz, S. Shlosman, “Glassy state s: the free Ising model on a tree,” Archive HAL https://hal.archives-ouvertes.fr/hal-01648385/document. 2019, to appear

work page 2019

[6] [6]

On free energies of the Ising model on the Cayley tree,

D. Gandolfo, M.M. Rakhmatullaev, U.A. Rozikov, J. Ruiz, “On free energies of the Ising model on the Cayley tree,” Jour. Stat. Phys. 150(6), 1201–1217 (2013). 12 A. LE NY, L. LIAO, U. A. ROZIKOV

work page 2013

[7] [7]

On p-adic Gibbs measures for hard core model on a Cayley tree,

D. Gandolfo, U.A. Rozikov, J. Ruiz, “On p-adic Gibbs measures for hard core model on a Cayley tree,” Markov Processes Related Fields. 18(4), 701–720 (2012)

work page 2012

[8] [8]

Pha se Transitions in the Ising Model on Z over the p-adic Number Field,

N.N. Ganikhodjaev, F.M. Mukhamedov, U.A. Rozikov, “Pha se Transitions in the Ising Model on Z over the p-adic Number Field,” Uzb. Mat. Zh., 4, 23–29 (1998)

work page 1998

[9] [9]

Georgii, Gibbs Measures and Phase Transitions, 2nd ed

H.-O. Georgii, Gibbs Measures and Phase Transitions, 2nd ed. De Gruyter Studies in Mathematics, 9. (Walter De Gruyter, Berlin, 2011)

work page 2011

[10] [10]

(French) Théorie des nombres, Année 1991/1992, 107 pp., Publ

Gras Georges, Mesures p-adiques. (French) Théorie des nombres, Année 1991/1992, 107 pp., Publ. Math.Fac. Sci. Besancon, Univ. Franche-Comté, Besancon

work page 1991

[11] [11]

On the uni queness of Gibbs measures for p−adic non homogeneous λ− model on the Cayley tree,

M. Khamraev, F.M. Mukhamedov, U.A. Rozikov, “On the uni queness of Gibbs measures for p−adic non homogeneous λ− model on the Cayley tree,” Letters in Math. Phys. 70, 17–28 (2004)

work page 2004

[12] [12]

p-Adic Valued Probability Measures,

A. Yu. Khrennikov, “ p-Adic Valued Probability Measures,” Indag. Math., New Ser. 7, 311–330 (1996)

work page 1996

[13] [13]

A. Yu. Khrennikov, p-Adic Valued Distributions in Mathematical Physics (Kluwer, Dordrecht, 1994)

work page 1994

[14] [14]

A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical S ystems and Biological Models (Kluwer, Dordrecht, 1997)

work page 1997

[15] [15]

The measure -theoretical approach to p-adic probability theory,

A. Yu. Khrennikov, S. Yamada, A. van Rooij, “The measure -theoretical approach to p-adic probability theory,” Ann. Math. Blaise Pascal. 6, 21–32 (1999)

work page 1999

[16] [16]

Koblitz, p-Adic Numbers, p-adic Analysis, and Zeta-Functions (Springer, Berlin, 1977)

N. Koblitz, p-Adic Numbers, p-adic Analysis, and Zeta-Functions (Springer, Berlin, 1977)

work page 1977

[17] [17]

Stochastic processe s on non-Archimedean spaces with values in non- Archimedean ﬁelds,

S. Ludkovsky, A. Yu. Khrennikov, “Stochastic processe s on non-Archimedean spaces with values in non- Archimedean ﬁelds,” Markov Processes Relat. Fields. 9, 131–162 (2003)

work page 2003

[18] [18]

On the p-adic Five-Point Function,

E. Marinari, G. Parisi, “On the p-adic Five-Point Function,” Phys. Lett. B 203, 52–54 (1988)

work page 1988

[19] [19]

On Gibbs Measures of p-adic Potts Model on the Cayley Tree,

F.M. Mukhamedov, U.A. Rozikov, “On Gibbs Measures of p-adic Potts Model on the Cayley Tree,” Indag. Math., New Ser. 15, 85–100 (2004)

work page 2004

[20] [20]

Mukhamedov, U.A

F.M. Mukhamedov, U.A. Rozikov, On Inhomogeneous p-adic Potts Model on a Cayley Tree, Inﬁn. Dimens. Anal. Quantum Probab. Relat. Top. 8, (2005), 277-290

work page 2005

[21] [21]

On Phas e Transitions for p-adic Potts Model with Com- peting Interactions on a Cayley Tree,

F.M. Mukhamedov, U.A. Rozikov, J.F.F. Mendes, “On Phas e Transitions for p-adic Potts Model with Com- peting Interactions on a Cayley Tree,” in p-Adic Mathematical Physics : Proc. 2nd Int. Conf., Belgrade, 2005 (Am. Inst. Phys., Melville, NY, 2006), AIP Conf. Proc. 826, p p. 140-150

work page 2005

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A. C. M. van Rooij, Non-Archimedean Functional Analysis (M. Dekker, New York, 1978)

work page 1978

[23] [23]

Representation of trees and their appli cations,

U.A. Rozikov, “Representation of trees and their appli cations,” Math. Notes. 72(3-4), 479–488 (2002)

work page 2002

[24] [24]

Rozikov, Gibbs measures on Cayley trees , (World Sci

U.A. Rozikov, Gibbs measures on Cayley trees , (World Sci. Publ. Singapore. 2013)

work page 2013

[25] [25]

p-adic Gibbs measures an d Markov random ﬁelds on countable graphs,

U.A. Rozikov, O.N. Khakimov,“p-adic Gibbs measures an d Markov random ﬁelds on countable graphs,” Theor. Math. Phys. 175(1), 518–525 (2013)

work page 2013

[26] [26]

On weak periodic Gi bbs measures of Ising model on Cayley trees,

U.A. Rozikov, M.M. Rakhmatullaev, “On weak periodic Gi bbs measures of Ising model on Cayley trees,” Theor. Math. Phys. 156(2), 1218–1227 (2008)

work page 2008

[27] [27]

Construction of a set of p-adic distributions,

U.A. Rozikov, Z.T. Tugyonov, “Construction of a set of p-adic distributions,” Theor. Math. Phys. 193(2), 1694–1702 (2017)

work page 2017

[28] [28]

Schikhof, Ultrametric Calculus (Cambridge Univ

W.H. Schikhof, Ultrametric Calculus (Cambridge Univ. Press, Cambridge, 1984)

work page 1984

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Shiryaev, Probability, 2nd ed

A.N. Shiryaev, Probability, 2nd ed. Graduate Texts in Mathematics, 95. (Springer, New York, 1996)

work page 1996

[30] [30]

Vladimirov, I

V.S. Vladimirov, I. V. Volovich, E. V. Zelenov, p-Adic Analysis and Mathematical Physics (Nauka, Moscow, 1994; World Sci., Singapore, 1994). A. Le Ny, Université P aris-Est, Labora toire d’Analyse et de Ma théma tiques Appliquées, LAMA UMR CNRS 8050, UPEC, 91 A venue du Général de Gaulle, 9401 0 Créteil cedex, France. E-mail address : arnaud.le-ny@u-pec.f...

work page 1994