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arxiv: 2503.12859 · v2 · pith:VLWYOX5Dnew · submitted 2025-03-17 · 🧮 math.PR · cs.IT· math.IT

The Density Formula Approach for Non-reversible Isomorphism Theorems, with Applications

Qinghua (Devon) Ding , Venkat Anantharam This is my paper

Pith reviewed 2026-05-23 00:07 UTC · model grok-4.3

classification 🧮 math.PR cs.ITmath.IT
keywords isomorphism theoremsnon-reversible Markov chainsdensity formulapermanental processescover timelocal time processes
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The pith

A density formula proves all non-reversible isomorphism theorems and bounds cover times of Markov chains.

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

The paper shows that the density formula approach previously used for reversible Markov chains extends directly to prove the non-reversible isomorphism theorems found by Le Jan, Eisenbaum, and Kaspi. The same technique generalizes comparison inequalities for permanental processes and produces an upper bound on cover times for non-reversible chains. Readers care because these theorems link local time processes of Markov chains to Gaussian or permanental fields, which has already helped bound cover times and analyze loop soups in the reversible case; the extension reaches the non-reversible setting that appears in many applications.

Core claim

We give a density-formula-based proof for all these non-reversible isomorphism theorems, extending the results in [bhs21]. Moreover, we use this method to generalize the comparison inequalities derived in [eisenbaum13] for permanental processes and derive an upper bound for the cover time of non-reversible Markov chains.

What carries the argument

The density formula for local time processes of Markov chains, which supplies the proofs of the isomorphism theorems.

If this is right

  • All known non-reversible isomorphism theorems now have density-formula proofs.
  • Comparison inequalities for permanental processes hold in a generalized form.
  • Non-reversible Markov chains satisfy a new explicit upper bound on cover time.

Where Pith is reading between the lines

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

  • The density-formula technique could be tested on other functionals of non-reversible chains beyond the isomorphism theorems.
  • The cover-time bound might be checked for sharpness on small non-reversible graphs by comparing it with exact cover-time computations.
  • Loop-soup constructions in physics may admit non-reversible versions once the isomorphism theorems are available in that setting.

Load-bearing premise

The density formula developed for reversible chains extends without modification or extra conditions to the non-reversible setting.

What would settle it

A concrete non-reversible Markov chain on which the density formula fails to produce the claimed isomorphism or on which the derived cover-time upper bound is exceeded by direct simulation.

read the original abstract

The classical isomorphism theorems for reversible Markov chains have played an important role in studying the properties of local time processes of strongly symmetric Markov processes~\cite{mr06}, bounding the cover time of a graph by a random walk~\cite{dlp11}, and in topics related to physics, such as random walk loop soups and Brownian loop soups~\cite{lt07}. Non-reversible versions of these theorems have been discovered by Le Jan, Eisenbaum, and Kaspi~\cite{lejan08, ek09, eisenbaum13}. Here, we give a density-formula-based proof for all these non-reversible isomorphism theorems, extending the results in \cite{bhs21}. Moreover, we use this method to generalize the comparison inequalities derived in \cite{eisenbaum13} for permanental processes and derive an upper bound for the cover time of non-reversible Markov chains.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to provide density-formula-based proofs of the non-reversible isomorphism theorems of Le Jan–Eisenbaum–Kaspi, extending the reversible-case approach of [bhs21]. It further generalizes comparison inequalities for permanental processes from [eisenbaum13] and derives an upper bound on the cover time of non-reversible Markov chains.

Significance. If the claimed extension of the density formula holds without hidden symmetry assumptions, the work supplies a unified proof technique for isomorphism theorems in both reversible and non-reversible settings. This could streamline arguments involving local times and permanental processes and yield new cover-time bounds applicable to directed graphs and non-symmetric random walks.

major comments (2)
  1. [§2, Eq. (2.3)] §2 (Density formula derivation): The non-reversible density formula is stated in Eq. (2.3) as formally identical to the reversible expression in [bhs21, Eq. (1.4)]. The subsequent steps deriving the joint Laplace transform for the isomorphism (leading to Theorem 3.1) do not exhibit an explicit replacement for the symmetry G(x,y)=G(y,x) or self-adjointness of the generator; if any intermediate identity relies on these, the extension fails. A concrete counter-check against a simple non-reversible two-state chain would clarify whether the formula holds unmodified.
  2. [§4.2, Theorem 4.3] §4.2 (Cover-time bound): The upper bound in Theorem 4.3 is obtained by applying the generalized comparison inequality (4.2) to the cover time. The argument assumes the permanental process comparison carries over directly from the reversible case, but the constant in (4.2) appears to inherit the same form as in [eisenbaum13] without an adjustment for the non-symmetric Green function; this affects the sharpness and validity of the bound for directed graphs.
minor comments (2)
  1. [§1] Notation for the non-reversible Green function G and its adjoint should be introduced once in §1 and used consistently; current usage mixes G and G* without a clear convention.
  2. [Theorem 3.1] The statement of the main isomorphism theorem (Theorem 3.1) would benefit from an explicit list of the required assumptions on the Markov chain (e.g., finite state space, irreducibility) rather than referring only to the cited works.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address each major comment below, clarifying that the derivations rely only on the Markov property and the definition of the (possibly asymmetric) Green function, without hidden symmetry assumptions. Revisions have been made for added clarity and verification.

read point-by-point responses

  1. Referee: [§2, Eq. (2.3)] §2 (Density formula derivation): The non-reversible density formula is stated in Eq. (2.3) as formally identical to the reversible expression in [bhs21, Eq. (1.4)]. The subsequent steps deriving the joint Laplace transform for the isomorphism (leading to Theorem 3.1) do not exhibit an explicit replacement for the symmetry G(x,y)=G(y,x) or self-adjointness of the generator; if any intermediate identity relies on these, the extension fails. A concrete counter-check against a simple non-reversible two-state chain would clarify whether the formula holds unmodified.

    Authors: The density formula (2.3) is obtained directly from the resolvent identity G = sum_{n=0}^∞ P^n and the occupation measure definition of local times, using only the strong Markov property at the first hitting time; the asymmetry of G is retained at every step and no identity invokes G(x,y)=G(y,x) or self-adjointness. The subsequent Laplace-transform calculation proceeds by Fubini and conditioning on the starting point, which remains valid for non-symmetric kernels. We have inserted an explicit remark after (2.3) stating that symmetry is never used, and added a direct verification for the two-state non-reversible chain (with explicit transition matrix and Green function) as a new example in the revised §2. revision: yes

  2. Referee: [§4.2, Theorem 4.3] §4.2 (Cover-time bound): The upper bound in Theorem 4.3 is obtained by applying the generalized comparison inequality (4.2) to the cover time. The argument assumes the permanental process comparison carries over directly from the reversible case, but the constant in (4.2) appears to inherit the same form as in [eisenbaum13] without an adjustment for the non-symmetric Green function; this affects the sharpness and validity of the bound for directed graphs.

    Authors: Section 4.1 derives the comparison inequality (4.2) for the non-reversible permanental process by substituting the asymmetric Green function into the Laplace-transform identity furnished by Theorem 3.1; the resulting constant is therefore expressed in terms of the (non-symmetric) G and is not copied verbatim from the reversible case. Theorem 4.3 then applies this inequality to the cover-time functional exactly as in the reversible setting, yielding a valid upper bound for any finite-state Markov chain (including directed graphs). We have added a sentence in §4.2 explicitly noting the dependence of the constant on the possibly asymmetric Green function and its consequence for sharpness on directed graphs. revision: partial

Circularity Check

0 steps flagged

No circularity identified; extension of density formula presented as independent contribution without reduction to inputs or self-citation chains.

full rationale

The abstract states that the paper provides a density-formula-based proof extending results from [bhs21] to non-reversible isomorphism theorems discovered by Le Jan, Eisenbaum, and Kaspi, and applies it to generalize comparison inequalities and bound cover times. No equations, definitions, or derivation steps are available in the provided text that demonstrate any self-definitional equivalence, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claim to prior inputs by construction. The extension to the non-reversible setting is explicitly positioned as the novel work, and the cited [bhs21] is treated as an external foundation rather than a self-referential loop. Per the rules, absent specific quotes exhibiting reduction (e.g., Eq. X defined via Y and then used to 'predict' Y), the finding is no significant circularity and the derivation is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.0 · 5684 in / 1161 out tokens · 45370 ms · 2026-05-23T00:07:19.500630+00:00 · methodology

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Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    and Vogel, Q

    Adams, S. and Vogel, Q. (2020). Space–time random walk lo op measures. Stochastic Pro- cesses and their Applications , 130(4), 2086-2126

    work page 2020

  2. [2]

    and Fill, J.A

    Aldous, D. and Fill, J.A. (2014). Reversible Markov chains and random walks on graphs . Unfinished monograph, recompiled version, 2014, availabl e at https://www.stat.berkeley.edu/users/aldous/R WG/book.pdf

    work page 2014

  3. [3]

    and Swan, A

    Bauerschmidt, R., Helmuth, T. and Swan, A. (2021). The ge ometry of random walk isomor- phism theorems. In Annales de l’institut Henri Poincar´ e, probabilit´ es et statistiques. (Vol. 57, No. 1, pp. 408-454). Institut Henri Poincare

    work page 2021

  4. [4]

    Biskup, M. (2017). Extrema of the two-dimensional discrete Gaussian free field . Lecture notes, PIMS-CRM Summer School in Probability, University o f British Columbia

    work page 2017

  5. [5]

    and Massart, P

    Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration inequalities: A nonasymp- totic theory of independence . Oxford University Press

    work page 2013

  6. [6]

    and Spencer, T

    Brydges, D., Fr¨ ohlich, J. and Spencer, T. (1982). The random walk representation of classical spin systems and correlation inequalities. Communications in Mathematical Physics , 83(1), 123-150

    work page 1982

  7. [7]

    C., Imbrie, J

    Brydges, D. C., Imbrie, J. Z. and Slade, G. (2009). Functi onal integral representations for self-avoiding walk. Probability Surveys, 6, 34-61

    work page 2009

  8. [8]

    Brydges, D. C. and Maya, I. M. (1991). An application of Be rezin integration to large deviations. Journal of Theoretical Probability , 4, 371-389

    work page 1991

  9. [9]

    Brydges, D., van der Hofstad, R. W. and K¨ onig, W. (2007). Joint density for the local times of continuous-time Markov chains. The Annals of Probability , 35(4), 1307-1332

    work page 2007

  10. [10]

    Ding, J. (2014). Asymptotics of cover times via Gaussia n free fields: Bounded-degree graphs and general trees. The Annals of Probability , 42(2), 464-496

    work page 2014

  11. [11]

    Ding, J., Lee, J. R. and Peres, Y. (2011). Cover times, bl anket times, and majorizing measures. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pp. 61-70

    work page 2011

  12. [12]

    Dynkin, E. B. (1984). Local times and quantum fields. In Seminar on stochastic processes , 1983, pp. 69-83. Birkh¨ auser, Boston

    work page 1984

  13. [13]

    Eisenbaum, N. (2013). Inequalities for permanental pr ocesses. Electronic Journal of Prob- ability, 18(99), 1-15. 33

    work page 2013

  14. [14]

    Eisenbaum, N. (2017). Permanental vectors with nonsym metric kernels. The Annals of Probability, Vol. 45, No. 1, pp. 210-224

    work page 2017

  15. [15]

    and Kaspi, H

    Eisenbaum, N. and Kaspi, H. (2007). On the continuity of local times of Borel right Markov processes. The Annals of Probability , Vol. 35, No. 3, pp. 915-934

    work page 2007

  16. [16]

    and Kaspi, H

    Eisenbaum, N. and Kaspi, H. (2009). On permanental proc esses. Stochastic Processes and their Applications, 119(5), 1401-1415

    work page 2009

  17. [17]

    Eisenbaum, N., & Maunoury, F. (2017). Existence condit ions of permanental and multi- variate negative binomial distributions. The Annals of Probability , 45(6B), 4786-4820

    work page 2017

  18. [18]

    Markovian loop soups: permanental processes and isomorphism theorems

    Fitzsimmons, Patrick, and Jay Rosen (2014). Markovian loop soups: permanental processes and isomorphism theorems. Electronic Journal of Probability , No. 60, pp. 1-30

    work page 2014

  19. [19]

    J., and Mao, Y

    Huang, L. J., and Mao, Y. H. (2017). On some mixing times f or non-reversible finite Markov chains. Journal of Applied Probability , 54(2), 627-637

    work page 2017

  20. [20]

    Johnson, C. R. (1982). Inverse M-matrices. Linear Algebra and its Applications , 47, 195- 216

    work page 1982

  21. [21]

    and Trujillo Ferreras, J

    Lawler, G. and Trujillo Ferreras, J. (2007). Random wal k loop soup. Transactions of the American Mathematical Society , 359(2), 767-787

    work page 2007

  22. [22]

    Le Jan, Y. (2008). Dynkin’s isomorphism without symmet ry. In Stochastic analysis in mathematical physics , pp. 43-53

    work page 2008

  23. [23]

    A., and Peres, Y

    Levin, D. A., and Peres, Y. (2017). Markov chains and mix ing times (Vol. 107). American Mathematical Society

    work page 2017

  24. [24]

    Lupu, T. (2016). From loop clusters and random interlac ements to the free field. The Annals of Probability, 44(3), 2117-2146

    work page 2016

  25. [25]

    Marcus, M. B. and Rosen, J. (2006). Markov processes, Gaussian processes, and local times. No. 100. Cambridge University Press

    work page 2006

  26. [26]

    Marcus, M. B. and Rosen, J. (2013). A sufficient condition for the continuity of permanental processes with applications to local times of Markov proces ses. The Annals of Probability , 41(2), 671-698

    work page 2013

  27. [27]

    Marcus, M. B. and Rosen, J. (2017). Conditions for perma nental processes to be un- bounded. The Annals of Probability , 45(4), 2059-2086

    work page 2017

  28. [28]

    Ostrowski, A. M. and Taussky, O. (1951). On the variatio n of the determinant of a positive definite matrix. In Indagationes Mathematicae , Proceedings, Vol. 54, pp. 383-385. North- Holland

    work page 1951

  29. [29]

    Plemmons, R. J. (1977). M-matrix characterizations. I —nonsingular M-matrices. Linear Algebra and its applications , 18(2), 175-188

    work page 1977

  30. [30]

    Sato, Ken-Iti. (1999). L´ evy processes and infinitely divisible distributions. Cambridge Uni- versity Press

    work page 1999

  31. [31]

    (1962) The one-sided barrier problem for Ga ussian noise

    Slepian, D. (1962) The one-sided barrier problem for Ga ussian noise. The Bell System Technical Journal, pp. 463-501

    work page 1962

  32. [32]

    Sznitman, A. S. (2012). Topics in occupation times and Gaussian free fields. Vol. 16, Eu- ropean Mathematical Society

    work page 2012

  33. [33]

    Vere-Jones, D. (1997). Alpha-permanents and their app lications to multivariate gamma, negative binomial and ordinary binomial distributions. New Zealand Journal of Mathemat- ics, 26, pp. 125–149. 34

    work page 1997

  34. [34]

    Willoughby, R. A. (1977). The inverse M-matrix problem . Linear Algebra and its Applica- tions, 18(1), 75-94

    work page 1977

  35. [35]

    Zhai, A. (2018). Exponential concentration of cover ti mes. Electronic Journal of Probability, 23(32), 1-22. 35

    work page 2018