$\Gamma$-expansion of the measure-current large deviations rate functional of non-reversible finite-state Markov chains

Claudio Landim; Seonwoo Kim

$\Gamma$-expansion of the measure-current large deviations rate functional of non-reversible finite-state Markov chains

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Pith's one-line read The paper proves that the measure-current large deviations rate functional of a finite-state Markov chain admits a hierarchical Γ-expansion whose terms are the Donsker–Varadhan rate functionals of the effective chains at each metastable…

desk verdict A careful, conditional Γ-expansion theorem that completes a program; the proof leans on prior work from the same group, but the reduction is explicit and the honest scope restrictions keep it credible. read the letter →

arxiv 2412.13515 v2 pith:7ZSIYCU7 submitted 2024-12-18 math.PR

Seonwoo Kim , Claudio Landim This is my paper · ORCID

classification math.PR MSC 60K3560F1082C22

keywords largedeviationsΓ-convergenceempiricalcurrentmeasuremetastabilityDonsker–Varadhanratefunctionalnon-reversibleMarkovchainsfinite-state

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

This paper studies sequences of continuous-time Markov chains on a fixed finite state space and asks what their large deviations rate functionals reveal about metastability. Its central claim is that, under a comparability condition on the jump rates, the measure-current rate functional $I_n$ admits the hierarchical $\Gamma$-expansion $I_n = I^{(0)} + \sum_{p=1}^{q} \theta_n^{(p)-1} I^{(p)}$, where the weights $\theta_n^{(p)}$ are the metastable time scales and each $I^{(p)}$ is the Donsker–Varadhan rate functional of an effective Markov chain describing transitions among metastable wells at that scale. If the claim is right, the entire metastable hierarchy — the wells, their relative weights, and the effective dynamics at every time scale — is encoded in a single functional, and the empirical current is forced to be the stationary flow of the limiting chain on all scales beyond the first. The paper also proves optimal conditions for recovering the generator from the measure or measure-current rate functional, and derives first and second derivatives of the measure rate functional.

What carries the argument

The carrying object is the rooted tree of partitions of $V$ constructed recursively from the comparability hypothesis: the leaves are the recurrent classes of the limiting chain $X_t^{(0)}$, each generation is a partition of $V$, and each step coarsens the partition by lumping the recurrent classes of the effective chain $X_t^{(p)}$. The construction generates the time scales $\theta_n^{(p)}$, the effective chains $X_t^{(p)}$, and the measures $\pi_j^{(p)}$ used in (2.16). The expansion is proved term by term: the $p=0$ term is the rate functional of the limiting chain, and for $p\ge1$ the argument uses the identity (1.7) that the measure-current functional minimized over currents equals the Donsker–Varadhan functional, so the measure-current $\Gamma$-limit is inherited from the known level-2 $\Gamma$-convergence $\theta_n^{(p)} I_n \to I^{(p)}$.

What would settle it

Take $V=\{1,2,3\}$ with rates $R_n(1,2)=R_n(2,1)=R_n(2,3)=R_n(3,2)=1$, $R_n(3,1)=a_n$, $R_n(1,3)=b_n$, where $a_n/b_n$ oscillates between two values; the products $a_n$ and $b_n$ appearing in (2.5) for $m=2$ are then not comparable. If the trace-rate expressions (2.18) oscillate and $\theta_n^{(1)} I_n$ has no $\Gamma$-limit, the comparability hypothesis is doing the claimed work; if a $\Gamma$-limit still exists, the theorem's hypothesis is not necessary.

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Extended reading notes

Core claim

The main theorem states that for each $1\le p\le q$ the scaled functional $\theta_n^{(p)} I_n$ $\Gamma$-converges to $I^{(p)}$, where $I^{(p)}(\mu,J)$ is finite exactly when $J=J_{\mu,R_0}$ and $\mu=\sum_j \omega_j \pi_j^{(p)}$ for a probability vector $\omega$, and in that case equals the Donsker–Varadhan rate $I^{(p)}(\omega)$ of the effective chain $X_t^{(p)}$. Thus the level-2.5 functional determines, at every metastable scale, both the asymptotic stationary weights $\pi_j^{(p)}$ and the effective chain $X_t^{(p)}$ through its Donsker–Varadhan functional. The proof reduces the $\Gamma$-limsup to the known $\Gamma$-convergence of level-2 functionals by projecting onto the unique optimal current, and the $\Gamma$-liminf follows from the nested zero structure $I^{(p)}(\mu,J)<\infty \iff I^{(p-1)}(\mu,J)=0$.

Load-bearing premise

The argument requires the comparability hypothesis (2.5): for every $m\ge1$, the sequences formed by products of $m$ jump rates along directed edges must be pairwise comparable, so none of their ratios may oscillate; without this, the trace rates in (2.18) can fail to converge and the recursive tree, time scales, and effective chains used in Theorem 2.3 may not be well defined.

Editorial extensions

If this is right

Corollary 1.2: at each scale the rescaled log-probability of any closed or open set of empirical measure-current pairs is controlled by the infimum of $I^{(p)}$, so the expansion gives quantitative estimates for the time the chain spends near or away from each metastable well.
The zeros of the $I^{(p)}$ form a nested hierarchy: $I^{(p)}(\mu,J)<\infty$ if and only if $I^{(p-1)}(\mu,J)=0$, so each additional scale only refines the set of pairs that are costless at the previous scale.
For $p\ge1$, finiteness forces $J=J_{\mu,R_0}$; the empirical current is asymptotically the stationary flow of the limiting chain, and any other current is exponentially unlikely at long time scales.
The rate functional determines the dynamics exactly when the state space is completely recurrent (or reversible for the measure-only functional), with explicit three-state counterexamples showing the conditions cannot be weakened.
The second derivative of the DV rate functional at the stationary state is expressible through the generator's symmetric part and the asymptotic variance, extending the i.i.d. relation between rate curvature and variance to Markov chains.

Reading between the lines

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

Because condition (2.5) only involves finite products of rates, one could verify it computationally for concrete model families; models satisfying it are exactly those for which the expansion's time scales are well-defined.
Remark 2.6's example shows there can be intermediate scales where the flow structure changes but the empirical measure does not; a finer expansion that keeps track of currents at every intermediate scale would capture these flow-only transitions.
The recoverability results imply an identifiability criterion for statistical inference: if all states are recurrent, long-time observations of empirical measure and current determine the jump rates uniquely; with transient states, different generators share the same rate functional.
The derivative formulas suggest that the DV rate functional can be used as a generating function for cumulants of additive functionals of the chain, analogous to how the i.i.d. rate function generates cumulants via its derivatives.

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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

Punchline: this paper proves the Γ-expansion of the level-2.5 measure-current rate functional for non-reversible finite-state Markov chains, extending Landim's earlier Donsker-Varadhan Γ-expansion. It is a real result, but its strength is tied to the comparability hypothesis (2.5) and to heavy results from earlier papers, mostly by the same group.

What's new and good: the main theorem, Theorem 2.3, states that θ_n^(p) I_n Γ-converges to I^(p), with I^(p) built from the DV functional of the effective chain X^(p)_t and the current forced to the stationary flow J_{µ,R0}. The reduction in Remark 2.4 is elegant: at scales p≥1 the current is too costly to deviate, so the measure-current functional collapses to the DV functional. The proof of the limsup is careful about the optimal current J*_n, using the recovery bound to get I_n(ν_n)→0 and then (4.2)–(4.4) to force J*_n→J_{µ,R0}. The hierarchical zero lemma (Lemma 4.1) is a clean structural observation. Section 5 is a genuine bonus: optimal conditions for the DV and BFG functionals to determine the generator, with explicit counterexamples, and derivative formulas (Lemma 5.11, Proposition 5.13, Lemma 5.14) that generalize the i.i.d. results. Proposition 2.1, the p=0 case, is self-contained and well argued.

Soft spots: the main theorem is conditional on (2.5), comparability of all products of rates. That is a strong structural assumption; the paper cites many models that satisfy it, but it is the place where the expansion could fail. The proof also imports the entire tree construction and the DV Γ-expansion from [22,25,2], so a referee has to trust several heavy lemmas not reproduced here. There is a minor typo in Theorem 1.1: it writes P(V)×E where F_E should be intended. The abstract's 'finishes the project' claim is a little overconfident, since Remark 2.6 honestly says intermediate flow-only time-scales are not captured. That is a scope restriction, not a flaw.

This paper is for probabilists working in metastability and large deviations. The reader's take and the stress-test note align with my reading: I could not locate an internal gap, and the flow-convergence argument in Section 4 is internally consistent. I would send this to a serious referee. The main things to check are the uses of [22]'s lemmas and whether (2.5) is as benign as claimed. I'd cite it if I worked in this area.

Referee Report

0 major / 5 minor

Summary. This paper studies sequences of irreducible continuous-time Markov chains on a fixed finite state space with a common edge set, and proves a Gamma-expansion of the level-2.5 (measure-current) large deviations rate functional I_n. Under the comparability hypothesis (2.5) on products of jump rates, the main theorem (Theorem 2.3) states that for each metastable time-scale theta_n^(p) produced by the recursive construction of [25], the rescaled functional theta_n^(p) I_n Gamma-converges to I^(p), where I^(p) is finite only when the empirical measure is a convex combination of the effective stationary measures pi_j^(p) and the empirical current equals J_{mu,R0}; in that case I^(p) reduces to the Donsker-Varadhan rate functional of the effective Markov chain X^(p)_t. The paper also proves the p=0 case (Proposition 2.1), characterizes the zero set of I^(0) (Lemma 3.1), and in Section 5 gives conditions under which the Donsker-Varadhan or measure-current rate functional determines the generator, together with counterexamples showing sharpness and formulas for the first and second functional derivatives of the Donsker-Varadhan functional.

Significance. If correct, the main result completes the program initiated in [8] and [22] by extending the Gamma-expansion of level-2 rate functionals to the level-2.5 measure-current setting, thereby providing a hierarchical description of metastable behavior in terms of large deviations rate functionals. The proof is conditional on the comparability hypothesis (2.5) and relies heavily on prior published results for the tree construction and for the Donsker-Varadhan Gamma-expansion, but the reduction is explicit and the novel step--showing that the optimal current J_n^* associated to the recovery sequence converges to J_{mu,R0}--is carefully argued. The paper also contains clean, falsifiable statements in Section 5: reversibility is sufficient for the Donsker-Varadhan functional to determine the generator and is not necessary in general (Example 5.4), and recurrence is sufficient for the measure-current functional and not necessary in general (Example 5.9). The derivative identities in Lemmas 5.11 and 5.14 and Proposition 5.13 generalize the classical Cramer-type formulas and are of independent interest.

minor comments (5)

[Section 4, proof of Theorem 2.3] In the paragraph following Eq. (4.4), the sentence "As nu_n -> mu and I^(0)(mu)=0 by Lemma 3.1, lim_n I_n(nu_n)=I^(0)(mu)=0" is not a valid consequence of Gamma-convergence, since Proposition 2.1 is only a Gamma-limit and does not give pointwise convergence along arbitrary recovery sequences. The desired conclusion I_n(nu_n)->0 does follow from the preceding display, which gives theta_n^(p) I_n(nu_n) <= I^(p)(mu)<infinity together with theta_n^(p)->infinity; please rephrase the justification accordingly.
[Section 2, Eq. (2.3)] The symbol n is used both as the sequence index and as the number of closed irreducible classes V_1,...,V_n of the limiting chain; this collision is confusing, especially since n_p is later introduced for the number of wells at level p. Please rename the number of classes, for instance to N or m.
[Remark 2.6] The displayed stationary weights contain a typo: the text reads "pi_n(-1)=pi_n(-1)=a_n/n", but it should be "pi_n(-1)=pi_n(1)=a_n/n". In the same remark, "metastabe" should be "metastable".
[Abstract and Introduction] The compiled text contains spacing artifacts such as "Gamma -exp ansion" and "Mark ovian" in the abstract and title; these should be cleaned in the final version.
[Section 5, Lemma 5.1] The statement "One can derive from I(.) the values of lambda(z) and R(x,y)R(y,x)" could be made more precise by saying "from the full rate functional I on P(V)"; otherwise the reader might wonder whether the claim is about the values at a single measure rather than the whole functional.

Circularity Check

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No significant circularity: Theorem 2.3 is proven by reduction to a published DV Gamma-expansion plus an independent optimal-current argument; the definitions of I^(p) do not do the work of the proof.

full rationale

The claimed Gamma-expansion (1.10) is conditional on the comparability hypothesis (2.5), and its proof does not assume the conclusion. The measure-current Gamma-convergence theta_n^(p) I_n -> I^(p) is obtained in Section 4 by combining the published Donsker-Varadhan Gamma-expansion theta_n^(p) Ĥ_n -> I^(p) from [22, Theorem 2.5] with an independent argument that the optimal current J*_n associated to any recovery sequence nu_n satisfies J*_n -> J_{mu,R0} (Eqs. (4.1)-(4.4) and the surrounding text). Although I^(p) is defined in (2.16) to be finite only when J = J_{mu,R0}, that is the statement of the candidate limit, not the proof: the Gamma-liminf part must still show that sequences with J different from J_{mu,R0} cost theta_n^(p) I_n -> infinity, which is handled through Lemma 4.1 and the induction using theta_n^(p-1) << theta_n^(p). Lemma 4.1 cites [22, Lemma 5.1] for the zero-set hierarchy; that is a published, parameter-free result with stated assumptions, so it counts as independent support rather than a circular self-citation. The tree construction and time-scales theta_n^(p) are imported from [25] and [2] under the explicit hypothesis (2.5), and Remark 2.7 explains why (2.5) is the natural comparability condition; this is an assumption, not a conclusion smuggled in. Remark 2.6 openly states that intermediate flow-only time-scales are not captured, an honest scope limitation. No equation in the paper reduces to its own input, and no fitted parameter is relabelled as a prediction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted constants and no new physical entities. Its load is carried by the comparability assumption (2.5), the fixed-edge assumption (1.8), the existence of limiting rates (2.2), and imported theorems from [22,25,2,6]. All imports are published, parameter-free derivations with stated assumptions; they are listed here for transparency.

assumptions (6)

domain assumption Comparability hypothesis (2.5): for each m ≥ 1, the sequences (Π_{(x,y)∈E} R_n(x,y)^{k(x,y)}), k ∈ Σ_m, are pairwise comparable.
Used to define the hierarchy; without it the trace rates in (2.18) may oscillate and the tree construction in [25] fails. The authors flag this as assumption (2.5) and discuss it in Remark 2.7.
domain assumption Fixed edge set (1.8): R_n(x,y) > 0 if and only if (x,y) ∈ E for all n.
Imposed so the state space and the flow space F_E do not change with n; standard in this line of work.
domain assumption Existence of limiting rates R_0(x,y) = lim_n R_n(x,y) with E_0 ≠ ∅ (2.2).
Needed to define the initial-scale functional I^(0) and the closed classes V_j.
standard math Gamma-convergence of the Donsker-Varadhan functional I_n at scale θ_n^(p) ([22, Theorem 2.5]).
Used as a black box in the proof of Theorem 2.3 to obtain (4.5) and the recovery sequence ν_n. Published in Landim 2023 with assumptions matching (2.5).
standard math Tree construction and time-scale ordering θ_n^(p-1) ≺ θ_n^(p) ([25, Assertion 8.B], [2]).
Provides the partition hierarchy, the chains X^(p)_t, and the strict separation of scales used in (2.9) and in Lemma 4.1.
standard math Level 2.5 to level 2 projection and existence of optimal currents ([6, Theorem 1.6]).
Used to connect the measure-current functional to the Donsker-Varadhan functional and to control the flows in the Gamma-limsup proof.

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Pith. "Pith review of $\Gamma$-expansion of the measure-current large deviations rate functional of non-reversible finite-state Markov chains." pith.science (2026). https://pith.science/paper/7ZSIYCU7

@misc{pith2026241213515,
  author       = {Pith},
  title        = {Pith review of: $\Gamma$-expansion of the measure-current large deviations rate functional of non-reversible finite-state Markov chains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ZSIYCU7}},
  note         = {Machine review of arXiv:2412.13515}
}

abstract

Consider a sequence of continuous-time Markov chains $(X^{(n)}_t:t\ge 0)$ evolving on a fixed finite state space $V$. Let $I_n$ be the measure-current large deviations rate functional for $X^{(n)}_t$, as $t\to\infty$. Under a hypothesis on the jump rates, we prove that $I_n$ can be written as $I_n = \mathbf I^{(0)} \,+\, \sum_{1\le p\le \mathfrak q} (1/\theta^{(p)}_n) \, \mathbf I^{(p)}$ for some rate functionals $\mathbf I^{(p)}$. The weights $\theta^{(p)}_n$ correspond to the time-scales at which the sequence of Markov chains $X^{(n)}_t$ evolves among the metastable wells, and the rate functionals $\mathbf I^{(p)}$ characterise the asymptotic Markovian dynamics among these wells. This expansion provides therefore an alternative description of the metastable behavior of a sequence of Markovian dynamics. Together with the results in \cite{bgl-24,l-gamma}, this work finishes the project of characterising the hierarchical metastable behavior of finite-state Markov chains by means of the $\Gamma$-expansion of large deviations rate functionals. In addition, we present optimal conditions under which the measure (Donsker-Varadhan) or the measure-current large deviations rate functional determines the original dynamics, and calculate the first and second derivatives of the measure large deviations rate functional, thereby generalising the results for i.i.d. random variables.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [22]

Landim: Metastability from the large deviations poi nt of view: A Γ-expansion of the level two large deviations rate functional of non-reversible ﬁni te-state Markov chains

C. Landim: Metastability from the large deviations poi nt of view: A Γ-expansion of the level two large deviations rate functional of non-reversible ﬁni te-state Markov chains. Stoch. Proc. Appl. 165, 275–315 (2023)

work page 2023

[2] [25]

Landim, T

C. Landim, T. Xu: Metastability of ﬁnite state Markov ch ains: a recursive procedure to identify slow variables for model reduction. ALEA Lat. Am. J . Probab. Math. Stat. 13, 725–751 (2016)

work page 2016

[3] [8]

Bertini, D

L. Bertini, D. Gabrielli, C. Landim: Metastable expansi on of ﬁnite state Markov chains level two large deviations rate functions. Ann. Appl. Probab. 34, 3820–3869 (2024)

work page 2024

[4] [1]

Beltr´ an, C

J. Beltr´ an, C. Landim: Tunneling and metastability of c ontinuous time Markov chains. J. Stat. Phys. 140, 1065–1114 (2010)

work page 2010

[5] [2]

Beltr´ an, C

J. Beltr´ an, C. Landim: Metastability of reversible ﬁni te state Markov processes. Stoch. Proc. Appl. 121, 1633–1677 (2011)

work page 2011

[6] [3]

Beltr´ an, C

J. Beltr´ an, C. Landim: Tunneling and metastability of c ontinuous time Markov chains II. J. Stat. Phys. 149, 598–618 (2012)

work page 2012

[7] [4]

Beltr´ an, C

J. Beltr´ an, C. Landim: Tunneling of the Kawasaki dynami cs at low temperatures in two dimensions. Ann. Inst. H. Poincar´ e, Probab. Statist. 51, 59–88 (2015)

work page 2015

[8] [5]

Ben Arous, R

G. Ben Arous, R. Cerf: Metastability of the three-dimens ional Ising model on a torus at very low temperature. Electron. J. Probab. 1, Research Paper 10 (1996)

work page 1996