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Quasi-stationary and quasi-ergodic distributions in the Pelikan random map

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Pith's one-line read An open Pelikan-map chain is claimed to have a continuum of quasi-stationary distributions, each with its own escape rate.

desk verdict A concrete countable Markov chain with a plausible continuum of QSDs, but the continuum rests on numerically verified positivity and a hand-waved QED uniqueness proof. read the letter →

arxiv 2607.19084 v1 pith:RQLTXIHS submitted 2026-07-21 nlin.CD cond-mat.stat-mechmath-phmath.DSmath.MP

Samuel Brevitt , Rainer Klages This is my paper · ORCID
classification nlin.CDcond-mat.stat-mechmath-phmath.DSmath.MP MSC 60J1037A5037H12
keywords quasi-stationarydistributionsquasi-ergodicPelikanrandommapcountableMarkovchainssubstochasticmatricesescaperatesR-recurrenceresetting
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

The paper studies a biased random walk on the nonnegative integers with resets and escape, derived from an open version of the Pelikan random map. It claims that this countable substochastic Markov chain has not one but infinitely many quasi-stationary distributions for generic parameter values, indexed by a free parameter α, with each distribution tied to its own eigenvalue λ = 1 − pεα and therefore its own long-term escape rate. If the claim is correct, the eventual escape rate is not fixed by the system alone but selected by the initial distribution. The paper also shows analytically and numerically that a unique quasi-ergodic distribution exists in the R-positive regime and does not exist in the R-transient regime.

What carries the argument

The central object is the generating function A(z) = Σ μ_k z^k for candidate invariant measures. Imposing M[A] = λA reduces the QSD equation to a rational function with free parameter α = μ_0, and the coefficients μ_k are obtained from a linear recurrence. The eigenvalue is pinned by normalization to λ = 1 − pεα. The same generating-function apparatus yields the Koopman eigenvector B(z), whose product with the principal QSD gives the QED.

What would settle it

Choose rational p, ε inside the claimed valid region and α just below α_u; iterate the recurrence (44) exactly, or with rigorous interval arithmetic, until the dominant-root criterion (48) applies. A single negative μ_k before that cutoff would disprove the continuum for that parameter set.

Watch

Extended reading notes

Core claim

The transfer operator of the chain is shown to admit a family of positive eigenvectors μ(α) satisfying Mμ = λ(α)μ, with λ(α) = 1 − pεα. The paper derives these from a generating function A(z) whose coefficients are generated by a three-term recurrence. Positivity of the coefficients is verified numerically up to a tail criterion, and simulations confirm that each such measure is preserved under the conditional dynamics and produces the predicted exponential escape rate. This continuous family of QSDs appears in both R-recurrent and R-transient parameter regions. In the R-positive region, the paper explicitly constructs the unique quasi-ergodic distribution as the product of the principal QSD

Load-bearing premise

The claimed continuum of quasi-stationary distributions exists only if every coefficient μ_k computed from the recurrence is non-negative for all k on an interval of α; the paper verifies this numerically up to a tail criterion but does not prove it.

Editorial extensions

If this is right

  • If the continuum of QSDs is genuine, the long-term escape rate of this system can be tuned continuously by choosing different initial distributions, without changing the parameters p or ε.
  • Non-uniqueness of quasi-stationary distributions occurs not only in exotic countable matrices but in a simple random-walk model with resetting and escape.
  • The asymptotic tail shape of a distribution is conserved by the normalized dynamics, partitioning distribution space into uncountably many invariant classes, each containing at most one QSD.
  • A unique normalizable quasi-ergodic distribution exists exactly in the R-positive regime; in the R-transient regime the time-averaged history of surviving trajectories does not converge.
  • Non-principal QSDs appear marginally stable to common smooth noise, while finite-sample noise eventually drives the system toward the principal QSD.

Reading between the lines

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

  • If the positivity interval for α is confirmed rigorously, similar generating-function families may yield QSD continua in other resetting random walks with geometric reset distributions and biased noise.
  • The result suggests that other observables, such as Lyapunov exponents or metric entropy, if defined for the open system, may inherit initial-condition dependence; the paper leaves that question open.
  • A direct experimental probe would be to initialize an ensemble with a prescribed tail shape and compare the measured long-term survival probability to λ(α); a match for non-principal α would provide independent evidence for the continuum.
  • Because the paper verifies positivity by iterating to a numerical cutoff, a symbolic or interval-arithmetic proof for rational parameter values could settle the existence of the continuum without exhaustive simulation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies a countable-state substochastic Markov chain, motivated by an open variant of the Pelikan random map, with transition operator given by Eq. (9). The authors derive a one-parameter family of formal eigenmeasures of the transfer operator, whose generating function is given by Eq. (32) and whose eigenvalue is λ(α)=1−pεα. They claim that, for generic parameter values, there is a continuum of positive normalizable quasi-stationary distributions (QSDs), each with its own escape rate, and they test this claim in simulations. The paper also analyzes the asymptotic shape of these QSDs, studies their stability under random perturbations, and constructs quasi-ergodic distributions (QEDs) for the principal eigenvalue in the R-positive region.

Significance. If the main existence claim were fully proved, this would be a valuable and explicit example of non-unique QSDs and initial-condition-dependent escape rates in a simple random dynamical system, with implications for metastability, stochastic resetting, and open dynamical systems. The paper contains several strengths: a clean generating-function derivation, an explicit recurrence for the coefficients, an independent derivation of the principal escape rate from singularity analysis, and extensive numerical simulations with large ensembles. The shape-conservation argument in Sec. 4.2 is also suggestive. However, the central theorem is not actually proved: the existence of a continuum of QSDs rests on a numerical positivity check, and the claimed uniqueness of QEDs rests on an unproved assertion.

major comments (2)
  1. [Sec. 3.2, Eqs. (44)–(48)] The central claim—that for generic (p, ε) there is a continuum of QSDs parameterized by α∈(0, α_u]—requires proving that the coefficients μ_k(α) defined by the recurrence (44) are non-negative for every k and for every α in an interval. The manuscript verifies this only numerically: Fig. 3 is generated by iterating (44) until either μ_k<0 or k>k* from criterion (48), with the coefficients C_i and threshold k* computed numerically on a grid of α. The statement 'we rather observe numerically that they are valid for α≤α_u and invalid otherwise' is an empirical observation, not a proof. If some subinterval of [0, α_u] had a negative μ_k, the infinite family would collapse to isolated QSDs. An analytic sign analysis of the three exponential contributions in (45), or a rigorous dominance argument for the coefficients, is required to establish the paper's main conclusion.
  2. [Sec. 6, Eq. (62) and footnote 5] The abstract and conclusion claim that unique QEDs are established. However, the uniqueness assertion is supported only by 'for technical reasons we believe this is in fact the only eigenvalue which produces a normalisable QED.' This is not a proof. To claim uniqueness, the authors must show that for every eigenvalue λ≠λ_u of the transfer operator, the product b_i(λ) μ_i(λ) is not summable, or that no corresponding positive left eigenvector exists. The cancellation argument in footnote 5 is local to λ_u and does not exclude other eigenvalues. Without a rigorous argument, the uniqueness claim should be downgraded to a conjecture.
minor comments (4)
  1. [Eq. (42)] In the recurrence (42), the value p_1 = α(2−ε)/2 appears to have the wrong sign. The numerator in (41) is α(1 − z(2−ε)/2), so p_1 should be −α(2−ε)/2. The explicit expression for μ_1 in (43) is consistent with the negative sign. Please correct.
  2. [Sec. 3.2, paragraph after Eq. (48)] The phrase 'enables us to rigorously determine the non-negativity of μ numerically in finite time' overstates what is done: the coefficients C_i in (48) and the threshold k* are computed numerically, and the α-axis is sampled on a grid. The procedure gives strong numerical evidence, but the word 'rigorously' should be replaced by a more qualified statement.
  3. [Abstract and Sec. 1.2] The phrase 'continuous spectrum' is likely to be confused with the spectral-theory notion of continuous spectrum. Since the paper establishes (or aims to establish) a continuum of eigenvalues, 'continuum of QSDs' would be less ambiguous.
  4. [Sec. 5.1] The stability conclusions are based on numerical experiments at five parameter points and a few noise models. The paper does state this, but the concluding sentence in Sec. 7 ('attested analytically and in simulations') goes beyond the analytic content, which is limited to the shape-conservation argument of Sec. 4.2. Please calibrate the wording.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the QSD continuum is an eigenvector solution with free parameter α, not a fit; the unproven positivity check is a correctness gap, not a circular reduction.

full rationale

The central derivation is self-contained. The generating function A(z) in Eq. (32) is obtained by solving the eigenvalue equation M[A]=λA, with λ(α)=1−pεα following from the mass balance in Eq. (31); α=μ0 is a free variable of the eigenvector equation, not a parameter fitted to any target. The principal rate λ_u=1/R is computed independently from the singularity analysis of G_00(s) in Eqs. (25)-(28). Positivity of the coefficients μ_k is the only step that is not proved: Sec. 3.2 states 'This, it turns out, is very difficult to determine by inspection of the generating function alone' and 'we rather observe numerically that they are valid for α≤α_u and invalid otherwise' (Fig. 3). That is an unproven assertion / numerical conjecture, so the continuous-spectrum claim carries a correctness risk, but it is not circular: the α-interval is not fitted to make the claim true, and the eigenvalue relation is not imposed to match simulated escape rates. The QED construction similarly solves the transposed eigenvector equation and specializes to λ_u by a stated cancellation argument; no output is used as an input. Self-citations to [13] are contextual (e.g. 'A more thorough investigation ... was given in [13]') and are not load-bearing for the QSD/QED derivations.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The derivation is largely self-contained, but the central claim rests on a numerically observed positivity threshold (alpha <= alpha_u), an unproved uniqueness assertion for the QED eigenvalue, and standard R-theory/generating-function tools from [8,36]. No new entities are postulated; alpha is a free index of the family, not fitted to data.

free parameters (1)
  • alpha (also written mu_0) = free; observed valid range 0 < alpha <= alpha_u; lambda = 1 - p*epsilon*alpha
    alpha is the probability mass at state 0 and appears as the free variable in solving the eigenvector equation (32)-(39). It parameterises the candidate QSD family and is not fitted to data; its allowed range is inferred numerically via the positivity check in Sec. 3.2.
assumptions (5)
  • ad hoc to paper The coefficients mu_k from recurrence (42)-(44) are non-negative for all k whenever 0 <= alpha <= alpha_u, and only then.
    Sec. 3.2 states this is 'observed numerically' (Fig. 3) rather than proved. This is the premise that turns the formal one-parameter family into a genuine continuum of probability QSDs.
  • ad hoc to paper Only lambda = lambda_u yields a normalisable QED after multiplying the Koopman eigenvector b by the QSD mu*.
    Sec. 6 footnote 5 justifies uniqueness by a cancellation argument described as 'for technical reasons we believe'; the uniqueness of the QED rests on this unproved assertion.
  • domain assumption Vere-Jones R-theory (R-positivity, R-transience, R-null recurrence) applies to this countable nonnegative matrix.
    Sec. 2.2 uses generating functions F_ii(R), G_ii(R) and the classification criteria of [8] to locate dynamical regimes; this is standard but imported theory for countable absorbing Markov chains.
  • domain assumption The open Pelikan map (12) reduces exactly to the Markov chain (9) on the Markov partition (14).
    Sec. 2 cites [13] for this reduction. It motivates the chain but the core Markov-chain results do not depend on the map equivalence.
  • standard math Generating-function singularity analysis determines coefficient asymptotics: mu_k decays according to the smallest singularity of A(z).
    Used throughout Secs. 3.2, 4.2, and 6 following Flajolet-Sedgewick [36]; it is standard analytic combinatorics.

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Pith. "Pith review of Quasi-stationary and quasi-ergodic distributions in the Pelikan random map." pith.science (2026). https://pith.science/paper/RQLTXIHS

@misc{pith2026260719084,
  author       = {Pith},
  title        = {Pith review of: Quasi-stationary and quasi-ergodic distributions in the Pelikan random map},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RQLTXIHS}},
  note         = {Machine review of arXiv:2607.19084}
}
read the original abstract

In this paper we present a concrete example of a substochastic discrete-time Markov chain on a countable state space producing a spectrum of infinitely many quasi-stationary distributions (QSDs) for generic parameter values, with each QSD supporting a distinct escape rate. Our system is motivated by an open variant of the Pelikan dynamical system, a random map introduced in the 1980s. These QSDs, and their stability to perturbative random noise, are tested in numerical simulations. The existence of unique QEDs is also established for some parameter values.

Figures

Figures reproduced from arXiv: 2607.19084 by the authors.

Figure 1
Figure 1. Left: The ‘open’ Pelikan map (12), with expanding and contracting branches shown in red and blue, respectively. The Markov partition (14) is shown in thin green dashed lines. The escape region is indicated in grey. Right: A schematic diagram of the Markov chain (9), which is the object of this paper’s study. 2.1 Case ε = 0 Thanks to results in [9, 11], we know the following information for the ‘closed’ Pelikan map (… view at source ↗
Figure 2
Figure 2. Left: R as a function of (p, ε) shown as a colour gradient, from either side of the dividing line (27). Right: Survival probability P[Xn ∈ Ω] as a function of time n, obtained from simulations (solid lines), plotted against predictions obtained from (28) (dashed lines), for the four identified sample points shown left. Sample size N = 2 × 107 . In the R-transient regime, simulations are affected by a non-exponential… view at source ↗
Figure 3
Figure 3. ‘Valid’ and ‘invalid’ values of α, for a range of p, ε. Values of α producing µk ≥ 0 for all k are shown in green. Invalid α, for which µk < 0 for some k, are shown in red. Plot titles are coloured to indicate the dynamical regions discussed in Sec. 2.2, with R-recurrent parameters shown in blue and R-transient parameters in red. The critical value αc is shown in black, and, where it differs from this, the asymptoti… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Top: Histograms of QSDs for a variety of α, for the sample parameters (p, ε) = (0.8, 0.2), which are preserved under the time evolution (29), normalised. The index k of each state is plotted from right to left, with the height of the bar indicating the probability mass…
Figure 5
Figure 5. Figure 5: Left: A map showing the five sample points in parameter space used in Sec. 5.1. Regions of R-recurrence and R-transience are demarcated by a solid line, while regions where the principal QSD has µk ∼ 2−k or not are demarcated by a dashed line. The colour gradient shows…
Figure 6
Figure 6. Figure 6: from top to bottom): • visual inspection of the evolved distribution (M∗ ) n[µ †#], by comparison with the unperturbed distribution µ † and the principal QSD µ ∗ , in the top row of plots – in [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Convergence to the principal QSD µ ∗ from an initial density µk when µk ≪ µ ∗ k . In each row, µk ∼ 2−k while in each of these three regions µ ∗ k ≫ 2−k. In the first two columns, higher iterations are shown in higher frequency hues, see text for details. The Koopman o…
Figure 8
Figure 8. Figure 8: From left to right, shown as colour gradients as a function of (p, ε): decay rate (‘shape’) of the principal QSD µ ∗ k wrt. k (darker is faster decay); growth rate of the corresponding Koopman eigenvector bk wrt. k (darker is faster growth); decay rate of the resulting…
Figure 9
Figure 9. Figure 9: QEDs estimated from numerical simulations, for increasing measurement times up to between t = 50 and t = 300 (coloured histograms); compared against theoretical QEDs calculated analytically in Sec. 6 (black dashed lines). Regions are colour coded according to R-recurre…

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