Partition Function Zeros of Paths and Normalization Zeros of ASEPS

Desmond A. Johnston; Zdzislaw Burda

REVIEW 2 major objections 6 minor 44 references

Partition Function Zeros of Paths and Normalization Zeros of ASEPS

T0 review · 2 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A conformal map gives the exact thermodynamic limit of the ASEP normalization zeros as images of a circle.

desk verdict Elegant but non-novel conformal-map re-derivation of ASEP zero loci; the two-pole step is asserted not proven, but the paper deserves referee time. read the letter →

arxiv 2501.14953 v3 pith:KMYYFEJY submitted 2025-01-24 cond-mat.stat-mech math-phmath.MP

Zdzislaw Burda , Desmond A. Johnston This is my paper · ORCID

classification cond-mat.stat-mechmath-phmath.MP MSC 82B2682C2205A15

keywords ASEPnormalizationpartitionfunctionzerosLee-YangconformalmapDyckpathsrandomallocationmodelthermodynamiclimitnon-equilibriumsteadystates

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 establishes that, in the thermodynamic limit, the zeros of the normalization $Z_N(\alpha,\beta)$ of the Asymmetric Simple Exclusion Process (ASEP) - viewed as a function of $\alpha$ for fixed $\beta$ - accumulate on explicit curves: $\alpha = f_D(e^{is}/4)$ for $\beta \geq 1/2$, and $\alpha = f_D(\beta(1-\beta)e^{is})$ for $\beta < 1/2$, where $f_D(z) = (1 - \sqrt{1-4z})/2$ is the generating function of a single Dyck excursion. The curves are images of circles under this conformal map, and the density of zeros is the image of the uniform circle density. The derivation rests on the factorization of the grand-canonical ASEP normalization into two Dyck-walk generating functions, so the normalization zeros are exactly the partition-function zeros of a pair of non-interacting adsorbing Dyck walks. The result matters because it gives an analytically solvable example of how the zeros of a non-equilibrium steady-state normalization encode phase boundaries, in direct analogy to Lee-Yang zeros in equilibrium.

What carries the argument

The central object is $f_D(z) = (1 - \sqrt{1-4z})/2$, the generating function for a single Dyck excursion (a positive lattice path returning to the axis), which maps the disc $|z| < 1/4$ conformally onto the interior of a cardioid. The grand-canonical ASEP normalization factorizes as $Z(z,\alpha,\beta) = 1/[(1 - f_D(z)/\alpha)(1 - f_D(z)/\beta)]$, so the $N$-site normalization is the coefficient of $z^N$ in this product of two Dyck-walk generating functions. The zero locus is then read off from the earlier formula $u = 1/f(\sigma e^{is})$ for the random allocation model: the critical curve is the image of the convergence circle $|z| = \sigma$ under the conformal map, with $\sigma = 1/4$ when the square-root branch point is dominant and $\sigma = \beta(1-\beta)$ when the $\beta$-pole is dominant. This machinery converts the zero-finding problem into singularity analysis of a generating function.

What would settle it

Take $\beta = 0.6$ and $N = 2000, 4000$, compute the zeros of $Z_N(\alpha,\beta)$, and check whether any accumulate on a curve other than $\alpha = f_D(e^{is}/4)$. A complementary check is to search numerically on the integrand's Riemann sheet for solutions of $f_D(z) = \beta$ with $|z| < 1/4$ when $\beta > 1/2$; finding one would invalidate the branch-point-only locus.

Watch

Extended reading notes

Core claim

The paper's central claim is that the ASEP normalization zeros are governed by the same conformal map that solves the adsorbing Dyck walk. For $\beta \geq 1/2$ the locus is $\gamma(s) = f_D(e^{is}/4)$, the image of the circle $|z| = 1/4$; for $\beta < 1/2$ it is $\gamma(s) = f_D(\beta(1-\beta)e^{is})$, the image of the circle of radius $\beta(1-\beta)$. At the critical points the curves meet the real axis at angle $\pm 3\pi/4$ for the second-order transition at $\alpha = 1/2$, and at angle $\pi/2$ for the first-order line at $\alpha = \beta$. The line density of zeros follows from the uniformity of the source circle under the map. These loci coincide with the curves obtained by matching the real parts of the ASEP free energies in the three phases, namely $|\alpha(1-\alpha)| = 1/4$, $|\beta(1-\beta)| = 1/4$, and $|\alpha(1-\alpha)| = |\beta(1-\beta)|$.

Load-bearing premise

For $\beta \geq 1/2$, the argument assumes that the $\beta$-pole contributes no singularity inside $|z| < 1/4$ on the sheet selected by the contour integral, so the zero locus is fixed entirely by the square-root branch point; if a hidden $\beta$-dependent singularity were present there, the predicted curve would be different.

Editorial extensions

If this is right

For $\beta \geq 1/2$, the zero locus of the ASEP normalization is the same cardioid-like curve as for adsorbing Dyck walks, so the second-order transition at $\alpha = 1/2$ shows zeros meeting the real axis at $\pm 3\pi/4$.
For $\beta < 1/2$, the zeros lie on the image of the smaller circle of radius $\beta(1-\beta)$ and meet the real axis at right angles, signalling the first-order transition at $\alpha = \beta$.
The density of zeros vanishes linearly near the second-order critical point (zeros are sparse there) and is finite at the first-order critical point, with value $\mu_{cr} = 4/(3\pi)$ for $\beta = 1/4$.
The conformal-map loci reproduce the electrostatic free-energy matching conditions $|\alpha(1-\alpha)| = 1/4$, $|\beta(1-\beta)| = 1/4$, and $|\alpha(1-\alpha)| = |\beta(1-\beta)|$, so the two derivations agree.
The same formula $u = 1/f(\sigma e^{is})$ gives the zero locus for the random allocation model, showing the ASEP result is part of a single conformal-map family.

Reading between the lines

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

Beyond the paper, the same conformal-map method should apply to other pair-factorized steady-state normalizations and urn models with multiple constraints, a direction the paper only flags as open; a concrete test would be a two-constraint urn model whose generating function has two dominant singularities.
Beyond the paper, since the density formula is exact, the spacing between consecutive zeros at finite $N$ could be predicted by integrating the pushforward density; the paper compares loci but does not compute finite-size spacings.
Beyond the paper, the factorization into two independent Dyck walks suggests studying the joint zeros of $Z_N(\alpha,\beta)$ in both parameters at once, where the product structure may produce interacting rather than superimposed curves.

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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

Quick take: this is a nice paper to have on the shelf, but it is not a new result. The loci for the ASEP normalization zeros were already in [29] numerically and in [30,42] analytically. What this paper adds is a compact conformal-map derivation that ties those loci to the Dyck-walk and random-allocation pictures, and a clear discussion of the electrostatic matching. The exposition is good, the numerics for N=1000 look convincing, and the authors are upfront about the prior work. I'd treat it as a solidly written re-derivation with a pleasant unifying perspective.

The real soft spot is the two-pole step. Formula (32) was derived for a single pole factor 1/(1 - u f(z)). Here you have two factors, one for α and one for β. For β ≥ 1/2 the paper simply asserts the β-pole does not affect the limiting zeros, and for β < 1/2 it asserts the β-pole dominates. That's plausible, and the numerical checks back it up, but it is asserted rather than shown. A partial-fraction decomposition or a saddle-point estimate would settle it. Since the final loci match the earlier electrostatic results exactly, I'm confident the answer is correct; it's a rigor gap rather than a correctness gap. I'd ask the authors to add a few lines on this before I'd be comfortable with it as the derivation of record.

Minor: Eq. (10) has a typo (the exponent should be -(n+1) ln z, not -(n+1) z). Also, the paper leans on [23] for the key formula, which is fine since [23] is published and the current text sketches the regimes, but a reader new to this method will need to have [23] open.

Who's this for? People working on Lee-Yang zeros in nonequilibrium steady states or on the ASEP-Dyck path connection. They'll find a clean, useful summary of how the conformal map and electrostatic approaches are equivalent. It's not essential reading if you already know [30,42], but it's a good tutorial.

My recommendation: if it crosses your desk, send it out. It's worth referee time, and with the two-pole gap patched and the typo fixed it would be a solid, if modest, contribution. Just don't let the authors claim novelty for the loci themselves—they're careful not to, and the referee should keep them honest on that.

Referee Report

2 major / 6 minor

Summary. The paper studies the thermodynamic limit of the locus of zeros of the ASEP normalization Z_N(alpha,beta), viewed as a polynomial in alpha for fixed beta. It reviews analytic-combinatorics methods for random allocation models and for adsorbing Dyck walks, and uses the conformal-map formula (32) from the authors' earlier work [23] to write the Dyck-walk zero locus as alpha = f_D(e^{is}/4). It then extends this to the ASEP by observing that the grand-canonical normalization (58) is a product of two Dyck generating functions with fugacities 1/alpha and 1/beta. The central results are the zero loci alpha = f_D(e^{is}/4) for beta >= 1/2 and alpha = f_D(beta(1-beta)e^{is}) for beta < 1/2, together with the zero density obtained from Eq. (35). The paper also argues that these loci agree with the electrostatic matching of the real parts of the known ASEP free energies (61)-(64).

Significance. If correct, the paper gives a concise conformal-map derivation of the ASEP normalization zero locus and its density, unifying the ASEP case with the random-allocation and Dyck-walk cases. The results are consistent with earlier numerical and electrostatic calculations, and the manuscript includes direct numerical root checks at N=1000 for beta=3/4 and beta=1/4, which is a useful cross-check. The novelty is moderate: the ASEP zero locus itself was already known from [29,30,42], but the conformal-map route and the explicit density formula are presented here in a unified and accessible way. The main value of the paper is pedagogical and unifying rather than the discovery of an entirely new locus.

major comments (2)

[Section VI, Eq. (59)] The central claim that the ASEP zero locus is given by the single-pole formula (32) is not derived for the two-pole integrand (59). The statement that the factor (1 - f_D(z)/beta)^{-1} "does not affect the limiting distribution of zeros" for beta >= 1/2 is justified neither by a Riemann-sheet argument nor by a saddle-point estimate. A partial-fraction decomposition of the integrand, 1/[(1-f_D/alpha)(1-f_D/beta)] = alpha beta/(beta-alpha)[(1/alpha)/(1-f_D/alpha) - (1/beta)/(1-f_D/beta)], shows that Z_N(alpha,beta) is a linear combination of the single-pole Dyck partition functions; the limiting zeros then follow from balancing the exponential rates of the two terms, i.e., from Re psi_D(alpha) = Re psi_D(beta). The authors should supply this balance analysis, or an explicit proof that for beta > 1/2 the beta-pole lies on a non-principal sheet and is therefore absent from the contour in (59), and that for beta < 1/2 the beta-pole term dominates. The numerical check at N=1000 for two values of beta is suggestive but is not a substitute for this step, which is load-bearing for the main result.
[Section VI, after Eq. (59)] The boundary case beta = 1/2 is included in the formula alpha = f_D(e^{is}/4) for beta >= 1/2, but at beta = 1/2 the pole of (1 - f_D(z)/beta)^{-1} coincides with the branch point at z = 1/4. The argument that the beta-pole does not affect the locus for beta > 1/2 does not extend automatically to this degenerate point. Please either exclude beta = 1/2 or treat it separately, for example by a limiting argument or by an explicit analysis of the double singularity, and state the result at the triple point.

minor comments (6)

[Section II, Eq. (10)] Equation (10) has a typographical error: the exponent should read ln f(z) - (n+1) ln z, not ln f(z) - (n+1) z.
[Section VII, Eq. (60)] The definition of the free energy is missing the logarithm; it should read F = lim_{N -> infinity} (1/N) ln Z_N(alpha,beta), consistently with the logarithmic expressions in Eqs. (61)-(64).
[Section V, Eq. (54)] The phrase "probabilistic weights (39)" after Eq. (54) is confusing: Eq. (39) defines the combinatorial weights w_D(s), while the probabilistic weights are defined in Eq. (54). Please correct the cross-reference.
[Section VI, Figures 5 and 6] Please clarify in the captions that the variable alpha in the ASEP plays the role of the variable v in the Dyck-walk calculation, so that the locus in Figure 5 is the same curve as in Figure 2.
[Section IV, Eq. (32)] Since formula (32) is imported from [23] and is the key tool of the paper, please state explicitly the regularity conditions under which it applies (for example injectivity of f on |z| < sigma and the precise meaning of the inverse f^{-1}) so that the two-pole extension in Section VI can be checked against these hypotheses.
[Section IV and Eq. (57)] The summation index p in Eq. (57) conflicts with the pressure variable p introduced in Section IV; please rename one of them for clarity.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the ASEP zero locus is independently checked against numerical roots and electrostatic free-energy matching; the main caveat (unproved two-pole dominance) is a correctness gap, not a circular step.

full rationale

The central derivation applies the conformal-map formula (32), taken from the authors' earlier work [23], to the ASEP generating function. This is a self-citation, but the paper also re-derives the same formula within the electrostatic framework in Section VII, and the ASEP-specific application is not obtained by fitting parameters or by definition. The predicted zero loci for β ≥ 1/2 and β < 1/2 are benchmarked against numerically computed roots of the exact polynomial Z_N(α, β) from (57) at N = 1000 for β = 3/4 and β = 1/4, and they agree with the independent electrostatic free-energy matching conditions (62)-(64). No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work. The main genuine weakness is that the paper asserts, rather than proves, that the fixed-β pole factor in (59) does not alter the limiting zero distribution for β ≥ 1/2: 'The term 1/(1 − β−1fD(z)) does not affect the limiting distribution of zeros in this case.' This is a statement about asymptotic dominance that could in principle be settled by a partial-fraction or saddle-point analysis; it is a correctness risk rather than a circular step, because the claimed locus is not an input to that assertion. The self-citations to [23], [30], and [42] point to parameter-free results with stated assumptions that do not include the target ASEP locus, and the numerical and electrostatic cross-checks provide independent support. Overall, the derivation is self-contained against external benchmarks, so the circularity score is low, reflecting only the presence of load-adjacent self-citations and a stated-but-unproved dominance claim.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The derivation rests on the analytic-combinatorics saddle point method, the locus formula (32) from the authors' prior paper [23], the Dyck-ASEP equivalence (58), and the equilibrium free-energy treatment of the ASEP normalization (flagged in the Discussion). No free parameters or invented entities are introduced.

assumptions (4)

domain assumption Formula (32): the locus of zeros is the image of the boundary circle |z|=sigma under u=1/f(z) when f is injective.
Quoted from the authors' companion paper [23] and used to read off the loci in Sections IV-VI, with only a sketch of the saddle-point derivation in this paper.
domain assumption The ASEP normalization equals the product of two Dyck walk generating functions (Eq. (58)).
Taken from [30,42]; this mapping is the bridge that lets the Lee-Yang apparatus apply to the non-equilibrium ASEP normalization.
domain assumption The ASEP normalization free energy can be treated as an equilibrium free energy for the purpose of matching real parts on phase boundaries.
Explicitly flagged in the Discussion after Eq. (60): 'Assuming that this "free energy" derived from ASEP normalization can be treated like the free energy of an equilibrium model, which can be justified post hoc by the Dyck path equivalence.'
standard math Saddle point evaluation of contour integrals (31) and (59) is valid in the thermodynamic limit.
Standard analytic-combinatorics saddle point method following [24]; the paper assumes subleading terms vanish as S,N approach infinity.

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Cite this review

Pith. "Pith review of Partition Function Zeros of Paths and Normalization Zeros of ASEPS." pith.science (2026). https://pith.science/paper/KMYYFEJY

@misc{pith2026250114953,
  author       = {Pith},
  title        = {Pith review of: Partition Function Zeros of Paths and Normalization Zeros of ASEPS},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KMYYFEJY}},
  note         = {Machine review of arXiv:2501.14953}
}

read the original abstract

We exploit the equivalence between the partition function of an adsorbing Dyck walk model and the Asymmetric Simple Exclusion Process (ASEP) normalization to obtain the thermodynamic limit of the locus of the ASEP normalization zeros from a conformal map. We discuss the equivalence between this approach and using an electrostatic analogy to determine the locus, both in the case of the ASEP and the random allocation model.

Figures

Figures reproduced from arXiv: 2501.14953 by the authors.

**Figure 2.** FIG. 2. The analytically calculated locus of zeros for adsorb [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. (Totally) Asymmetric Exclusion Process on a line [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. The ASEP phase diagram. Region (1) is a low [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 6.** Figure 6: FIG. 6. Analytically calculated locus of zeros in the ther [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 5.** Figure 5: FIG. 5. Analytically calculated locus of zeros in the ther [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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free energy

In this case, the critical curve impacts the real axis at an angle π/2, which is characteristic of a first-order transition. The density of zeros at this first-order phase FIG. 6. Analytically calculated locus of zeros in the ther- modynamic limit from Z(z, α, β) and numerical...

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Reviewed August 10, 2026 · model on record in the stance chip above.

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