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REVIEW 2 major objections 5 minor 93 references

Two fluctuating interfaces with sticking interactions: Invariant measures and dynamics

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Pith's one-line read This paper proves that when two fluctuating interfaces obey identical Edwards-Wilkinson or Kardar-Parisi-Zhang dynamics with sticking at contacts, the exact steady-state measure equals the equilibrium Poland-Scheraga measure of DNA…

desk verdict Exact steady state in the a=a' subspace is real and well supported; the 'always entangled for any s' phase claim drops a loop-initiation prefactor and is likely wrong near s=1. read the letter →

arxiv 2507.20350 v1 pith:VKMLGLOR submitted 2025-07-27 cond-mat.stat-mech cond-mat.softphysics.bio-ph

Samvit Mahapatra , Malay Bandyopadhyay , Mustansir Barma This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.softphysics.bio-ph
keywords stickinginteractiontwofluctuatinginterfacessingle-stepsurfacesexactsteadystatePoland-ScheragamodelpairwisebalanceultraslowdynamicsKardar-Parisi-Zhanguniversality
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 minimal nonequilibrium model in which two single-step surfaces in one dimension fluctuate independently except at sites where they cross, where the update probability is reduced by a detachment factor $s$. Its central result is exact for the subspaces where both surfaces evolve by the same dynamics—both Edwards-Wilkinson ($a=a'=0$) or both Kardar-Parisi-Zhang ($a=a'>0$): the steady-state probability of a configuration with $k$ stuck sites is $P(C_s^k)=p_0 s^{L-k}$, an inhomogeneous product measure that assigns weight 1 to each stuck site and $s$ to each detached site. This measure is shown to be exactly the equilibrium Poland-Scheraga measure of DNA denaturation, with loop weight $g=4s$, loop exponent $c=3/2$, and stuck-segment weight $v=2$, so the two-surface steady state is always entangled in the thermodynamic limit. The same model also produces qualitatively different approach-to-steady-state dynamics depending on the biases: zipper-like diffusive closure in the 2EW case, ballistic faceting in the 2KPZ case, diamond-shaped jammed bubbles with lifetimes $\sim e^{\lambda L^2}$ in the 2OKPZ case, and a coexistence of stuck and detached histories in the EWKPZ case. The paper therefore connects a driven two-interface system to an equilibrium statistical-mechanics model and identifies regimes where the steady state is effectively unreachable.

What carries the argument

The carrying object is the mapping of each single-step surface to a sequence of tilts, equivalently particles and holes in an exclusion process, and then to a 'brackets' notation in which only the local hills and valleys of the two-surface configuration are kept; stuck bends are angular brackets and detached bends are round brackets. The proof of the steady state uses pairwise balance of probability fluxes: incoming and outgoing fluxes at every bend are paired so that the master equation splits into terms $A(w_{C''\to C}P(C'')-w_{C\to C'}P(C))$, and the product measure $P(C)=p_0\prod_i p_i$ with $p_i=1$ for a stuck site and $p_i=s$ for a detached site makes each paired flux difference vanish. The identification with the Poland-Scheraga model is carried by the exact configurational counting of bubbles, which gives a generating function whose large-length behavior is $B(l)\sim 4s^l/l^{3/2}$, with stuck-segment weight $v=2$. For the ultraslow dynamics, the machinery is an Arrhenius activation picture: diamonds have area $\sim O(L^2)$, and their collapse is treated as crossing a super-extensive energy barrier set by the Hamiltonian $H=\epsilon\sum_i|h_{1i}-h_{2i}|$ of an auxiliary non-crossing model that satisfies detailed balance.

What would settle it

Simulate the 2OKPZ model with $s=0$ for system sizes $L=64,128,256,512$ and measure the mean lifetime of the largest diamond from the initial configuration; if the lifetime grows as a power of $L$ rather than as $e^{\lambda L^2}$, the activation picture is wrong. For the exact steady-state claim, sample the full configuration measure at fixed $k$ for a small system such as $L=16$ and test whether $P(C_s^k)/p_0$ equals $s^{L-k}$ for every configuration with $k$ stuck sites, and whether the measured loop weight is $4s$.

Watch

Extended reading notes

Core claim

Within the (2EW + 2KPZ) subspace $a=a'\ge 0$, the paper proves by pairwise balance of probability fluxes that the exact invariant measure is $P(C_s^k)=p_0 s^{L-k}$: all configurations with the same number $k$ of stuck sites are equally likely, and each detached site costs a factor $s$ relative to a stuck site. This measure does not depend on the bias $a$, so the interacting two-surface problem inherits the well-known $a$-invariance of a single single-step surface. Written in terms of alternating bubbles and stuck segments, the measure factorizes into loop weights $B(l)\sim g^l/l^c$ with $g=4s$ and $c=3/2$ and stuck-segment weights $R(r)\sim v^r$ with $v=2$, which is precisely the Poland-Scheraga partition function for DNA denaturation in the regime $v>v_c(s)$. Consequently, for any $s$ in $(0,1)$ the thermodynamic-limit steady state of the identical-dynamics subspaces is entangled, with a finite fraction of stuck sites. The paper further argues that outside this subspace the dynamics is controlled by large bubbles inherited from the initial condition: they close diffusively (2EW), ballistically as facets (2KPZ), deform into long-lived diamonds whose lifetimes scale as $e^{\lambda L^2}$ (2OKPZ, $s=0$), or produce two competing evolutions—near-complete sticking or complete detachment—for an EW surface against a KPZ surface.

Load-bearing premise

The load-bearing premise for the ultra-slow timescale claim is that the diamond-closing time in the original sticking model is the same as in an auxiliary exclusion model with an area-energy Hamiltonian, so that closing is an Arrhenius activation over an $O(L^2)$ energy barrier; the paper asserts this equivalence heuristically rather than proving it.

Editorial extensions

If this is right

  • For all $s$ in $(0,1)$, the exact steady state in the 2EW and 2KPZ subspaces has the same form regardless of the KPZ bias $a$; any observable such as the bubble-size distribution or the stuck fraction can be computed from the Poland-Scheraga partition function.
  • In the thermodynamic limit the identical-dynamics steady state is always entangled for any $s>0$, so there is no detached phase on this locus; the fraction of stuck sites is set by the PS loop weight $g=4s$ and stuck weight $v=2$.
  • For the 2OKPZ case with irreversible sticking ($s=0$), the largest initial bubbles become diamonds whose closing time grows as $e^{\lambda L^2}$, making the final stuck state unreachable on any practical timescale for large $L$.
  • For an EW surface against a KPZ surface at small $s$, the same initial configuration can either end detached after a ballistic time $\sim L/s$ or end stuck after the EW surface diffuses onto the KPZ facet in time $\sim L^2$; the two outcomes occur with order-one fractions when $s$ scales as $L^{-0.75}$.
  • In the 2EW case with $s=0$, the transient width collapse gives a dynamic exponent $z\simeq 1.5$, different from the single-surface EW value $z\simeq 2$, and the same exponent controls short-time transients when detachments are allowed.

Reading between the lines

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

  • If the exact measure holds, the 2EW and 2KPZ models provide a dynamical way to generate equilibrium Poland-Scheraga statistics; one could measure critical exponents of DNA denaturation from a driven system rather than from a thermal ensemble.
  • The pairwise-balance bracket construction is likely to extend to other height-difference interactions, such as asymmetric sticking or short-range repulsion, producing exact measures of the same product form with modified site weights; that would generalize the set of solvable nonequilibrium coupled-interface models.
  • The claimed $e^{\lambda L^2}$ diamond lifetime implies pronounced aging and history dependence in the 2OKPZ system; autocorrelation functions should exhibit two-step relaxation, and the steady state should be unattainable even at times far beyond ordinary equilibration.
  • The unexplained $z\simeq 1.5$ in the 2EW $s=0$ case might be a universal property of the closure of the largest excursion in a random landscape; testing the closure-time distribution of the largest bubble in a simpler model, such as a single interface against a fixed wall, would show whether this exponent is specific to two-surface sticking or generic.
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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 / 5 minor

Summary. The paper studies two single-step interfaces in 1+1 dimensions with short-range sticking at contact sites; the update probabilities are (1/2±a) per site, multiplied by a detachment probability s at sites where the surfaces cross. In the subspace a=a'≥0 (two EW or two KPZ surfaces) the authors derive an exact invariant measure P(C_s^k)=p0 s^{L-k} by pairwise balance and identify it with the equilibrium Poland–Scheraga measure, concluding that the steady state is always entangled. The remaining sections analyze transient dynamics: 2EW shows a numerical dynamic exponent z≈1.5, 2KPZ forms facets, 2OKPZ produces long-lived diamond bubbles with claimed lifetimes T∼exp(λL^2), and EWKPZ exhibits two competing evolutions with a detachment fraction boundary s*∼L^{-0.75}.

Significance. The exact steady-state derivation is a valuable and nontrivial result: an out-of-equilibrium two-surface model with sticking interactions is shown, by a pairwise-balance construction, to have an inhomogeneous product measure, and the numerical support in Fig. 2 (a-invariance of the width and bubble-size distributions) is convincing. The identification with the Poland–Scheraga measure is conceptually appealing and, once the phase-boundary calculation is corrected, should provide a useful bridge between non-equilibrium interface models and DNA-denaturation models. The dynamical observations (z≈1.5 in 2EW, facet formation in 2KPZ, diamond stability in 2OKPZ, and the two-mode EWKPZ evolution) are interesting and well presented, though some of the more dramatic claims, especially T∼exp(λL^2), are heuristic at present.

major comments (2)
  1. [Sec. I ('The steady state in the 2EW and 2KPZ cases'), Eq. (3) and following paragraph] The conclusion following Eq. (3) that v>v_c(s) for every s<1, and hence that the steady state is always entangled, drops the loop-initiation prefactor of the bubble weight. Using the authors' exact bubble count from the Supplemental Material, B(l)=C_{l-1} (Catalan numbers), the rescaled loop weight is w(l)=C_{l-1}/4^l, and its unit-fugacity generating function is A=Σ_{l≥2} C_{l-1}/4^l = 1/4, not ζ(3/2)≈2.61. Replacing ζ(3/2) by A in the critical condition used by the authors gives v_c = g/(1+A) = 4s/(1+1/4) = 16s/5; with v=2 the bound phase therefore requires s<5/8≈0.625 (using only the asymptotic prefactor q=1/(4√π) gives s_c≈0.68). Thus for s near 1 the predicted steady state is denatured rather than entangled. The paper's own Supplemental Fig. S4(d) is consistent with this: at s=0.9 the bubble-size distribution is already close to the noninteracting l^{-3/2} law. Please recompute the Poland–Scheraga phase boundary with the exact loop generating function and revise the abstract, the bullet list, and the phase-diagram claims accordingly.
  2. [End Matter, 'Ultraslow timescales in the 2OKPZ case when no detachments are allowed (s=0)'] The claimed diamond lifetime T∼exp(λL^2) is presented as a property of the original sticking model, but the derivation is made for an auxiliary non-crossing exclusion model with Hamiltonian H=εΣ_i |h1_i−h2_i|. The statement that the longest closing timescale of a diamond in the original model equals that of the auxiliary model is asserted, not derived, and the activation picture over an O(L^2) area barrier is heuristic. Because the exp(λL^2) scaling is a headline claim in the abstract and in the fourth bullet point, please provide direct numerical measurements of diamond closing times as a function of L in the original model, or explicitly label the scaling as a conjecture.
minor comments (5)
  1. [Supplemental Material, Fig. S2(b) and accompanying text] The text states that the detachment probability s(L) follows a power law ∼ L^η with η≈0.75, but the plotted fit is s*∼L^{-0.75}; the exponent in the text should carry a minus sign.
  2. [End Matter, Hamiltonian H for the non-crossing model] The text defines H=εΣ_i |h1_i−h2_i| but then uses signed height differences in the energy-change formulas; please clarify whether the potential is the absolute value or a signed sum, since the claimed area barrier depends on this distinction.
  3. [References, [91]] The numerical dynamic exponent z≈1.5 in the 2EW case and the analysis of the closing of the largest bubbles are attributed to an unpublished work; please provide a publicly available preprint or include the numerical data and fitting procedure in the Supplemental Material.
  4. [End Matter, 'Pairing scheme for the Fluxes in the 2EW and 2KPZ steady state'] The pairwise-balance proof leading to Eq. (1) is compressed into Fig. 7 and a short paragraph; since this is the central exact result, a complete enumeration of the flux-pairing cases in an appendix or in the Supplemental Material would strengthen the paper.
  5. [Fig. 2 caption] The caption mixes the description of width-versus-time plots and bubble-size distributions in a single parenthetical phrase, making it difficult to tell which panels correspond to which observable; please separate the panel descriptions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the invariant measure is derived from the update rules by explicit pairwise balance, and the Poland-Scheraga comparison is an external benchmark.

full rationale

The paper's central derivation is self-contained. Eq. (2), P(C_s^k)=p0 s^{L-k}, is obtained by pairing incoming and outgoing probability fluxes and imposing pairwise balance, with the pairing scheme spelled out in the End Matter (Figs. 6-7); no parameter entering the measure is fitted to the target conclusion. The subsequent identification with the Poland-Scheraga measure uses independently computed inputs: the bubble loop exponent c=3/2 from exact enumeration of bubble configurations in the Supplemental Material, g=4s, and v=2 from the random-walk count of stuck segments. The PS critical condition v_c = g/(1+ζ(c)) is an external standard result, not an input of this paper. Self-citations ([72] for brackets notation, [80-82] for the pairwise-balance method, [91] for the z≈1.5 explanation) are not load-bearing in the sense of forcing the result: the brackets construction is re-derived in the End Matter, and pairwise balance is an established technique that the paper verifies locally. The manuscript itself flags open points: the origin of z≈1.5 (main text: 'remains to be understood'), the full entangled-detached separatrix ('remains to be evaluated'), and the ultra-slow 2OKPZ timescale, which is asserted heuristically via an auxiliary non-crossing model in the End Matter rather than derived. These are limitations or correctness risks, not circularity. Likewise, Eq. (3) drops the non-unit amplitude of the exact bubble weight when invoking the standard PS threshold; this is an approximation concern in the external benchmark, not a circular reduction, because Eq. (2) itself is not used to define the phase boundary.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central steady-state result depends only on the model parameters s, a, a' and on standard random-walk combinatorics. The only ad hoc element is the auxiliary non-crossing model used to estimate diamond lifetimes, which is not a new physical entity but a heuristic modeling variant.

free parameters (3)
  • bias a
    Bias for surface 1 update probabilities; a control parameter, not fitted. In the exact steady state, the measure is independent of a in the a=a' subspace.
  • bias a'
    Bias for surface 2 update probabilities; a control parameter, not fitted. In the exact steady state, the measure is independent of a' in the a=a' subspace.
  • detachment probability s
    Probability factor for updates at stuck sites; a control parameter (0<s<1). The steady state measure P(C)=p0 s^{L-k} depends on s, so s is not fitted but is the tunable interaction strength.
assumptions (3)
  • standard math Single-step surfaces with periodic boundary conditions map to biased exclusion processes (SEP/ASEP).
    Used in the Supplemental to establish the model's relation to known exact solutions; standard mapping.
  • standard math The configurational weight of a bubble of length l is B(l) ~ (4s)^l / l^{3/2}, and of a stuck segment R(r) ~ 2^r, based on random-walk combinatorics.
    Used to identify the partition function with the Poland-Scheraga model; derived from exact counting of bubble configurations.
  • ad hoc to paper The longest closing timescale of a diamond in the original sticking model equals that of the non-crossing exclusion variant with Hamiltonian H=εΣ|h1_i-h2_i|.
    Used in End Matter to argue T~exp(λL^2) for 2OKPZ s=0; this equivalence is asserted heuristically, not proven.

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Cite this review

Pith. "Pith review of Two fluctuating interfaces with sticking interactions: Invariant measures and dynamics." pith.science (2026). https://pith.science/paper/VKMLGLOR

@misc{pith2026250720350,
  author       = {Pith},
  title        = {Pith review of: Two fluctuating interfaces with sticking interactions: Invariant measures and dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VKMLGLOR}},
  note         = {Machine review of arXiv:2507.20350}
}
read the original abstract

We introduce and study a non-equilibrium stochastic model of two fluctuating interfaces which interact through short-range attractive interactions at their points of contact. Beginning from an entangled state, the system exhibits diverse dynamics -- ranging from fast transients with small lifetimes to ultraslow evolution through quasi-stationary states -- and reaches stuck, entangled, or detached steady states. Near the stuck-detached transition, two distinct dynamical modes of evolution co-occur. When the two surfaces evolve through similar dynamics (both Edwards-Wilkinson or both Kardar-Parisi-Zhang), the invariant measure is determined and found to have an inhomogeneous product form. This exact steady state is shown to be the measure of the equilibrium Poland-Scheraga model of DNA denaturation.

Figures

Figures reproduced from arXiv: 2507.20350 by the authors.

Figure 1
Figure 1. (a) (i)-(iii) Update rules of one of the surfaces with bias [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. (a) Collapse of the transient minimum in rescaled plots of width vs. time in the 2EW case for different system sizes [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. In the 2OKPZ case for s = 0, the sticking fraction vs. time trajectories for individual configurations show (a) ultraslow evolution, because diamonds persist without clos￾ing for several decades of time. (b) With time the bubble size distribution recedes slowly from the small l region as the smaller diamonds close, while surviving diamonds of size l remain distributed as in the initial state P(l) ∼ l −3/2 . In the E… view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: In the EWKPZ case for small non-zero s and the 2OKPZ case for small s and finite system size, we observe two distinct evolutions of the two surfaces beginning from the same initial random configuration. The differences in evolu￾tions arise from the orientation of the b…
Figure 7
Figure 7. Figure 7: Illustrating paired fluxes in the 2EW and 2KPZ [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

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    The change in the height difference is0, 0, +2, and−2, respectively. Within every bubble configuration, the two surfaces have different tilts at the edge sites. For a bubble of size l = n + 2, where +2 accounts for the bubble’s edge sites, its exact number of configurations is...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.