An invariance principle for biased voter model interfaces

Jan M. Swart; Jinjiong Yu; Rongfeng Sun

An invariance principle for biased voter model interfaces

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A one-dimensional biased voter model interface converges, in the weak-bias diffusive limit, to a sharp boundary moving as a drifted Brownian path.

desk verdict A careful and correct-looking proof of the diffusive scaling limit for biased voter model interfaces; the stress-test concern about the state-space definition is a false alarm. read the letter →

arxiv 1908.02944 v3 pith:U5ORP4X7 submitted 2019-08-08 math.PR

Rongfeng Sun , Jan M. Swart , Jinjiong Yu This is my paper · ORCID

classification math.PR MSC 82C2282C2482C4160K35

keywords biasedvotermodelinterfacetightnessinvarianceprincipleweightedmidpointbranchingandcoalescingrandomwalksmeasure-valuedprocessdriftingBrownianmotionone-dimensionalinteractingparticlesystems

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 establishes a scaling limit for the interface of a one-dimensional biased voter model: when the bias is small and space and time are rescaled diffusively, the fuzzy hybrid zone between an infinite population of 0's and an infinite population of 1's collapses to a sharp boundary that moves as a Brownian motion with drift. The theorem covers every initial interface state and every interaction kernel that is irreducible, has mean zero, and has finite second moment. This matters because it reduces a hard interacting-particle problem to a one-dimensional diffusion statement, extending the well-understood unbiased voter model to the biased case. The proof works by showing that the weighted midpoint of the interface is a random walk with a random clock, and that the clock, after rescaling, becomes deterministic with speed $\sigma^2$, the second moment of the kernel.

What carries the argument

The central object is the weighted midpoint $M(x)$, the unique half-integer position where the number of 1's to the left equals the number of 0's to the right. The paper shows that $M(X^\varepsilon_t)$ is a random time-changed random walk: it jumps left with rate $\tfrac12 \sum_k a(k) I_k(x)$ and right with rate $\tfrac12(1-\varepsilon) \sum_k a(k) I_k(x)$, where $I_k(x)$ counts the $k$-boundaries of the interface. The random clock is $S^\varepsilon_t = \int_0^t \sum_k a(k) I_k(X^\varepsilon_s)\, ds$, and the proof's key step (Lemma 2.2) is that $\varepsilon^2 S^\varepsilon_{\varepsilon^{-2}t}$ converges in probability to $\sigma^2 t$, uniformly on compact time intervals. That convergence is derived from an equilibrium identity for the unbiased invariant law, $\mathbb{E}\left[\sum_k a(k) I_k(X^0_\infty)\right] = \sigma^2$, combined with renewal estimates that make the convergence uniform as the bias vanishes. Once the clock is deterministic, the weighted midpoint converges to a drifted Brownian motion, and comparisons with the unbiased voter model supply the tightness that upgrades midpoint convergence to convergence of the full measure-valued process.

What would settle it

For a fixed kernel satisfying (i)--(iii), such as $a(\pm1) = a(\pm2) = \tfrac14$, simulate the biased voter model for small $\varepsilon$ and estimate $\varepsilon M(X^\varepsilon_{\varepsilon^{-2}t})$ at several times $t$; if the empirical distribution does not approach a Brownian motion with variance $\sigma^2 t$ and drift $-\tfrac12\sigma^2 t$, Theorem 1.1 is false, and a systematic deviation in $\varepsilon^2 \int_0^{\varepsilon^{-2}t} \sum_k a(k) I_k(X^\varepsilon_s)\,ds$ from $\sigma^2 t$ would pinpoint the failure.

Watch

Extended reading notes

Core claim

Under assumptions (i)--(iii) on the kernel $a$---irreducibility, mean zero, and finite second moment $\sigma^2 := \sum_k a(k) k^2$---Theorem 1.1 asserts that the measure-valued process $\mu^\varepsilon_t := \sum_{i \in \mathbb{Z}} \varepsilon X^\varepsilon_{\varepsilon^{-2}t}(i)\, \delta_{\varepsilon i}$ converges weakly, as $\varepsilon \downarrow 0$, to $\mu_t(dx) = 1_{\{x \ge B_t\}} dx$, where $B_t = W_{\sigma^2 t} - \tfrac{1}{2}\sigma^2 t$ is a Brownian motion with drift $-\sigma^2/2$ and diffusion coefficient $\sigma^2$. In words, after diffusive rescaling the interface appears as a sharp Heaviside boundary whose location follows a drifted Brownian motion; at any fixed macroscopic time, the fraction of sites of the wrong type on either side of the boundary vanishes with $\varepsilon$. The same Brownian motion arises as the limit of the weighted midpoint, and the left and right boundaries of the interface converge to it in finite-dimensional distributions. The claim is new for biased voter models and reduces to the known unbiased result when $\varepsilon = 0$.

Load-bearing premise

The proof stands on the earlier interface-tightness results holding uniformly as the bias $\varepsilon$ tends to zero: specifically, that return times to Heaviside states and their boundary integrals converge in mean as $\varepsilon \downarrow 0$, which converts the random clock into deterministic time $\sigma^2 t$; if that uniformity failed, the derivation of the limiting drift and diffusion would collapse.

Editorial extensions

If this is right

The interface location, measured by the weighted midpoint, converges after diffusive rescaling to a Brownian motion with drift $-\sigma^2/2$ and diffusion coefficient $\sigma^2$, so the bias survives in the limit only as a deterministic drift equal to half the diffusivity.
At every fixed macroscopic time, the rescaled configuration is asymptotically a half-line of 1's on the right of the Brownian path and 0's on the left, with the fraction of misclassified sites going to zero as $\varepsilon \downarrow 0$.
The left and right boundaries of the interface have the same finite-dimensional limiting law as the midpoint, so the entire interface is concentrated around a single curve in the limit.
The limit does not depend on the initial interface state in $S^{01}_{\mathrm{int}}$, so memory of the initial configuration is lost at the diffusive scale.

Reading between the lines

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

Editorial inference: the mechanism suggests an Einstein-relation-like identity, $\sigma^2 = \mathbb{E}[\sum_k a(k) I_k(X^0_\infty)]$, which one could test directly by measuring the stationary average number of $k$-boundaries in the unbiased interface; the same quantity sets both the diffusivity and, through the weak-bias asymmetry, the drift.
Editorial inference: if the result extends to kernels whose range grows as $\varepsilon$ shrinks, the time-change argument indicates the limiting drift would remain $-\sigma^2/2$, but the proof of clock convergence would require a second-moment condition that is uniform in the growing range.
Editorial inference: the paper leaves open whether the left and right boundaries converge as processes, not just in finite-dimensional distributions; by analogy with the unbiased case, one expects path-level tightness to hold for kernels with a finite $(3+\delta)$-th moment, a conjecture the authors state explicitly.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Here's the short version: this is a solid, careful proof of the diffusive scaling limit for one-dimensional biased voter model interfaces, and the stress-test note's central objection is mistaken. The definition of S01_int in (1.2) uses ordinary limits, so every configuration is eventually 0 to the left and eventually 1 to the right. That makes the state space countable and makes the weighted midpoint in (1.6) well-defined. The 'infinite interface' states the note worries about are not in S01_int. So no domain correction is needed.

What's new: Theorem 1.1 extends the unbiased invariance principle of [AS11] to the weakly biased case, with the expected drift -σ^2/2 in the limiting Brownian motion. The proof is not a routine tweak. The weighted midpoint is identified as a time-changed random walk, and the time change is controlled via renewal arguments using the equilibrium equation from the authors' earlier paper [SSY19]. The tightness argument is clever: it compares the biased process to the unbiased one, and treats the two directions asymmetrically because upward (1→0) invasions are controlled by persistence. The appendix fills in the technical lemmas (weak law of large numbers, uniform ergodicity, convergence of inverses), so the paper is close to self-contained apart from the cited interface tightness results.

Soft spots, in proportion: the paper leans heavily on [SSY19] for uniform interface tightness, convergence of invariant laws, and the equilibrium equation. That's a heavy load, but the results are published and do not assume the current theorem, so it is not circular. Lemma 2.15 extends the unbiased invariance principle to arbitrary initial configurations with only a sentence of justification; I believe it's true, but a referee might want a citation or a short argument. Some parts of the renewal proof (e.g., the handling of the inverse interpolation around (2.36)-(2.38)) are sketched and left to the appendix; the appendix does cover them, though.

The citation pattern is fine: self-citations are to [SSY19], which is genuinely the source of the key input, and to [AS11] for the unbiased case. No post-hoc exclusions, no fitted parameters. I did not find a concrete error; my confidence in the main theorem is moderate, mostly because I did not verify every inequality in the tightness section line by line.

Who this is for: anyone working on voter model interfaces, interacting particle systems, or scaling limits of measure-valued processes. It deserves a serious referee. I would recommend accepting after a careful check of the tightness estimates and a request to expand Lemma 2.15.

Referee Report

0 major / 4 minor

Summary. The paper proves an invariance principle for the interface of a one-dimensional biased voter model in the weak-bias diffusive scaling. For a kernel satisfying irreducibility, mean zero, and finite second moment, and for any initial state in S01_int, the rescaled measure-valued process (1.3) is shown to converge weakly on D([0,∞), M(R)) to the measure-valued process (1.4), whose density is the indicator of a half-line with boundary B_t = W_{σ^2 t} - (1/2)σ^2 t. The proof proceeds by first proving convergence of the weighted midpoint via a time-changed random walk representation, then establishing convergence of the time change through renewal arguments based on the authors' earlier paper [SSY19], and finally deriving finite-dimensional convergence and tightness through comparisons with the unbiased voter model and Jakubowski's tightness criterion.

Significance. If correct, this is a substantial and natural extension of the unbiased voter model interface invariance principle of [AS11] to biased models, and it rigorously confirms the heuristic that under diffusive scaling the interface is a sharp drifted Brownian boundary. The limiting parameters are explicit and parameter-free: the diffusion coefficient is the second moment σ^2 and the drift is -σ^2/2. The paper is carefully structured, includes complete proofs of the auxiliary lemmas in an appendix, and builds on published results [SSY19] for interface tightness and the equilibrium equation; these results are used without assuming the target theorem, so I do not see a circularity concern. The proof is technically demanding, especially the tightness argument, and the paper represents a solid contribution to the interface scaling literature.

minor comments (4)

[2.5, Eq. (2.66)] In the estimate for σ^{n,-}_s, the event in the probability on the right-hand side should be {⟨ν^{εn}_{s+δ}, f⟩ − ⟨ν^{εn}_s, f⟩ ≤ −η/2} rather than {≥ η/2}. Applying the argument of (2.60) to the negative of the increment yields the lower-tail event; the displayed upper-tail event would only be obtained after an additional symmetry argument in the Brownian limit. Please correct the sign and spell out the symmetry step, since as written the verification of (2.59)(ii) is not correct.
[2.5, Eq. (2.66)] Lemma 2.15 is stated for deterministic initial states with ν^{εn}_0 converging vaguely to 1_{y≥0}dy. In (2.66) the unbiased process is started at time s from the random state X^{εn}_s, whose rescaled measure converges only in distribution to 1_{x≥B_s}dx. Please add a justification (for example by Skorohod representation and translation invariance, or by stating a conditional version of Lemma 2.15) for replacing the pre-limit probability by the Brownian probability with initial boundary B_s.
[1.1, Eq. (1.2)] The definition (1.2) in fact implies that the set of sites where x differs from a Heaviside configuration is finite; consequently S01_int and its quotient are countable and the weighted midpoint in (1.6) is well defined for every x in S01_int. A brief remark to this effect would prevent the possible misreading that configurations with infinitely many 1's on the left or infinitely many 0's on the right are included.
[2.3, proof of (2.41)] In the nearest-neighbor case, the multi-type argument would benefit from one additional sentence: after the finitely many initial types between L(x) and R(x) have died out, the monotonicity of the ancestor map in the one-dimensional voter model implies that the surviving left-tail and right-tail types occupy a left and a right half-line, respectively, so the configuration is Heaviside. As written, the conclusion that X_t is in a Heaviside state is somewhat abrupt.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the invariance principle is derived from independent prior interface-tightness results; the drift and diffusion constants are computed, not fitted.

full rationale

The central derivation is not circular. Theorem 1.2 is obtained by combining Lemma 2.1, a time-changed random walk representation proved in this paper, with Lemma 2.2, which proves that the time change converges to sigma^2 t. Lemma 2.2 is proved from the renewal identities in Section 2.3, whose key input is the equilibrium equation E[sum_k a(k) I_k(X^0_infinity)] = sigma^2 (Proposition 2.4). That equation is cited from [SSY19, Prop. 3.7], a separate published result proved under the same kernel assumptions (i)-(iii); it does not assume the invariance principle or the target Theorem 1.1. The paper's own equations (2.13)-(2.15) show that the time change is exactly sum_k a(k) I_k, while the constants sigma^2 and -sigma^2/2 are not fitted: they are computed from the kernel a(k) and from the equilibrium expectation of I_k. Similarly, the tightness argument uses [AS11]'s unbiased voter model invariance principle (Lemma 2.15), which is an independent earlier result; R. Sun is a co-author of [AS11], but the result is not the present biased statement and does not use it. Hence the dependence on [SSY19] and [AS11] is self-citation, but it is real evidence rather than circular reasoning. The only notable concern is mathematical, not circular: the definition of S01_int in (1.2) includes configurations for which the weighted midpoint M(x) in (1.6) need not be defined and for which S01_int would not be countable, contrary to Section 1.2. This is a domain/correctness issue in the theorem statement, not a reduction of the conclusion to the hypotheses.

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

The paper introduces no fitted parameters or invented entities. The only inputs are the kernel a and the bias epsilon, and the limit constant sigma^2 is defined directly from a. The cited results from [SSY19] are published theorems that the current proof relies on.

assumptions (3)

domain assumption The kernel a is a probability measure on Z with a(0)=0, irreducible, mean zero, and finite second moment (Assumptions (i)-(iii), Section 1.1).
This is the standing hypothesis of the paper; it guarantees interface tightness and the finiteness of sigma^2.
domain assumption The biased voter model modulo translations is positive recurrent with unique invariant law pi^epsilon, and pi^epsilon converges weakly to pi^0 as epsilon goes to zero ([SSY19, Thm 1.3]).
Invoked to control return times and the time-change limit in Lemma 2.2 and Theorem 1.2.
domain assumption The equilibrium equation E[sum_k a(k) I_k(X^0_infinity)] = sigma^2 holds for the unbiased invariant law ([SSY19, Prop 3.7], cited as Proposition 2.4 here).
This identity identifies the slope sigma^2 of the time-change limit and hence the drift and diffusion of the limiting Brownian motion.

how reviews work

0 comments

Cite this review

Pith. "Pith review of An invariance principle for biased voter model interfaces." pith.science (2026). https://pith.science/paper/U5ORP4X7

@misc{pith2026190802944,
  author       = {Pith},
  title        = {Pith review of: An invariance principle for biased voter model interfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U5ORP4X7}},
  note         = {Machine review of arXiv:1908.02944}
}

read the original abstract

We consider one-dimensional biased voter models, where 1's replace 0's at a faster rate than the other way round, started in a Heaviside initial state describing the interface between two infinite populations of 0's and 1's. In the limit of weak bias, for a diffusively rescaled process, we consider a measure-valued process describing the local fraction of type 1 sites as a function of time. Under a finite second moment condition on the rates, we show that in the diffusive scaling limit there is a drifted Brownian path with the property that all but a vanishingly small fraction of the sites on the left (resp. right) of this path are of type 0 (resp. 1). This extends known results for unbiased voter models. Our proofs depend crucially on recent results about interface tightness for biased voter models.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

D. Aldous. Stopping times and tightness. Ann.\ Probab. 6 (1978), 335--340

work page 1978
[2]

Athreya and R

S.R. Athreya and R. Sun. One-dimensional voter model interface revisited. Electron.\ Commun.\ Probab. 16 (2011), Article 70, 792--800

work page 2011
[3]

Asmussen

S. Asmussen. Applied Probability and Queues , 2nd ed. Springer-Verlag, New York, 2003

work page 2003
[4]

Billingsley

P. Billingsley. Convergence of Probability Measures , 2nd edition. John Wiley & Sons, 1999

work page 1999
[5]

Belhaouari, T

S. Belhaouari, T. Mountford, R. Sun and G. Valle. Convergence results and sharp estimates for the voter model interfaces. Electron.\ J.\ Prob. 11 (2006), Paper 30, 768--801

work page 2006
[6]

Belhaouari, T

S. Belhaouari, T. Mountford, and G. Valle. Tightness for the interfaces of one-dimensional voter models. Proc.\ London Math.\ Soc. 94(3) (2007), 421--442

work page 2007
[7]

Cox and R

J.T. Cox and R. Durrett. Hybrid zones and voter model interfaces. Bernoulli 1 (1995), 343--370

work page 1995
[8]

Dawson, Measure-valued Markov processes

D.A. Dawson, Measure-valued Markov processes. \'Ecole d'\'Et\'e de Probabilit\'es de Saint-Flour XXI--1991 , 1--260, Lecture Notes in Math., 1541, Springer, Berlin, 1993

work page 1991

Show all 14 references

[9]

Ethier and T.G

S.N. Ethier and T.G. Kurtz. Markov Processes; Characterization and Convergence. John Wiley & Sons, New York, 1986

1986
[10]

Fontes, M

L.R.G. Fontes, M. Isopi, C.M. Newman, and K. Ravishankar. The Brownian web: characterization and convergence. Ann.\ Probab. 32(4) (2004), 2857--2883

2004
[11]

Newman, K

C.M. Newman, K. Ravishankar and R. Sun. Convergence of coalescing nonsimple random walks to the Brownian web. Electron. J. Prob. 10 (2005), 21--60

2005
[12]

Sun and J.M

R. Sun and J.M. Swart. The Brownian net. Ann. Probab. 36(3) (2008), 1153--1208

2008
[13]

Sturm and J.M

A. Sturm and J.M. Swart. Tightness of voter model interfaces. Electron.\ Commun.\ Probab. 13, No. 16, 165--174, 2008

2008
[14]

Sun, J.M

R. Sun, J.M. Swart and J. Yu. Equilibrium interfaces of biased voter models. Ann.\ App.\ Probab. 29, No. 4, 2556--2593, 2019

2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

D. Aldous. Stopping times and tightness. Ann.\ Probab. 6 (1978), 335--340

work page 1978

[2] [2]

Athreya and R

S.R. Athreya and R. Sun. One-dimensional voter model interface revisited. Electron.\ Commun.\ Probab. 16 (2011), Article 70, 792--800

work page 2011

[3] [3]

Asmussen

S. Asmussen. Applied Probability and Queues , 2nd ed. Springer-Verlag, New York, 2003

work page 2003

[4] [4]

Billingsley

P. Billingsley. Convergence of Probability Measures , 2nd edition. John Wiley & Sons, 1999

work page 1999

[5] [5]

Belhaouari, T

S. Belhaouari, T. Mountford, R. Sun and G. Valle. Convergence results and sharp estimates for the voter model interfaces. Electron.\ J.\ Prob. 11 (2006), Paper 30, 768--801

work page 2006

[6] [6]

Belhaouari, T

S. Belhaouari, T. Mountford, and G. Valle. Tightness for the interfaces of one-dimensional voter models. Proc.\ London Math.\ Soc. 94(3) (2007), 421--442

work page 2007

[7] [7]

Cox and R

J.T. Cox and R. Durrett. Hybrid zones and voter model interfaces. Bernoulli 1 (1995), 343--370

work page 1995

[8] [8]

Dawson, Measure-valued Markov processes

D.A. Dawson, Measure-valued Markov processes. \'Ecole d'\'Et\'e de Probabilit\'es de Saint-Flour XXI--1991 , 1--260, Lecture Notes in Math., 1541, Springer, Berlin, 1993

work page 1991