Completing the Ermakov–Tóth–Werner scaling-limit program for the one-dimensional coalescing ideal gas: particle paths for general profile functions and the proof of the coalescence-time invariance principle

The 1998 paper of Ermakov, Tóth and Werner derives the scaling limits of the one-dimensional coalescing ideal gas — point particles with velocities $\pm 1$ that merge and pick a fresh velocity at each collision — through a deterministic particle-path construction $\xi(\Phi,x,t)$ read off from a càdlàg profile function $\Phi$, developed under the standing condition $\liminf_{x\to\pm\infty}\Phi(x)=-\infty$. Two steps of the program were explicitly left unwritten: the adaptation of the path construction and of the subsequent convergence proofs to arbitrary profile functions $\Phi\in D(\mathbb{R})$ not satisfying the liminf condition, which the authors said requires only 'very slight modifications', and the complete proof of Theorem 15, the invariance principle stating that rescaled discrete coalescence times converge in distribution to the coalescence time $T_{x,y}$ of the coalescing flight process, which the paper says can be proven 'along similar lines as Theorem 13' and leaves to the reader. The problem — to write out both missing pieces of the proof route in the paper's intended generality, or to demonstrate precisely where the announced adaptations fail — remains open.

Background

In the one-dimensional coalescing ideal gas, point-like particles sit on the lattice $\mathbb{Z}$ (or on the line), move ballistically with velocities $\pm1$, and when two particles collide they merge into one particle which chooses a fresh velocity $\pm1$, each with probability $1/2$; a standard initial condition assigns to each site an independent exponential mass with a random sign, the sign fixing the initial velocity. The model was introduced by Ermakov, who derived exact identities in law for the infinitely extended system. Its macroscopic theory was derived by Ermakov, Tóth and Werner in the paper from which this problem is taken; the annihilating counterpart of the model, and the scaling limit of its height function, had been treated earlier by Belitsky and Ferrari. Coalescing systems of this kind go back to Arratia's coalescing random walks — random walks on the lattice that merge into a single walk when they meet — and their diffusion-scale limiting point processes.

The paper's proof route is deterministic at its core. A configuration of the system is encoded by a càdlàg profile function $\Phi\in D(\mathbb{R})$ ($D(\mathbb{R})$ denotes the space of right-continuous real functions with left limits): $\Phi$ records the signed mass accumulated between points, the initial profile being $\hat\Phi_0(x)=\sum_{i\in\mathbb{Z}\cap(0,x]}m_i$ for $x>0$ (and analogously for $x<0$, with $\hat\Phi_0(0)=0$), where $m_i$ is the signed charge of the particle at site $i$; hats distinguish discrete, lattice quantities from their continuum counterparts. Under this encoding the time evolution is the semigroup of operators $S_t(\Phi)(x)=\inf\{\Phi(x+y):y\in[-t,t)\},\qquad x\in\mathbb{R},\ t\ge0,$ an 'erosion' of the profile at unit speed: the profile of the gas at time $t$ is $S_t(\Phi)$, which for $t>0$ has locally bounded variation and decomposes into alternating monotone pieces, the particles of the deterministic picture.

Individual particle trajectories are read off from the profile as follows. For a level $h$ let $l_x(h)=\sup\{y<x:\Phi(y)<h\}$ and $r_x(h)=\inf\{y>x:\Phi(y)<h\}$, and write $\theta_x(h)=\tfrac12(r_x(h)-l_x(h))$ and $\chi_x(h)=\tfrac12(r_x(h)+l_x(h))$; for a target time $t$ let $h^*=\sup\{h:\theta_x(h)>t\}$. For $x$ at which $\Phi$ is not constant in any neighbourhood, the particle started at $x$ is at time $t$ located at $\xi(\Phi,x,t)$, defined by interpolating linearly between the points $(\theta_x(h),\chi_x(h))$ as the level $h$ decreases past $h^*$. The construction yields 1-Lipschitz paths which coalesce upon meeting, with coalescence time $T_{x,y}(\Phi)=\theta_x(\inf_{z\in(x,y)}\Phi(z))$ for $x<y$. This construction, and the proofs built on it, are made under the standing condition $\liminf_{x\to\pm\infty}\Phi(x)=-\infty,$ which guarantees that $l_x(h)$ and $r_x(h)$ are finite.

The main results are invariance principles (functional central limit theorems in the spirit of Donsker's theorem for partial sums). Under hyperbolic rescaling of space and time ($x\mapsto Nx$, $t\mapsto Nt$) with diffusive normalization of the amplitude, the random initial profile of the gas converges to a two-sided Brownian motion $B=(B_x)_{x\in\mathbb{R}}$ (Brownian motion indexed by the whole line with $B_0=0$), and, by continuity of the map $\Phi\mapsto S_t(\Phi)$, the evolved profile converges to $S_t(B)$. The tagged-particle trajectories (a tagged particle is one whose individual trajectory is followed through the evolution) converge jointly — for any finite set of tagged particles — to the coalescing flight process: a collection of unit-speed, piecewise-linear paths which coalesce upon meeting, a single one being distributed as the 'Brownian flight' $\eta(t)=\int_0^t\operatorname{sgn}(W_s)\,ds$ built from a Brownian motion $W$, so that its turning instants are the zeros of $W$. The paper also identifies the law of the limiting coalescence time between two flights: $T_{x,y}/(y-x)\stackrel{d}{=}T_{0,1}\stackrel{d}{=}\tfrac12\bigl(1+(W_1-2I_1)^2Z\bigr)$, where $I_1=\inf_{s\in[0,1]}W_s$ and $Z$ is an independent positive stable(1/2) random variable, i.e. a variable with density $(2\pi t^3)^{-1/2}e^{-1/(2t)}$ on $(0,\infty)$.

Two parts of this program were not written out. First, the whole path construction is developed only under the liminf condition above; the authors remark that it could be adapted to a general $\Phi$ 'with very slight modifications', but no such adaptation is given. Second, Theorem 13 (the joint invariance principle for finitely many tagged trajectories) is proved via an almost-sure continuity lemma for $\Phi\mapsto\xi(\Phi,x,\cdot)$ and Donsker convergence of the rescaled profiles, while the analogous invariance principle for coalescence times, Theorem 15 — asserting that the rescaled discrete coalescence time of the two particles started near $nx$ and $ny$ converges in distribution to $T_{x,y}$ — is stated with the remark that it can be proven along similar lines as Theorem 13 and is left to the reader.

Problem Statement

Complete the rigorous exposition of the Ermakov–Tóth–Werner scaling-limit program for the one-dimensional coalescing ideal gas by carrying out the two pieces of the particle-path proof route that the source paper explicitly leaves unwritten. The two pieces serve one objective — that the paper's announced invariance principles hold with full written proofs in the generality the paper intends — and they are linked: the first widens the class of initial data for which the route is valid, while the second closes the one stated theorem whose proof the route was announced to yield but was never written.

(1) General profile functions. Write out, with full proofs, the adaptation of the particle-path construction $\xi(\Phi,x,t)$ (defined through the level-set quantities $l_x(h)$, $r_x(h)$, $\theta_x(h)$, $\chi_x(h)$ and the level $h^*$) to arbitrary càdlàg profile functions $\Phi\in D(\mathbb{R})$ that do not satisfy the standing condition $\liminf_{x\to\pm\infty}\Phi(x)=-\infty$ — for example bounded, eventually increasing, or compactly supported profiles. This deterministic adaptation must establish, for such profiles and with full proofs, each of the following three components, all of which are required: the path properties (the 1-Lipschitz property, the positive homogeneity, i.e. the scale-invariance $\xi(\lambda\Phi,x,t)=\xi(\Phi,x,t)$ for $\lambda>0$, and the coalescence-time formula $T_{x,y}(\Phi)=\theta_x(\inf_{z\in(x,y)}\Phi(z))$); the identification of the discrete gas trajectories with $\xi(\hat\Phi_0,x,t)$; and the continuity property of $\Phi\mapsto\xi(\Phi,x,\cdot)$ that replaces the paper's Lemma 14 and is the general-profile ingredient of the convergence arguments for the invariance principles.

(2) Proof of Theorem 15. Provide a complete written proof of the invariance principle for coalescence times: for the coalescing ideal gas on $\mathbb{Z}$ with i.i.d. exponential masses and independent signs, and fixed $x<y$, the rescaled coalescence time $\hat T_{\lfloor nx\rfloor,\lfloor ny\rfloor}/n$ of the two particles started at $\lfloor nx\rfloor$ and $\lfloor ny\rfloor$ converges in distribution as $n\to\infty$ to $T_{x,y}$, the coalescence time of the two coalescing flights started at $x$ and $y$ in the Brownian profile. The paper states that this can be proven along similar lines as Theorem 13; no proof or sketch is given.

Either of the two deliverables may in fact be impossible as claimed. A refutation of (1) must show that, for some càdlàg profile without the liminf condition, the construction or one of its required properties — the path properties, the identification of the discrete gas trajectories, or the continuity property — fails, and must delimit the maximal class of càdlàg profiles for which the construction and those properties do hold. A refutation of (2) must show that the stated convergence of the rescaled lattice coalescence times fails, and must identify the class of initial conditions, or the corrected limiting statement, that replaces it.

References

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