Infinite-volume mean-field and large-mass limits of interacting Bose gases beyond the cluster-expansion regime

Fröhlich–Knowles–Schlein–Sohinger (Ann. Henri Poincaré 24 (2023), 1439–1503) represented the grand-canonical Gibbs states of interacting lattice Bose gases as ensembles of interacting random loops (the Ginibre ensemble) and proved convergence of these states, in the mean-field regime and in the large-mass regime, to classical limits: a quartic scalar field theory and a classical interacting particle gas, respectively. On the infinite lattice their theorems hold only in the cluster-expansion regime: the interaction must be small in $\ell^1(\mathbb{Z}^d)$ relative to the chemical-potential parameter $\kappa$ (mean-field case) or the activity small (large-mass case), and it must be nonnegative, of positive type, and radially decreasing. Later work has settled the continuum analogues in finite volume — including explicit convergence rates and sign-changing stable potentials — but the most recent treatment (2026) re-imposes the same smallness and nonnegativity hypotheses for every infinite-volume statement. The open problem is to determine the true domain of validity of the infinite-volume convergence and of the interchange of the time-step and thermodynamic limits: extend these theorems beyond the small-interaction regime, weaken the positivity hypotheses, quantify convergence rates, and delimit any obstruction from phase transitions or Bose–Einstein condensation.

Background

An interacting Bose gas on a finite box $\Lambda\subset\mathbb{Z}^d$ with periodic boundary conditions is described, in the grand-canonical ensemble at unit inverse temperature, by the family of $n$-particle Hamiltonians $H_n^{\nu,\lambda}=-\frac{\nu}{2}\sum_{i=1}^n\Delta_i+\frac{\lambda}{2}\sum_{i,j=1}^n v(x_i-x_j)$ acting on symmetric (bosonic) wave functions, where $\nu>0$ plays the role of Planck's constant and of the time step of the associated random walk, $v$ is the two-body interaction potential, $\lambda$ the coupling, and $\kappa$ (with activity $\mathrm{e}^{-\kappa}$) the chemical-potential parameter. The basic observables are the relative partition function $\mathcal{Z}$ (the interacting one normalized by the free one), the reduced $p$-particle density matrices $\Gamma_p$ (partial traces of the Gibbs state over all but $p$ particles), and the specific Gibbs potential $g=|\Lambda|^{-1}\log\mathcal{Z}$, the free-energy density. A classical result of Ginibre (J. Math. Phys. 6 (1965), 238–251) represents such Gibbs states exactly as statistical ensembles of interacting random loops: closed space-time random walks on $\Lambda\times[0,\infty)$ with durations in $\nu\mathbb{N}$, whose pairwise interaction is the potential $v$ integrated over pairs of loop times. The mirror image on the classical side is the Symanzik loop representation of scalar Euclidean field theories, reached through the Hubbard–Stratonovich transform — a Gaussian linearization of a quartic self-interaction that requires the interaction to be of positive type, meaning that its Fourier transform is nonnegative.

Fröhlich, Knowles, Schlein, and Sohinger used these loop representations to prove two quantum-to-classical limit theorems for lattice Bose gases. In the mean-field regime one sets $\lambda=\nu^2$ with $\kappa>1$ fixed: as $\nu\to0$ the relative partition function and the rescaled reduced density matrices converge to those of a classical complex scalar field theory with quartic self-interaction. In the large-mass regime one sets $\lambda=1$ and $\kappa=\kappa_0/\nu$: all loops collapse to points weighted by occupation numbers, and the limit is a classical gas of interacting weighted particles (with hard-core exclusions when $v(0)=\infty$, since multiple occupancy of a site then costs infinite energy). Both statements are proved first on a finite torus (Theorems 1.2 and 1.4 of the source paper) and then extended to the infinite lattice $\mathbb{Z}^d$ (Theorems 1.6 and 1.8). The infinite-volume mean-field theorem assumes that $v$ is pointwise nonnegative, of positive type, in $\ell^1(\mathbb{Z}^d)$, and radially decreasing, and requires $\|v\|_{\ell^1(\mathbb{Z}^d)}$ to be sufficiently small depending on $\kappa$ (or $\kappa$ large depending on $\|v\|_{\ell^1}$); the infinite-volume large-mass theorem assumes a nonnegative interaction with hard core of radius $R\in\{0,1\}$ and $\ell^1$-decay off the core, and requires the activity parameter $\kappa_0$ to be sufficiently large. In both cases the conclusion is that the specific Gibbs potentials and the locally projected reduced density matrices have infinite-volume limits and that the double limits $\lim_{\nu\to0}\lim_{L\to\infty}$ coincide with the infinite-volume limits of the corresponding classical theories. The smallness conditions originate in the method: the proofs pass through cluster expansions — convergent reorganizations of the logarithm of the partition function in powers of the interaction around the free gas — which converge only in a small-interaction or dilute regime. By contrast, the convergence of the thermodynamic pressure as $L\to\infty$ alone is known for all temperatures by subadditivity arguments, so the open difficulty concerns the correlation functions and the interchange of the two limits. The positivity hypotheses also have a structural role, identified in Remark 1.11 of the paper: positive type is needed to apply the Hubbard–Stratonovich formula at the start of the Symanzik representation, and pointwise nonnegativity enters the convergence proof of the Ginibre representation; both are described there as needed purely for mathematical reasons. The restriction of the hard-core radius to $R\in\{0,1\}$ is explicitly declared by the authors to be a simplification, its extension to arbitrary $R\ge0$ being routine.

A large parallel literature treats the continuum (Brownian-loop) versions of these limits in finite volume, that is, on a torus: the mean-field convergence of the interacting Bose gas to renormalized nonlinear Schrödinger Gibbs measures in dimensions $d\le3$ was proved by Lewin, Nam, and Rougerie via variational methods and quantum de Finetti arguments (the quantum de Finetti theorem is an exchangeability statement identifying large-system reduced density matrices as mixtures of product states), and by Fröhlich, Knowles, Schlein, and Sohinger via functional-integral methods, first with a modified Gibbs state in $d=2,3$ and later without it; subsequent work reached the local $\Phi^4_2$ and $\Phi^4_3$ theories, unbounded and one-dimensional focusing interactions, and renormalized three-body interactions. In $d=2,3$ these continuum derivations require Wick renormalization — subtracting the divergent diagonal part of the interaction, since the limiting field lives on distributions — to control ultraviolet (short-distance) singularities that are absent on the lattice. Most recently, Garouniatis, Saksida, and Sohinger (2026) proved the continuum large-mass limit: in finite volume for stable (possibly sign-changing) Hölder-continuous potentials with explicit convergence rates, and in infinite volume — but only for nonnegative, differentiable interactions decaying at infinity as $|v|+|\nabla v|\le C\langle x\rangle^{-(d+\varepsilon_{0})}$ (polynomial decay faster than $|x|^{-d}$, a class containing all Schwartz potentials) and satisfying the smallness condition $\frac{1}{\kappa-1}(\|v\|_{L^1(\mathbb{R}^d)}+1)\le c_d$, again via cluster expansion. No work in this program treats the interacting infinite-volume limit outside the cluster-expansion regime.

Problem Statement

For the interacting lattice Bose gas studied by Fröhlich–Knowles–Schlein–Sohinger (Ann. Henri Poincaré 24 (2023), 1439–1503), the infinite-volume convergence theorems are confined to the cluster-expansion regime. The problem — the common objective underlying the items below — is to determine the true domain of validity of these theorems, on the lattice and in the continuum, and in particular:

(1) Mean-field regime. Prove or disprove that, for fixed $\kappa>1$ and every nonnegative interaction $v\in\ell^1(\mathbb{Z}^d)$ of positive type with no smallness assumption on $\|v\|_{\ell^1(\mathbb{Z}^d)}$, the infinite-volume limits $\lim_{L\to\infty}$ of the specific Gibbs potential and of the locally projected reduced density matrices exist and satisfy the double-limit identities of Theorem 1.6 of the source paper; identify which auxiliary hypotheses (radial decrease, positive type) are genuinely necessary.

(2) Large-mass regime. Prove or disprove the analogous extension of Theorem 1.8 to fixed activity parameter $\kappa_0>0$ and general nonnegative $\ell^1$ interactions with hard core; the extension of the hard-core radius from $R\in\{0,1\}$ to arbitrary $R\ge0$ is declared routine by the authors of the source paper and is not itself part of the open problem.

(3) Positivity. Determine how far the positivity hypotheses can be weakened — in particular whether stable interactions (not pointwise nonnegative, not of positive type) can be admitted.

(4) Rates. Obtain quantitative convergence rates in $\nu$ (and, for the interchange of limits, in the side length $L$) for the lattice theorems, where presently only convergence without rates is known.

These four items are facets of the single objective above rather than a checklist that must be settled jointly: items (1) and (2) are independently refutable claims, while (3) and (4) are partial objectives, and any one of them — resolved for the lattice theorems or for their continuum analogues below — is a valid answer to the problem. Item (3) also absorbs the clause of (1) asking whether positive type is genuinely necessary, and the continuum analogues are judged by the same criteria, forming part of the same objective rather than separate deliverables.

The same questions apply to the continuum analogues of Garouniatis–Saksida–Sohinger, whose infinite-volume theorems currently require $v\ge0$, decay at infinity of order $|v|+|\nabla v|\le C\langle x\rangle^{-(d+\varepsilon_{0})}$, and the smallness condition $\frac{1}{\kappa-1}(\|v\|_{L^1(\mathbb{R}^d)}+1)\le c_d$. Since beyond the smallness regime the infinite-volume Gibbs state need not be unique — Bose–Einstein condensation (macroscopic occupation of the zero-momentum state) at high activity and ordering phase transitions for repulsive or hard-core interactions at low temperature are expected — a complete answer may legitimately take the form of a sharp delimitation: proofs of the identities in the largest regime where they hold, together with a rigorous characterization of the parameter region where they fail or become boundary-condition dependent. An answer restricted to re-deriving the infinite-volume convergence already established under the existing cluster-expansion hypotheses does not solve the problem, whereas new consequences drawn under those same hypotheses — such as convergence rates or the necessity or removal of an auxiliary hypothesis — do.

References

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