The non-conserved assisted exclusion model (NAEM), exactly solved in two dimensions by a matrix-product ansatz, has a steady state that maps exactly onto the hard-square lattice gas with nearest-neighbor exclusion under particle-vacancy exchange (z=1/αz = 1/\alpha, ρ=1−η\rho = 1 - \eta). The exact density curves of strips of finite width MM appear to pass through a common crossing point that coincides with the hard-square critical point (zc≃3.7962z_c \simeq 3.7962, ηc≃0.3677\eta_c \simeq 0.3677), a coincidence the source authors state they neither understand nor know how to exploit. The task is to (i) derive quantitatively why the finite-width density corrections nearly vanish and change sign at zcz_c, and (ii) determine whether the crossing property yields a systematic, assumption-independent finite-size determination of the hard-square critical parameters.

Background

The hard-square lattice gas (HSLG), also called the nearest-neighbor-exclusion or 1-NN model, places particles on a square lattice so that no two adjacent sites are both occupied; every allowed configuration has equal energy, so the thermodynamics is controlled by a single parameter, the fugacity zz, which weights a configuration with NN particles by zNz^N (Baxter, Enting and Tsang 1980). The particle density η\eta rises from 00 at z=0z=0 to the close-packed value 1/21/2 at z→∞z\to\infty, where one of the two checkerboard sublattices is fully occupied. At an intermediate fugacity the model undergoes an entropy-driven order-disorder transition in which one sublattice becomes spontaneously preferred; it is continuous and lies in the two-dimensional Ising universality class, the universality class of the Ising model's thermal transition, whose critical exponents are exactly known (Runnels 1965; Fernandes, Arenzon and Levin 2007; Guo and Blöte 2002). Despite the model's simplicity, and in contrast to the exactly solved hard-hexagon model on the triangular lattice, the hard-square model is believed to be non-integrable (not exactly solvable by integrable-systems methods, which for hard hexagons yield the critical point in closed form) and its critical fugacity zcz_c and critical density ηc\eta_c are not known exactly (Assis et al. 2014). Numerically they have been pinned down with increasing precision: series analysis and the corner transfer matrix method (a technique that factorizes a lattice model's partition function into matrices associated with the quadrants of the lattice, yielding highly accurate series expansions) gave zc=3.7962z_c = 3.7962, ηc=0.3677\eta_c = 0.3677 (Baxter, Enting and Tsang 1980); Monte Carlo simulation confirmed a continuous Ising-class transition at chemical potential μc≃1.33\mu_c \simeq 1.33, i.e. zc=eμc≃3.8z_c = e^{\mu_c} \simeq 3.8, with ηc≃0.37\eta_c \simeq 0.37 (Fernandes, Arenzon and Levin 2007); a 92-term low-density series gave zc=3.79635(9)z_c = 3.79635(9) (Chan 2012); and transfer-matrix finite-size-scaling analyses (extracting bulk critical behavior from the system-size dependence of observables) on strips of widths up to 34 (Todo and Suzuki 1996; Guo and Blöte 2002), assuming the correction-to-scaling spectrum of the Ising model (the set of exponents governing the leading subleading corrections to finite-size scaling) and conformal invariance, determined the critical chemical potential mc=1.33401510027774(1)m_c = 1.33401510027774(1), i.e. zc=3.79625517391234(4)z_c = 3.79625517391234(4), together with the critical density ηc=0.3677429990410(3)\eta_c = 0.3677429990410(3). The off-critical value η(z=1)\eta(z=1) is known to much higher precision still, η(1)=0.2265708154…\eta(1) = 0.2265708154\ldots to 38 digits, from corner-transfer-matrix work (Baxter 1999). A traditional tool for this model is the strip transfer matrix: for a strip of width MM (periodic across the strip) and infinite length, the partition function is dominated by the largest eigenvalue of a finite matrix acting on the allowed column configurations, giving the exact strip density ηM(z)\eta_M(z); properties of the infinite lattice follow by extrapolating M→∞M \to \infty (Runnels 1965; Ree and Chesnut 1966; Bellemans and Nigam 1966).

The problem originates in a 2026 work that extends the matrix product ansatz (MPA), the representation of a stationary state as a trace of a product of non-commuting matrices associated with local occupation variables and a workhorse of exact solutions of one-dimensional stochastic models, to two dimensions (Ray, Banerjee and Mohanty 2026). As an application the authors introduce and exactly solve the non-conserved assisted exclusion model (NAEM), in which a particle hops to a neighboring site only when exactly one of its neighbors is vacant, and particles are created or annihilated only at sites whose neighbors are all occupied, with birth and death rates pp and qq. Its two-dimensional steady state is exactly the uniform measure over configurations containing no adjacent vacancies, each particle carrying a weight α=p/q\alpha = p/q; upon exchanging particles and vacancies this is precisely the hard-square lattice gas with fugacity z=1/αz = 1/\alpha and densities related by ρ=1−η\rho = 1 - \eta. The authors compute the exact density curves ρM(α)\rho_M(\alpha) of strips of width M=4,6,8M = 4, 6, 8 and larger (their transfer matrices coincide with the hard-square strip transfer matrices) and observe that these curves appear to pass through a common crossing point located at αc=1/zc\alpha_c = 1/z_c and ρc=1−ηc\rho_c = 1 - \eta_c, the critical point of the hard-square model. They state that they have no clear understanding of this behavior and do not know how to exploit it to determine ρc\rho_c more accurately.

Problem Statement

For the hard-square lattice gas with nearest-neighbor exclusion on the square lattice, equivalently via the exact steady-state mapping of the two-dimensional non-conserved assisted exclusion model (NAEM; particles exchanged with vacancies, z=1/αz = 1/\alpha, ρ=1−η\rho = 1 - \eta), the exact densities ηM(z)\eta_M(z) of infinite strips of finite width MM (equal to the NAEM curves ρM(α)\rho_M(\alpha)) pass through a near-common crossing point that coincides, up to their residual width dependence, with the critical point (zc,ηc)(z_c, \eta_c) of the order-disorder transition: the finite-width corrections to the density nearly vanish and change sign at zcz_c. Two linked questions jointly constitute the problem. (i) Explain this quantitatively: why the leading width-dependent corrections to the density change sign at zcz_c and are anomalously small there, and characterize how the pairwise crossing points of successive widths approach zcz_c. (ii) Determine whether this crossing property can be exploited as a systematic finite-size determination of (zc,ηc)(z_c, \eta_c) with controlled errors, in particular one that does not rest on the assumed Ising correction-to-scaling spectrum and conformal invariance underlying the most precise existing estimates, or establish that the coincidence carries no exact content beyond standard pseudocritical finite-size behavior.

References

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