Monomer-dimer Model Research Articles

Limit theorems in the imitative monomer-dimer mean-field model via Stein’s method

We consider the imitative monomer-dimer model on the complete graph introduced in the work of Alberici et al. [J. Math. Phys. 55, 063301-1–063301-27 (2014)]. It was shown that this model is described by the monomer density and has a phase transition along certain coexistence curve, where the monomer and dimer phases coexist. More recently, it was understood [D. Alberici et al., Commun. Math. Phys. (published online, 2016)] that the monomer density exhibits the central limit theorem away from the coexistence curve and enjoys a non-normal limit theorem at criticality with normalized exponent 3/4. By reverting the model to a weighted Curie-Weiss model with hard core interaction, we establish the complete description of the fluctuation properties of the monomer density on the full parameter space via Stein’s method of exchangeable pairs. Our approach recovers what were established in the work of Alberici et al. [Commun. Math. Phys. (published online, 2016)] and furthermore allows to obtain the conditional central limit theorems along the coexistence curve. In all these results, the Berry-Esseen inequalities for the Kolmogorov-Smirnov distance are given.

Spatial mixing and the connective constant: optimal bounds

We study the problem of deterministic approximate counting of matchings and independent sets in graphs of bounded connective constant. More generally, we consider the problem of evaluating the partition functions of the monomer-dimer model (which is defined as a weighted sum over all matchings where each matching is given a weight $$\gamma ^{|V| - 2 |M|}$$ in terms of a fixed parameter $$\gamma $$ called the monomer activity) and the hard core model (which is defined as a weighted sum over all independent sets where an independent set I is given a weight $$\lambda ^{|I|}$$ in terms of a fixed parameter $$\lambda $$ called the vertex activity). The connective constant is a natural measure of the average degree of a graph which has been studied extensively in combinatorics and mathematical physics, and can be bounded by a constant even for certain unbounded degree graphs such as those sampled from the sparse Erdős–Renyi model $$\mathcal {G}(n, d/n)$$ . Our main technical contribution is to prove the best possible rates of decay of correlations in the natural probability distributions induced by both the hard core model and the monomer-dimer model in graphs with a given bound on the connective constant. These results on decay of correlations are obtained using a new framework based on the so-called message approach that has been extensively used recently to prove such results for bounded degree graphs. We then use these optimal decay of correlations results to obtain fully polynomial time approximation schemes (FPTASs) for the two problems on graphs of bounded connective constant. In particular, for the monomer-dimer model, we give a deterministic FPTAS for the partition function on all graphs of bounded connective constant for any given value of the monomer activity. The best previously known deterministic algorithm was due to Bayati et al. (Proc. 39th ACM Symp. Theory Comput., pp. 122–127, 2007), and gave the same runtime guarantees as our results but only for the case of bounded degree graphs. For the hard core model, we give an FPTAS for graphs of connective constant $$\varDelta $$ whenever the vertex activity $$\lambda < \lambda _c(\varDelta )$$ , where $$\lambda _c(\varDelta ) :=\frac{\varDelta ^\varDelta }{(\varDelta - 1)^{\varDelta + 1}}$$ ; this result is optimal in the sense that an FPTAS for any $$\lambda > \lambda _c(\varDelta )$$ would imply that NP=RP (Sly and Sun, Ann. Probab. 42(6):2383–2416, 2014). The previous best known result in this direction was in a recent manuscript by a subset of the current authors (Proc. 54th IEEE Symp. Found. Comput. Sci., pp 300–309, 2013), where the result was established under the sub-optimal condition $$\lambda < \lambda _c(\varDelta + 1)$$ . Our techniques also allow us to improve upon known bounds for decay of correlations for the hard core model on various regular lattices, including those obtained by Restrepo et al. (Probab Theory Relat Fields 156(1–2):75–99, 2013) for the special case of $$\mathbb {Z}^2$$ using sophisticated numerically intensive methods tailored to that special case.

Monomer–dimer problem on some networks

Abstract Zhang et al. (2012) obtained the exact formula for the number of all possible monomer–dimer arrangements and the asymptotic growth constant on a scale-free small-world network. In this note, we generalize this result and obtain the exact solution on the monomer–dimer model on many networks. Particularly, we prove that these networks have the same asymptotic growth constant of the number of monomer–dimer arrangements.

Non-Gaussian fluctuations in monomer-dimer models

A hard-core monomer-dimer mean-field model is considered with the addition of an attraction potential between similar particles. We find that in the curve where the monomer and dimer phases coexist, the equilibrium state, due to the lack of gauge symmetry, turns out to be a superposition with unequal weights. We show, moreover, that at the critical point the number of monomers has non-Gaussian, quartic exponential, fluctuations.

Limit Theorems for Monomer–Dimer Mean-Field Models with Attractive Potential

The number of monomers, in a monomer-dimer mean-field model with an attractive potential, fluctuates according to the central limit theorem when the parameters are outside the critical curve. At the critical point the model belongs to the same universality class of the mean-field ferromagnet. Along the critical curve the monomer and dimer phases coexist.

A Cluster Expansion Approach to the Heilmann–Lieb Liquid Crystal Model

A monomer-dimer model with a short-range attractive interaction favoring colinear dimers is considered on the lattice $\mathbb{Z}^2$. Although our choice of the chemical potentials results in more horizontal than vertical dimers, the horizontal dimers have no long-range traslational order - in agreement with the Heilmann-Lieb conjecture.

Independence polynomial and matching polynomial of the Koch network

The lattice gas model and the monomer-dimer model are two classical models in statistical mechanics. It is well known that the partition functions of these two models are associated with the independence polynomial and the matching polynomial in graph theory, respectively. Both polynomials have been shown to belong to the “[Formula: see text]-complete” class, which indicate the problems are computationally “intractable”. We consider these two polynomials of the Koch networks which are scale-free with small-world effects. Explicit recurrences are derived, and explicit formulae are presented for the number of independent sets of a certain type.

Order–disorder transition in the two-dimensional interacting monomer-dimer model: Ising criticality

We study the order–disorder transition of the two-dimensional interacting monomer-dimer model (IMD) which has two symmetric absorbing states. To be self-contained, we first estimate numerically the dynamic exponent z of the two-dimensional Ising model. From the relaxation dynamics of the magnetization at the critical point, we obtain , or , where and are exactly known exponents. We then compare the critical relaxation of the order parameter at the transition point of the IMD with that of the Ising model. We find that the critical relaxation exponent is in good agreement with the Ising model, unlike the recent claim by Nam et al (2014 J. Stat. Mech. P08011). We also claim that the Binder cumulant is not an efficient quantity to locate the order–disorder transition point of the model with two symmetric absorbing states.

Dimer–monomer model on the Towers of Hanoi graphs

The number of dimer–monomers (matchings) of a graph [Formula: see text] is an important graph parameter in statistical physics. Following recent research, we study the asymptotic behavior of the number of dimer–monomers [Formula: see text] on the Towers of Hanoi graphs and another variation of the Sierpiński graphs which is similar to the Towers of Hanoi graphs, and derive the recursion relations for the numbers of dimer–monomers. Upper and lower bounds for the entropy per site, defined as [Formula: see text], where [Formula: see text] is the number of vertices in a graph [Formula: see text], on these Sierpiński graphs are derived in terms of the numbers at a certain stage. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of the entropy can be evaluated with more than a hundred significant figures accuracy.

Spin-Liquid Ground State in the Spin 1/2 Distorted Diamond Chain Compound K3Cu3AlO2(SO4)4

The new compound K3Cu3AlO2(SO4)4, representing a spin 1/2 distorted diamond chain system, was synthesized and its magnetic properties were studied by magnetic susceptibility, specific heat, and high-field magnetization measurements. Magnetic Cu2+ ions were found to form a highly one-dimensional distorted diamond chain with strong intrachain interactions. Short-range magnetic correlations were observed to develop in a two-stage process without magnetic long-range ordering by magnetic susceptibility and specific heat measurements. The exchange constant ratio J2/J1, where J2 is the magnetic interaction of the diamond backbone and J1 is the magnetic interaction of the monomers along the chain, was estimated to be 2.5, demonstrating that the magnetic ground state is composed of an alternating dimer–monomer spin-liquid phase. The absence of an M = 0 plateau in the high-field magnetization curve is also consistent with an alternating dimer–monomer model. These results strongly suggest that K3Cu3AlO2(SO4)4 is the first material exhibiting an alternating dimer–monomer spin-liquid ground state.

A Mean-Field Monomer–Dimer Model with Randomness: Exact Solution and Rigorous Results

Independent random monomer activities are considered on a mean-field monomer–dimer model. Under very general conditions on the randomness the model is shown to have a self-averaging pressure density that obeys an exactly solvable variational principle. The dimer density is exactly computed in the thermodynamic limit and shown to be a smooth function.

A Statistical Model of Current Loops and Magnetic Monopoles

We formulate a natural model of current loops and magnetic monopoles for arbitrary planar graphs, which we call the monopole-dimer model, and express the partition function of this model as a determinant. We then extend the method of Kasteleyn and Temperley-Fisher to calculate the partition function exactly in the case of rectangular grids. This partition function turns out to be a square of the partition function of an emergent monomer-dimer model when the grid sizes are even. We use this formula to calculate the local monopole density, free energy and entropy exactly. Our technique is a novel determinantal formula for the partition function of a model of vertices and loops for arbitrary graphs.

Positivity of the virial coefficients in lattice dimer models and upper bounds on the number of matchings on graphs

Using a simple relation between the virial expansion coefficients of the pressure and the entropy expansion coefficients in the case of the monomer–dimer model on infinite regular lattices, we have shown that, on hypercubic lattices of any dimension, the virial coefficients are positive through the 20th order. We have observed that all virial coefficients so far known for this system are positive also on infinite regular lattices with different structure. We are thus led to conjecture that the virial expansion coefficients mk are always positive.These considerations can be extended to the study of related bounds on finite graphs generalizing the infinite regular lattices, namely the finite grids and the regular biconnected graphs. The validity of the bounds Δkln(i!N(i))≤0 for k≥2, where N(i) is the number of configurations of i dimers on the graph and Δ is the forward difference operator, is shown to correspond to the positivity of the virial coefficients.Our tests on many finite lattice graphs indicate that on large lattices these bounds are satisfied, giving support to the conjecture on the positivity of the virial coefficients. Moreover, in an exhaustive survey of some classes of regular biconnected graphs with a not too large number v of vertices, we observe only few violations of these bounds. We conjecture that the frequency of the violations vanishes as v→∞.Using an inequality by Heilmann and Lieb, we find rigorous upper bounds on N(i) valid for arbitrary graphs and for regular graphs. The similarity between this inequality and the one conjectured above suggests that one study the stricter inequality mk≥12k for the virial coefficients, which is valid for all the known coefficients of the infinite regular lattice models.

Kosterlitz-Thouless transitions and phase diagrams of the interacting monomer-dimer model on a checkerboard lattice.

Using the tensor network approach, we investigate the monomer-dimer models on a checkerboard lattice, in which there are interactions (with strength v) between the parallel dimers on half of the plaquettes. For the fully packed interacting dimer model, we observe a Kosterlitz-Thouless (KT) transition between the low-temperature symmetry breaking and the high-temperature critical phases; for the doped monomer-dimer case with finite chemical potential μ, we also find an order-disorder phase transition which is of second order instead. We use the boundary matrix product state approach to detect the KT and second-order phase transitions and obtain the phase diagrams v-T and μ-T. Moreover, for the noninteracting monomer-dimer model (setting μ=ν=0), we get an extraordinarily accurate determination of the free energy per site (negative of the monomer-dimer constant h_{2}) as f=-0.662798972833746 with the dimer density n=0.638123109228547, both of 15 correct digits.

Nonequilibrium critical dynamics of two dimensional interacting monomer-dimer model: non-Ising criticality

We investigate the nonequilibrium relaxation dynamics of an interacting monomer-dimer model with nearest neighbor repulsion on a square lattice, which possesses two symmetric absorbing states. The model is known to exhibit two nearby continuous transitions: the Z2 symmetry-breaking order-disorder transition and the absorbing transition with directed percolation criticality. We performed a more detailed analysis of our extensive simulations on bigger lattice systems which reaffirms that the symmetry-breaking transition exhibits a non-Ising critical behavior with β ≃ 0.149(2) and η ≃ 0.30(1) that are distinct from those values of a pure two dimensional Ising model. Finite size scaling of dimer density near the symmetry breaking transition gives logarithmic scaling (α = 0.0) which is consistent with the hyperscaling relation but the corresponding exponent of νB ≃ 1.37(2) exhibits a conspicuous deviation from the pure Ising value of 1. The value of dynamic critical exponent z, however, is found to be close to that of the kinetic Ising model as 1/z ≃ 0.466(5) from the relaxation of staggered magnetization (and also similar but slightly smaller values from coarsening).

Solution of the Monomer–Dimer Model on Locally Tree-Like Graphs. Rigorous Results

We consider the monomer–dimer model on sequences of random graphs locally convergent to trees. We prove that the monomer density converges almost surely, in the thermodynamic limit, to an analytic function of the monomer activity. We characterise this limit as the expectation of the solution of a fixed point distributional equation and we give an explicit expression of the limiting pressure per particle.

Grassmannian representation of the two-dimensional monomer-dimer model.

We present an application of the Grassmann algebra to the problem of the monomer-dimer statistics on a two-dimensional square lattice. The exact partition function, or total number of possible configurations, of a system of dimers with a finite set of n monomers at fixed positions can be expressed via a quadratic fermionic theory. We give an answer in terms of a product of two pfaffians and the solution is closely related to the Kasteleyn result of the pure dimer problem. Correlation functions are in agreement with previous results, both for monomers on the boundary, where a simple exact expression is available in the discrete and continuous case, and in the bulk where the expression is evaluated numerically.

A mean-field monomer-dimer model with attractive interaction: Exact solution and rigorous results

A mean-field monomer-dimer model which includes an attractive interaction among both monomers and dimers is introduced and its exact solution rigorously derived. The Heilmann-Lieb method for the pure hard-core interacting case is used to compute upper and lower bounds for the pressure. The bounds are shown to coincide in the thermodynamic limit for a suitable choice of the monomer density m. The computation of the monomer density is achieved by solving a consistency equation in the phase space (h, J), where h tunes the monomer potential and J the attractive potential. The critical point and exponents are computed and show that the model is in the mean-field ferromagnetic universality class.

Stochastic delayed monomer-dimer surface reaction model with various dimer adsorption

In this paper, we study the impacts of noise and time delay in the monomer-dimer (MD) surface reaction model by using theoretical analysis. The MD models for various dimer adsorption mechanisms (namely, local and random adsorption models), which exhibit the first- and second-order phase transitions (PTs), are analyzed. By comparison with two various dimer adsorption, it is found that the qualitative predictions of the well-known first- and second-order PTs are better for the random adsorption model than that for the local adsorption model. The validity of approximate analytic results is checked by numerical simulations.

Lee–Yang Theorems and the Complexity of Computing Averages

We study the complexity of computing average quantities related to spin systems, such as the mean magnetization and susceptibility in the ferromagnetic Ising model, and the average dimer count (or average size of a matching) in the monomer-dimer model. By establishing connections between the complexity of computing these averages and the location of the complex zeros of the partition function, we show that these averages are #P-hard to compute, and hence, under standard assumptions, computationally intractable. In the case of the Ising model, our approach requires us to prove an extension of the famous Lee–Yang Theorem from the 1950s.