The author has recently presented two different expressions for lambda_d(p) of the monomer-dimer problem involving a power series in p, the first jointly with Shmuel Friedland. These two expressions are certainly equal, but this has not yet been proven rigorously. The first is naturally developed from quantities J_i, cluster expansion kernels. The second from the Mayer (or Virial) series of a dimer gas, in particular from the b_i coefficients in the Mayer series. The sets {b_i} and {J_i} can be derived from each other. Given an arbitrary set of values for either the b_i or the J_i, both expressions may be given in terms of a formal sum. The master algebraic conjecture is that these two expressions are equivalent. This is detailed in the special case all J_i are zero.
A two populations mean-field monomer-dimer model including both hard-core and attractive interactions between dimers is considered. The pressure density in the thermodynamic limit is proved to satisfy a three-dimensional variational principle. A detailed analysis is made in the limit in which one population is much smaller than the other and a ferromagnetic mean-field phase transition is found.
We consider the monomer-dimer model on weighted graphs embedded in surfaces with boundary, with the restriction that only monomers located on the boundary are allowed. We give a Pfaffian formula for the corresponding partition function, which generalises the one obtained by Giuliani, Jauslin and Lieb for graphs embedded in the disk. Our proof is based on an extension of a bijective method mentioned in their paper, together with the Pfaffian formula for the dimer partition function of Cimasoni-Reshetikhin.
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 consistency equation characterising m is studied 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 th…
In the past few years we have derived asymptotic expansions for lambda_d of the dimer problem and lambda_d(p) of the monomer-dimer problem. The many expansions so far computed are collected herein. We shine a light on results in two dimensions inspired by the work of M. E. Fisher. Much of the work reported here was joint with Shmuel Friedland.
The author and Shmuel Friedland recently presented an expression for lambda_d(p) of the monomer-dimer problem involving a power series in p . Herein I present a simple way to derive this expression for lambda_d(p) from the Mayer (or Virial) series of a dimer gas. The derivation is an exercise in basic statistical mechanics. We expect the relationship to be very useful.
Independent random monomer activities are considered on a mean-feld monomer-dimer model. Under very general conditions on the randomness the model is shown to have a self-averaging pressure density that obeys a solvable variational principle. The dimer density is exactly computed in the thermodynamic limit and shown to be a smooth function.
We define the matching measure of a lattice L as the spectral measure of the tree of self-avoiding walks in L. We connect this invariant to the monomer-dimer partition function of a sequence of finite graphs converging to L. This allows us to express the monomer-dimer free energy of L in terms of the measure. Exploiting an analytic advantage of the matching measure over the Mayer series then leads to new, rigorous bounds on the monomer-dimer free energies of various Euclidean lattices. While our estimates use only the computational data given in previous papers, they improve the known bounds s…
Shmuel Friedland and the author recently presented a formal expansion for lambda_d(p) of the monomer-dimer problem. Herein we prove that if the terms in the expansion are rearranged as a power series in p, then for sufficiently small p this series converges.
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.
In a recent paper S. Friedland and the author presented a formal expression for lambda_d(p) of the monomer-dimer problem on a d-dimensional rectangular lattice, which involved a power series in p. Herein, we find simlar expressions for lambda_tri(p) and lambda_hex(p), the entropy per site for triangular and hexagonal lattices.
We study the number of dimer-monomers $M_d(n)$ on the Tower of Hanoi graphs $TH_d(n)$ at stage $n$ with dimension $d$ equal to 3 and 4. The entropy per site is defined as $z_{TH_d}=\lim_{v \to \infty} \ln M_d(n)/v$, where $v$ is the number of vertices on $TH_d(n)$. We obtain the lower and upper bounds of the entropy per site, and the convergence of these bounds approaches to zero rapidly when the calculated stage increases. The numerical value of $z_{TH_d}$ is evaluated to more than a hundred digits correct. Using the results with $d$ less than or equal to 4, we predict the general form of the…
The finite volume correction for a mean-field monomer-dimer system with an attractive interaction are computed for the pressure density, the monomer density and the susceptibility. The results are obtained by introducing a two-dimensional integral representation for the partition function decoupling both the hard-core interaction and the attractive one. The next-to-leading terms for each of the mentioned quantities is explicitly derived as well as the value of their sign that is related to their monotonic convergence in the thermodynamic limit.
The correlation functions of an arbitrary number of boundary monomers in the system of close-packed dimers on the square lattice are computed exactly in the scaling limit. The equivalence of the 2n-point correlation functions with those of a complex free fermion is proved, thereby reinforcing the description of the monomer-dimer model by a conformal free field theory with central charge c=1.
We obtain an asymptotic formula, as $n\to\infty$, for the monomer-monomer correlation function $K_2(x,y)$ in the classical dimer model on a triangular lattice, with the horizontal and vertical weights $w_h=w_v=1$ and the diagonal weight $w_d=t>0$, where $x$ and $y$ are sites $n$ spaces apart in adjacent rows. We find that $t_c=\frac{1}{2}$ is a critical value of $t$. We prove that in the subcritical case, $0<t<\frac{1}{2}$, as $n\to\infty$, $K_2(x,y)=K_2(\infty)\left[1-\frac{e^{-n/ξ}}{n}\,\Big(C_1+C_2(-1)^n+\mathcal O(n^{-1})\Big)\right]$, with explicit formulae for $K_2(\infty)$, $ξ$, $C_1$, …
We consider a monomer-dimer system with a strong attractive dimer-dimer interaction that favors alignment. In 1979, Heilmann and Lieb conjectured that this model should exhibit a nematic liquid crystal phase, in which the dimers are mostly aligned, but do not manifest any translational order. We prove this conjecture for large dimer activity and strong interactions. The proof follows a Pirogov-Sinai scheme, in which we map the dimer model to a system of hard-core polymers whose partition function is computed using a convergent cluster expansion.
We consider the monomer-dimer partition function on arbitrary finite planar graphs and arbitrary monomer and dimer weights, with the restriction that the only non-zero monomer weights are those on the boundary. We prove a Pfaffian formula for the corresponding partition function. As a consequence of this result, multipoint boundary monomer correlation functions at close packing are shown to satisfy fermionic statistics. Our proof is based on the celebrated Kasteleyn theorem, combined with a theorem on Pfaffians proved by one of the authors, and a careful labeling and directing procedure of the…
We compute the free energy of the planar monomer-dimer model. Unlike the classical planar dimer model, an exact solution is not known in this case. Even the computation of the low-density power series expansion requires heavy and nontrivial computations. Despite of the exponential computational complexity, we compute almost three times more terms than were previously known. Such an expansion provides both lower and upper bound for the free energy, and allows to obtain more accurate numerical values than previously possible. We expect that our methods can be applied to other similar problems.
The number of dimer-monomers (matchings) of a graph $G$ is an important graph parameter in statistical physics. Following recent research, we study the asymptotic behavior of the number of dimer-monomers $m(G)$ 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 $μ_{G}=\lim_{v(G)\rightarrow\infty}\frac{\ln m(G)}{v(G)}$, where $v(G)$ is the number of vertices in a graph $G$, on these Sierpiński …
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 for the limiting pressure per particle.
We present some promising ideas to treat the problem of making completely rigorous the development of our expression for $λ_d(p)$ of the monomer-dimer problem on a $d$-dimensional hypercubic lattice \begin{equation}\label{abstract1} λ_d(p)=\frac{1}{2}\Big(p\ln(2d)-p\ln(p)-2(1-p)\ln(1-p)-p\Big) +\sum_{k=2}a_k(d)p^k \end{equation} where $a_k(d)$ is a sum of powers $(1/d)^r$ for \begin{equation}\label{abstract2} k-1\leq r\leq k/2 \end{equation} In fact as we will point out one has allready rigorously established the convergence of the sum in expression for $λ$ for small $p$. It is the $d$ depende…
In a recent paper S. Friedland and the author presented a formal expression for Lambda_d(p) of the monomer-dimer problem in d dimensions involving a power series in p. We there presented the result of computations for the terms in the power series through the sixth power. I herein present the result for the seventh power term, in d = 2. An interesting feature of the new computation is that it would have been impossible without a new algorithm for computation of the Tutte polynomial, by Bjorklund, Husfeldt, Kaski, and Koivisto.
We outline the most recent theory for the computation of the exponential growth rate of the number of configurations on a multi-dimensional grid. As an application we compute the monomer-dimer constant for the 2-dimensional grid to 8 decimal digits, agreeing with the heuristic computations of Baxter, and for the 3-dimensional grid with an error smaller than 1.35%.
We consider the imitative monomer-dimer model on the complete graph introduced in [1]. It was understood that this model is described by the monomer density and has a phase transition along certain critical line. 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. We show that this quantity exhibits the central limit theorem away from the critical line and enjoys a non-normal limit theorem at criticality …
A collection of rigorous results for a class of mean-field monomer-dimer models is presented. It includes a Gaussian representation for the partition function that is shown to considerably simplify the proofs. The solutions of the quenched diluted case and the random monomer case are explained. The presence of the attractive component of the Van der Waals potential is considered and the coexistence phase coexistence transition analysed. In particular the breakdown of the central limit theorem is illustrated at the critical point where a non Gaussian, quartic exponential distribution is found f…
Let (lambda_d)(p) be the p monomer-dimer entropy on the d-dimensional integer lattice Z^d, where p in [0,1] is the dimer density. We give upper and lower bounds for (lambda_d)(p) in terms of expressions involving (lambda_(d-1))(q). The upper bound is based on a conjecture claiming that the p monomer-dimer entropy of an infinite subset of Z^d is bounded above by (lambda_d)(p). We compute the first three terms in the formal asymptotic expansion of (lambda_d)(p) in powers of 1/d. We prove that the lower asymptotic matching conjecture is satisfied for (lambda_d)(p).
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 with 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 exp…
In previous papers an asymptotic expansion for the dimer lambda_d of the form lambda_d ~ (1/2)ln(2d) - 1/2 + c_1/d + c_2/d^2 ... was developed. Kernels J_n were a key ingredient in the theory. Herein we prove J_n are of the form J_n = C_r/d^r + C_(r+1)/d^(r+1) + ... + C_(n-1)/d^(n-1) with r > (n/2)-1.
The correlation of gaps in dimer systems was introduced in 1963 by Fisher and Stephenson, who looked at the interaction of two monomers generated by the rigid exclusion of dimers on the closely packed square lattice. In previous work we considered the analogous problem on the hexagonal lattice, and we extended the set-up to include the correlation of any finite number of monomer clusters. For fairly general classes of monomer clusters we proved that the asymptotics of their correlation is given, for large separations between the clusters, by a multiplicative version of Coulomb's law for 2D ele…
Recently an expansion as a power series in 1/d has been presented for the specific entropy of a complete dimer covering of a d-dimensional hypercubic lattice. This paper extends from 3 to 10 the number of terms known in the series. Likewise an expansion for the entropy, dependent on the dimer-density p, of a monomer-dimer system, involving a sum sum_k a_k(d) p^k, has been recently offered. We herein extend the number of the known expansion coefficients from 6 to 20 for the hyper-cubic lattices of general dimension d and from 6 to 24 for the hyper-cubic lattices of dimensions d < 5 . We show th…
We study the classical dimer model on rhombic Penrose tilings, whose edges and vertices may be identified with those of a bipartite graph. We find that Penrose tilings do not admit perfect matchings (defect-free dimer coverings). Instead, their maximum matchings have a monomer density of $81-50φ\approx 0.098$ in the thermodynamic limit, with $φ=\left(1+\sqrt{5}\right)/2$ the golden ratio. Maximum matchings divide the tiling into a fractal of nested closed regions bounded by loops that cannot be crossed by monomers. These loops connect second-nearest neighbour even-valence vertices, each of whi…
In the development of a presumed asymptotic expansion for lambda_d in previous work, a basic step involved extracting the asymptotic behavior of a sum as being dominated by the largest term in the sum. But this argument only is valid when the terms in the sum are positive. In the case terms in the sum are not all positive (some of the J_n are negative) we replace the `largest term' argument by the route of this paper. The sum is replaced by a contour integral. We then find a stationary point of the integrand that we conjecture dominates the asymptotic behavior of the integral.
The dimer problem arose in a thermodynamic study of diatomic molecules, and was abstracted into one of the most basic and natural problems in both statistical mechanics and combinatoric mathematics. Given a rectangular lattice of volume V in d dimensions, the dimer problem loosely speaking is to count the number of different ways dimers (dominoes) may be layed down on the lattice to completely cover it. It is known that the number of such coverings is roughly exp(lambda_d V) for some number lambda_d. The first terms in the expansion of lambda_d have been known for about thirty years lambda_d ~…
We compute the asymptotics of a block Toeplitz determinant which arises in the classical dimer model for the triangular lattice when considering the monomer-monomer correlation function. The model depends on a parameter interpolating between the square lattice ($t=0$) and the triangular lattice ($t=1$), and we obtain the asymptotics for $0<t\le 1$. For $0<t<1$ we apply the Szegö Limit Theorem for block Toeplitz determinants. The main difficulty is to evaluate the constant term in the asymptotics, which is generally given only in a rather abstract form.
In this paper we review the asymptotic matching conjectures for $r$-regular bipartite graphs, and their connections in estimating the monomer-dimer entropies in $d$-dimensional integer lattice and Bethe lattices. We prove new rigorous upper and lower bounds for the monomer-dimer entropies, which support these conjectures. We describe a general construction of infinite families of $r$-regular tori graphs and give algorithms for computing the monomer-dimer entropy of density $p$, for any $p\in [0,1]$, for these graphs. Finally we use tori graphs to test the asymptotic matching conjectures for ce…
Working with a presumed asymptotic series for lambda_d developed in previous work, we make some intelligent guesses for lambda_d with d=3, 4, 5; and estimates for the corresponding errors. We present arguments in favor of these guesses, we earnestly believe they will turn out to be correct. Such approximate values may help stimulate people working on rigorous bounds. In addition to suggesting bounds to prove, there will be the strong motivation to prove me wrong.
We study various statistical properties of the double-dimer model, a generalization of the dimer model, on rectangular domains of the square lattice. We take advantage of the Grassmannian representation of the dimer model, first to calculate the probability distribution of the number of nontrivial loops around a cylinder, which is consistent with the previously known result, and then to calculate the expectation value of the number of loops surrounding two faces and the left-passage probability, both in the discrete and the continuum cases. We also briefly explain the calculation of some relat…
In this work, some classical results of the pfaffian theory of the dimer model based on the work of Kasteleyn, Fisher and Temperley are introduced in a fermionic framework. Then we shall detail the bosonic formulation of the model {\it via} the so-called height mapping and the nature of boundary conditions is unravelled. The complete and detailed fermionic solution of the dimer model on the square lattice with an arbitrary number of monomers is presented, and finite size effect analysis is performed to study surface and corner effects, leading to the extrapolation of the central charge of the …
In a previous paper an asymptotic expansion for lambda_d in powers of 1/d was developed. The results of computer computations for some terms in the expansion, as well as various quantities associated to the expansion, are herein presented. The computations (in integer arithmetic) are actually done only for d=1, d=2, and d=3, but some results for arbitrary dimension follow from the general structure of the theory. In particular we obtain the next term in powers of 1/d, the 1/d^3 term in the asymptotic expansion for lambda_d, as well as checking the correctness of the terms previously calculated…
