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Register a new userThe Widom-Rowlinson model, the hard-core model and the extremality of the complete graph
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Let $H_{mathrm{WR}}$ be the path on $3$ vertices with a loop at each vertex. D. Galvin conjectured, and E. Cohen, W. Perkins and P. Tetali proved that for any $d$-regular simple graph $G$ on $n$ vertices we have $$hom(G,H_{mathrm{WR}})leq hom(K_{d+1},H_{mathrm{WR}})^{n/(d+1)}.$$ In this paper we give a short proof of this theorem together with the proof of a conjecture of Cohen, Perkins and Tetali. Our main tool is a simple bijection between the Widom-Rowlinson model and the hard-core model on another graph. We also give a large class of graphs $H$ for which we have $$hom(G,H)leq hom(K_{d+1},H)^{n/(d+1)}.$$ In particular, we show that the above inequality holds if $H$ is a path or a cycle of even length at least $6$ with loops at every vertex.
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In 1990, Cvetkovi{c} and Rowlinson [The largest eigenvalue of a graph: a survey, Linear Multilinear Algebra 28(1-2) (1990), 3--33] conjectured that among all outerplanar graphs on $n$ vertices, $K_1vee P_{n-1}$ attains the maximum spectral radius. In 2017, Tait and Tobin [Three conjectures in extremal spectral graph theory, J. Combin. Theory, Ser. B 126 (2017) 137-161] confirmed the conjecture for sufficiently large values of $n$. In this article, we show the conjecture is true for all $ngeq2$ except for $n=6$.
In the hard-core model on a finite graph we are given a parameter lambda>0, and an independent set I arises with probability proportional to lambda^|I|. On infinite graphs a Gibbs distribution is defined as a suitable limit with the correct conditional probabilities. In the infinite setting we are interested in determining when this limit is unique and when there is phase coexistence, i.e., existence of multiple Gibbs states. On finite graphs we are interested in determining the mixing time of local Markov chains. On Z^2 it is conjectured that these problems are related and that both undergo a phase transition at some critical point lambda_c approx 3.79. For phase coexistence, much of the work to date has focused on the regime of uniqueness, with the best result being recent work of Restrepo et al. showing that there is a unique Gibbs state for all lambda < 2.3882. Here we give the first non-trivial result in the other direction, showing that there are multiple Gibbs states for all lambda > 5.3646. Our proof adds two significant innovations to the standard Peierls argument. First, building on the idea of fault lines introduced by Randall, we construct an event that distinguishes two boundary conditions and always has long contours associated with it, obviating the need to accurately enumerate short contours. Second, we obtain vastly improved bounds on the number of contours by relating them to a new class of self-avoiding walks on an oriented version of Z^2. We extend our characterization of fault lines to show that local Markov chains will mix slowly when lambda > 5.3646 on lattice regions with periodic (toroidal) boundary conditions and when lambda > 7.1031 with non-periodic (free) boundary conditions. The arguments here rely on a careful analysis that relates contours to taxi walks and represent a sevenfold improvement to the previously best known values of lambda.
The critical behavior of the Widom-Rowlinson mixture [J. Chem. Phys. 52, 1670 (1970)] is studied in d=3 dimensions by means of grand canonical Monte Carlo simulations. The finite size scaling approach of Kim, Fisher, and Luijten [Phys. Rev. Lett. 91, 065701 (2003)] is used to extract the order parameter and the coexistence diameter. It is demonstrated that the critical behavior of the diameter is dominated by a singular term proportional to t^(1-alpha), with t the relative distance from the critical point, and alpha the critical exponent of the specific heat. No sign of a term proportional to t^(2beta) could be detected, with beta the critical exponent of the order parameter, indicating that pressure-mixing in this model is small. The critical density is measured to be rho*sigma^3 = 0.7486 +/- 0.0002, with sigma the particle diameter. The critical exponents alpha and beta, as well as the correlation length exponent nu, are also measured and shown to comply with d=3 Ising criticality.
The Widom-Rowlinson model of a fluid mixture is studied using a new cluster algorithm that is a generalization of the invaded cluster algorithm previously applied to Potts models. Our estimate of the critical exponents for the two-component fluid are consistent with the Ising universality class in two and three dimensions. We also present results for the three-component fluid.
The $g$-girth-thickness $theta(g,G)$ of a graph $G$ is the minimum number of planar subgraphs of girth at least $g$ whose union is $G$. In this paper, we determine the $6$-girth-thickness $theta(6,K_n)$ of the complete graph $K_n$ in almost all cases. And also, we calculate by computer the missing value of $theta(4,K_n)$.
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