scholarly journals Hard Squares with Negative Activity and Rhombus Tilings of the Plane

10.37236/1093 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Jakob Jonsson
Keyword(s):  
Partition Function ◽  
Statistical Mechanics ◽  
Simplicial Complex ◽  
Transfer Matrix ◽  
Euler Characteristic ◽  
Independent Sets ◽  
Roots Of Unity ◽  
Hard Squares ◽  
Vertex Set

Let $S_{m,n}$ be the graph on the vertex set ${\Bbb Z}_m \times {\Bbb Z}_n$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,d\pm 1)$ or $(a,b) = (c \pm 1,d)$ modulo $(m,n)$. We present a formula for the Euler characteristic of the simplicial complex $\Sigma_{m,n}$ of independent sets in $S_{m,n}$. In particular, we show that the unreduced Euler characteristic of $\Sigma_{m,n}$ vanishes whenever $m$ and $n$ are coprime, thereby settling a conjecture in statistical mechanics due to Fendley, Schoutens and van Eerten. For general $m$ and $n$, we relate the Euler characteristic of $\Sigma_{m,n}$ to certain periodic rhombus tilings of the plane. Using this correspondence, we settle another conjecture due to Fendley et al., which states that all roots of $\det (xI-T_m)$ are roots of unity, where $T_m$ is a certain transfer matrix associated to $\{\Sigma_{m,n} : n \ge 1\}$. In the language of statistical mechanics, the reduced Euler characteristic of $\Sigma_{m,n}$ coincides with minus the partition function of the corresponding hard square model with activity $-1$.

scholarly journals Hard Squares with Negative Activity on Cylinders with Odd Circumference

10.37236/71 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Jakob Jonsson
Keyword(s):  
Partition Function ◽  
Statistical Mechanics ◽  
Simplicial Complex ◽  
Euler Characteristic ◽  
Independent Sets ◽  
Square Grid ◽  
Unit Square ◽  
Hard Squares ◽  
Vertex Set

Let $C_{m,n}$ be the graph on the vertex set $\{1, \ldots, m\} \times \{0, \ldots, n-1\}$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,d\pm 1)$ or $(a,b) = (c \pm 1,d)$, where the second index is computed modulo $n$. One may view $C_{m,n}$ as a unit square grid on a cylinder with circumference $n$ units. For odd $n$, we prove that the Euler characteristic of the simplicial complex $\Sigma_{m,n}$ of independent sets in $C_{m,n}$ is either $2$ or $-1$, depending on whether or not $\gcd(m-1,n)$ is divisble by $3$. The proof relies heavily on previous work due to Thapper, who reduced the problem of computing the Euler characteristic of $\Sigma_{m,n}$ to that of analyzing a certain subfamily of sets with attractive properties. The situation for even $n$ remains unclear. In the language of statistical mechanics, the reduced Euler characteristic of $\Sigma_{m,n}$ coincides with minus the partition function of the corresponding hard square model with activity $-1$.

scholarly journals Cooperative Colorings and Independent Systems of Representatives

10.37236/2488 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Ron Aharoni ◽  
Ron Holzman ◽  
David Howard ◽  
Philipp Sprüssel
Keyword(s):  
Simplicial Complex ◽  
Independent Sets ◽  
Choice Functions ◽  
Sample Result ◽  
List Colorings ◽  
Vertex Set ◽  
Degree Conditions

We study a generalization of the notion of coloring of graphs, similar in spirit to that of list colorings: a cooperative coloring of a family of graphs $G_1,G_2, \ldots,G_k$ on the same vertex set $V$ is a choice of independent sets $A_i$ in $G_i$ ($1 \le i \le k)$ such that $\bigcup_{i=1}^kA_i=V$. This notion is linked (with translation in both directions) to the notion of ISRs, which are choice functions on given sets, whose range belongs to some simplicial complex. When the complex is that of the independent sets in a graph $G$, an ISR for a partition of the vertex set of a graph $G$ into sets $V_1,\ldots, V_n$ is a choice of a vertex $v_i \in V_i$ for each $i$ such that $\{v_1,\ldots,v_n\}$ is independent in $G$. Using topological tools, we study degree conditions for the existence of cooperative colorings and of ISRs. A sample result: Three cycles on the same vertex set have a cooperative coloring.

Statement B and the Yang-Baxter Equation

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Statistical Mechanics ◽  
Dirichlet Series ◽  
Weyl Group ◽  
Transfer Matrix ◽  
Baxter Equation ◽  
Partition Functions ◽  
Alternate Proof ◽  
Crystal Basis ◽  
Exactly Solved Models ◽  
Multiple Dirichlet Series

This chapter reinterprets Statements A and B in a different context, and yet again directly proves that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 19.10. The p-parts of Weyl group multiple Dirichlet series, with their deformed Weyl denominators, may be expressed as partition functions of exactly solved models in statistical mechanics. The transition to ice-type models represents a subtle shift in emphasis from the crystal basis representation, and suggests the introduction of a new tool, the Yang-Baxter equation. This tool was developed to prove the commutativity of the row transfer matrix for the six-vertex and similar models. This is significant because Statement B can be formulated in terms of the commutativity of two row transfer matrices. This chapter presents an alternate proof of Statement B using the Yang-Baxter equation.

scholarly journals Statistical Mechanics of Discrete Multicomponent Fragmentation

2020 ◽  
Vol 5 (4) ◽  
pp. 64
Author(s):  
Themis Matsoukas
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Partition Function ◽  
Statistical Mechanics ◽  
Closed Form ◽  
Fixed Number ◽  
Arbitrary Degree ◽  
Fragment Distribution ◽  
The Mean ◽  
Fragmentation Event

We formulate the statistics of the discrete multicomponent fragmentation event using a methodology borrowed from statistical mechanics. We generate the ensemble of all feasible distributions that can be formed when a single integer multicomponent mass is broken into fixed number of fragments and calculate the combinatorial multiplicity of all distributions in the set. We define random fragmentation by the condition that the probability of distribution be proportional to its multiplicity, and obtain the partition function and the mean distribution in closed form. We then introduce a functional that biases the probability of distribution to produce in a systematic manner fragment distributions that deviate to any arbitrary degree from the random case. We corroborate the results of the theory by Monte Carlo simulation, and demonstrate examples in which components in sieve cuts of the fragment distribution undergo preferential mixing or segregation relative to the parent particle.

scholarly journals THE 8V CSOS MODEL AND THE sl2 LOOP ALGEBRA SYMMETRY OF THE SIX-VERTEX MODEL AT ROOTS OF UNITY

2002 ◽  
Vol 16 (14n15) ◽  
pp. 1899-1905 ◽  
Author(s):  
TETSUO DEGUCHI
Keyword(s):  
Transfer Matrix ◽  
Lattice Model ◽  
Algebraic Method ◽  
System Size ◽  
Spin Operator ◽  
Total Spin ◽  
Coupling Constants ◽  
Roots Of Unity ◽  
Vertex Model ◽  
Elliptic Quantum

We review an algebraic method for constructing degenerate eigenvectors of the transfer matrix of the eight-vertex Cyclic Solid-on-Solid lattice model (8V CSOS model), where the degeneracy increases exponentially with respect to the system size. We consider the elliptic quantum group Eτ,η(sl2) at the discrete coupling constants: 2N η = m1 + im2τ, where N, m1 and m2 are integers. Then we show that degenerate eigenvectors of the transfer matrix of the six-vertex model at roots of unity in the sector SZ ≡ 0 ( mod N) are derived from those of the 8V CSOS model, through the trigonometric limit. They are associated with the complete N strings. From the result we see that the dimension of a given degenerate eigenspace in the sector SZ ≡ 0 ( mod N) of the six-vertex model at Nth roots of unity is given by [Formula: see text], where [Formula: see text] is the maximal value of the total spin operator SZ in the degenerate eigenspace.

PHYSICAL STATES AND SCALING PROPERTIES OF W GRAVITIES AND W STRINGS

1992 ◽  
Vol 07 (10) ◽  
pp. 2295-2333 ◽  
Author(s):  
SUMIT R. DAS ◽  
AVINASH DHAR ◽  
S. KALYANA RAMA
Keyword(s):  
Partition Function ◽  
Euler Characteristic ◽  
Extra Dimensions ◽  
Minimal Models ◽  
Scaling Properties ◽  
Critical Dimensions ◽  
Chemical Potentials ◽  
Natural Way

We discuss some physical aspects of W gravities and W strings. We identify global characteristics in W gravities (in addition to the usual Euler characteristic) and show how the dependence of the partition function on the various chemical potentials involves these quantities. We find the operators which create physical states in W3 and W4 gravities and discuss their relationship with screening operators. W strings are discussed in the framework of a natural way of coupling “matter” to W gravity, and the issues of extra dimensions and critical dimensions are clarified. We find a remarkable relationship between pure W gravities and ordinary gravity coupled to c<1 unitary minimal models.

scholarly journals Hypergraph domination and strong independence

2009 ◽  
Vol 3 (2) ◽  
pp. 347-358 ◽  
Author(s):  
Bibin Jose ◽  
Zsolt Tuza
Keyword(s):  
Dominating Set ◽  
Independent Sets ◽  
Dominating Sets ◽  
Tight Bound ◽  
Open Problems ◽  
Odd Cycle ◽  
Vertex Set

We solve several conjectures and open problems from a recent paper by Acharya [2]. Some of our results are relatives of the Nordhaus-Gaddum theorem, concerning the sum of domination parameters in hypergraphs and their complements. (A dominating set in H is a vertex set D X such that, for every vertex x? X\D there exists an edge E ? E with x ? E and E?D ??.) As an example, it is shown that the tight bound ??(H)+??(H) ? n+2 holds in hypergraphs H = (X, E) of order n ? 6, where H is defined as H = (X, E) with E = {X\E | E ? E}, and ?? is the minimum total cardinality of two disjoint dominating sets. We also present some simple constructions of balanced hypergraphs, disproving conjectures of the aforementioned paper concerning strongly independent sets. (Hypergraph H is balanced if every odd cycle in H has an edge containing three vertices of the cycle; and a set S X is strongly independent if |S?E|? 1 for all E ? E.).

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