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Shu-Chiuan Chang

Publications and source records attributed to Shu-Chiuan Chang.

At least 19 recordsLinked to original sources
arXiv2025

Chromatic Zeros on the Limit $G^{(p,\ell)}_\infty$ of the Family $G^{(p,\ell)}_m$ of Hierarchical Graphs

We calculate the continuous accumulation set ${\cal B}_q(p,\ell)$ of zeros of the chromatic polynomial $P(G^{(p,\ell)}_m,q)$ in the limit $m \to \infty$, on a family of graphs $G^{(p,\ell)}_m$ defined such that $G^{(p,\ell)}_m$ is obtained from $G^{(p,\ell)}_{m-1}$ by replacing each edge (i.e., bond) on $G^{(p,\ell)}_m$ by $p$ paths each of length $\ell$ edges, starting with the tree graph $T_2$. Our method uses the property that the chromatic polynomial $P(G,q)$ of a graph $G$ is equal to the $v=-1$ evaluation of the partition function of the $q$-state Potts model, together with (i) the property that $Z(G^{(p,\ell)}_m,q,v)$ can be expressed via an exact closed-form real-space renormalization (RG) group transformation in terms of $Z(G^{(p,\ell)}_{m-1},q,v')$, where $v'=F_{(p,\ell),q}(v)$ is a rational function of $v$ and $q$ and (ii) ${\cal B}_q(p,\ell)(v)$ is the locus in the complex $q$-plane that separates regions of different asymptotic behavior of the $m$-fold iterated RG transformation $F_{(p,\ell),q}(v)$ in the $m \to \infty$ limit. Thus, our results involve calculations of region diagrams in the complex $q$-plane showing the type of behavior that occurs in the $m \to \infty$ limit of the $m$-fold iterated RG transformation mapping $F_{(p,\ell),q}(v)$ starting with the initial value $v=v_0=-1$. Calculations are presented of the maximal point $q_c(G^{(p,\ell)}_\infty)$ at which the locus ${\cal B}_q$ crosses the real-$q$ axis, as well as several other points at which, depending on $p$ and $\ell$, the locus ${\cal B}_q$ crosses this axis. We give explicit results for a variety of $(p,\ell)$ cases and observe a number of interesting features. Calculations of the ground-state degeneracy of the Potts antiferromagnet at $q_c(G^{(p,\ell)}_\infty)$ are presented. This work extends a previous study with R. Roeder of the $(p,\ell)=(2,2)$ case to higher $p$ and $\ell$ values.

cond-mat.stat-mech↗
arXiv2024

Potts Partition Function Zeros and Ground State Entropy on Hanoi Graphs

We study properties of the Potts model partition function $Z(H_m,q,v)$ on $m$'th iterates of Hanoi graphs, $H_m$, and use the results to draw inferences about the $m \to \infty$ limit that yields a self-similar Hanoi fractal, $H_\infty$. We also calculate the chromatic polynomials $P(H_m,q)=Z(H_m,q,-1)$. From calculations of the configurational degeneracy, per vertex, of the zero-temperature Potts antiferromagnet on $H_m$, denoted $W(H_m,q)$, estimates of $W(H_\infty,q)$, are given for $q=3$ and $q=4$ and compared with known values on other lattices. We compute the zeros of $Z(H_m,q,v)$ in the complex $q$ plane for various values of the temperature-dependent variable $v=y-1$ and in the complex $y$ plane for various values of $q$. These are consistent with accumulating to form loci denoted ${\cal B}_q(v)$ and ${\cal B}_v(q)$, or equivalently, ${\cal B}_y(q)$, in the $m \to \infty$ limit. Our results motivate the inference that the maximal point at which ${\cal B}_q(-1)$ crosses the real $q$ axis, denoted $q_c$, has the value $q_c=(1/2)(3+\sqrt{5} \, )$ and correspondingly, if $q=q_c$, then ${\cal B}_y(q_c)$ crosses the real $y$ axis at $y=0$, i.e., the Potts antiferromagnet on $H_\infty$ with $q=(1/2)(3+\sqrt{5} \, )$ has a $T=0$ critical point. Finally, we analyze the partition function zeros in the $y$ plane for $q \gg 1$ and show that these accumulate approximately along parts of the sides of an equilateral triangular with apex points that scale like $y \sim q^{2/3}$ and $y \sim q^{2/3} e^{\pm 2πi/3}$. Some comparisons are presented of these findings for Hanoi graphs with corresponding results on $m$'th iterates of Sierpinski gasket graphs and the $m \to \infty$ limit yielding the Sierpinski gasket fractal.

cond-mat.stat-mech↗
arXiv2023

Measures of Spin Ordering in the Potts Model with a Generalized External Magnetic Field

We formulate measures of spin ordering in the $q$-state ferromagnetic Potts model in a generalized external magnetic field that favors or disfavors spin values in a subset $I_s = \{1,...,s\}$ of the total set of $q$ values. The results are contrasted with the corresponding measures of spin ordering in the case of a conventional external magnetic field that favors or disfavors a single spin value out of total set of $q$ values. Some illustrative calculations are included.

cond-mat.stat-mech↗
arXiv2021

Exact Results for Average Cluster Numbers in Bond Percolation on Infinite-Length Lattice Strips

We calculate exact analytic expressions for the average cluster numbers $\langle k \rangle_{Λ_s}$ on infinite-length strips $Λ_s$, with various widths, of several different lattices, as functions of the bond occupation probability, $p$. It is proved that these expressions are rational functions of $p$. As special cases of our results, we obtain exact values of $\langle k \rangle_{Λ_s}$ and derivatives of $\langle k \rangle_{Λ_s}$ with respect to $p$, evaluated at the critical percolation probabilities $p_{c,Λ}$ for the corresponding infinite two-dimensional lattices $Λ$. We compare these exact results with an analytic finite-size correction formula and find excellent agreement. We also analyze how unphysical poles in $\langle k \rangle_{Λ_s}$ determine the radii of convergence of series expansions for small $p$ and for $p$ near to unity. Our calculations are performed for infinite-length strips of the square, triangular, and honeycomb lattices with several types of transverse boundary conditions.

cond-mat.stat-mech↗
arXiv2020

Exponential Growth Constants for Spanning Forests on Archimedean Lattices: Values and Comparisons of Upper Bounds

We compare our upper bounds on the exponential growth constant $ϕ(Λ)$ characterizing the asymptotic behavior of spanning forests on Archimedean lattices $Λ$ with recently derived upper bounds. Our upper bounds on $ϕ(Λ)$, which are very close to the respective values of $ϕ(Λ)$ that we have calculated, are shown to be significantly better for these lattices than the new upper bounds.

math.CO↗
arXiv2020

Asymptotic Behavior of Spanning Forests and Connected Spanning Subgraphs on Two-Dimensional Lattices

We calculate exponential growth constants $ϕ$ and $σ$ describing the asymptotic behavior of spanning forests and connected spanning subgraphs on strip graphs, with arbitrarily great length, of several two-dimensional lattices, including square, triangular, honeycomb, and certain heteropolygonal Archimedean lattices. By studying the limiting values as the strip widths get large, we infer lower and upper bounds on these exponential growth constants for the respective infinite lattices. Since our lower and upper bounds are quite close to each other, we can infer very accurate approximate values for these exponential growth constants, with fractional uncertainties ranging from $O(10^{-4})$ to $O(10^{-2})$. We show that $ϕ$ and $σ$, are monotonically increasing functions of vertex degree for these lattices.

cond-mat.stat-mech↗
arXiv2019

$q$-Plane Zeros of the Potts Partition Function on Diamond Hierarchical Graphs

We report exact results concerning the zeros of the partition function of the Potts model in the complex $q$ plane, as a function of a temperature-like Boltzmann variable $v$, for the $m$'th iterate graphs $D_m$ of the Diamond Hierarchical Lattice (DHL), including the limit $m \to \infty$. In this limit we denote the continuous accumulation locus of zeros in the $q$ planes at fixed $v = v_0$ as ${\mathcal B}_q(v_0)$. We apply theorems from complex dynamics to establish properties of ${\mathcal B}_q(v_0)$. For $v=-1$ (the zero-temperature Potts antiferromagnet, or equivalently, chromatic polynomial), we prove that ${\mathcal B}_q(-1)$ crosses the real-$q$ axis at (i) a minimal point $q=0$, (ii) a maximal point $q=3$ (iii) $q=32/27$, (iv) a cubic root that we give, with the value $q = q_1 = 1.6388969..$, and (v) an infinite number of points smaller than $q_1$, converging to $32/27$ from above. Similar results hold for ${\mathcal B}_q(v_0)$ for any $-1 < v < 0$ (Potts antiferromagnet at nonzero temperature). The locus ${\mathcal B}_q(v_0)$ crosses the real-$q$ axis at only two points for any $v > 0$ (Potts ferromagnet). We also provide computer-generated plots of ${\mathcal B}_q(v_0)$ at various values of $v_0$ in both the antiferromagnetic and ferromagnetic regimes and compare them to numerically computed zeros of $Z(D_4,q,v_0)$.

math-ph↗
arXiv2018

Dimer-monomer model on the generalized Tower of Hanoi graph

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 lower and upper bounds for $z_{TH_d}$ with arbitrary $d$.

math-ph↗
arXiv2018

Study of Exponential Growth Constants of Directed Heteropolygonal Archimedean Lattices

We infer upper and lower bounds on the exponential growth constants $α(Λ)$, $α_0(Λ)$, and $β(Λ)$ describing the large-$n$ behavior of, respectively, the number of acyclic orientations, acyclic orientations with a unique source vertex, and totally cyclic orientations of arrows on bonds of several $n$-vertex heteropolygonal Archimedean lattices $Λ$. These are, to our knowledge, the best bounds on these growth constants. The inferred upper and lower bounds on the growth constants are quite close to each other, which enables us to derive rather accurate values for the actual exponential growth constants. Combining our new results for heteropolygonal Archimedean lattices with our recent results for homopolygonal Archimedean lattices, we show that the exponential growth constants $α(Λ)$, $α_0(Λ)$, and $β(Λ)$ on these lattices are monotonically increasing functions of the lattice coordination number. Comparisons are made with the corresponding growth constants for spanning trees on these lattices. Our findings provide further support for the Merino-Welsh and Conde-Merino conjectures.

cond-mat.stat-mech↗
arXiv2018

Asymptotic Behavior of Acyclic and Cyclic Orientations of Directed Lattice Graphs

We calculate exponential growth constants describing the asymptotic behavior of several quantities enumerating classes of orientations of arrow variables on the bonds of several types of directed lattice strip graphs $G$ of finite width and arbitrarily great length, in the infinite-length limit, denoted {G}. Specifically, we calculate the exponential growth constants for (i) acyclic orientations, $α(\{G\})$, (ii) acyclic orientations with a single source vertex, $α_0(\{G\})$, and (iii) totally cyclic orientations, $β(\{G\})$. We consider several lattices, including square (sq), triangular (tri), and honeycomb (hc). From our calculations, we infer lower and upper bounds on these exponential growth constants for the respective infinite lattices. To our knowledge, these are the best current bounds on these quantities. Since our lower and upper bounds are quite close to each other, we can infer very accurate approximate values for the exponential growth constants, with fractional uncertainties ranging from $O(10^{-4})$ to $O(10^{-2})$. Further, we present exact values of $α(tri)$, $α_0(tri)$, and $β(hc)$ and use them to show that our lower and upper bounds on these quantities are very close to these exact values, even for modest strip widths. Results are also given for a nonplanar lattice denoted $sq_d$. We show that $α(\{G\})$, $α_0(\{G\})$, and $β(\{G\})$ are monotonically increasing functions of vertex degree for these lattices. We also study the asymptotic behavior of the ratios of the quantities (i)-(iii) divided by the total number of edge orientations as the number of vertices goes to infinity. A comparison is given of these exponential growth constants with the corresponding exponential growth constant $τ(\{G\})$ for spanning trees. Our results are in agreement with inequalities following from the Merino-Welsh and Conde-Merino conjectures.

cond-mat.stat-mech↗
arXiv2015

Exact Partition Functions for the $q$-State Potts Model with a Generalized Magnetic Field on Lattice Strip Graphs

We calculate the partition function of the $q$-state Potts model on arbitrary-length cyclic ladder graphs of the square and triangular lattices, with a generalized external magnetic field that favors or disfavors a subset of spin values $\{1,...,s\}$ with $s \le q$. For the case of antiferromagnet spin-spin coupling, these provide exactly solved models that exhibit an onset of frustration and competing interactions in the context of a novel type of tensor-product $S_s \otimes S_{q-s}$ global symmetry, where $S_s$ is the permutation group on $s$ objects.

cond-mat.stat-mech↗
arXiv2012

Zeros of the Potts Model Partition Function on Sierpinski Graphs

We calculate zeros of the $q$-state Potts model partition function on $m$'th-iterate Sierpinski graphs, $S_m$, in the variable $q$ and in a temperature-like variable, $y$. We infer some asymptotic properties of the loci of zeros in the limit $m \to \infty$ and relate these to thermodynamic properties of the $q$-state Potts ferromagnet and antiferromagnet on the Sierpinski gasket fractal, $S_\infty$.

cond-mat.stat-mech↗
arXiv2012

Some Exact Results on Bond Percolation

We present some exact results on bond percolation. We derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice $Λ$ by $\ell$ bonds connecting the same adjacent vertices, thereby yielding the lattice $Λ_\ell$. This relation is used to calculate the bond percolation threshold on $Λ_\ell$. We show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality $d \ge 2$ but changes it on a one-dimensional lattice and quasi-one-dimensional infinite-length strips. We also present analytic expressions for the average cluster number per vertex and correlation length for the bond percolation problem on the $N \to \infty$ limits of several families of $N$-vertex graphs. Finally, we explore the effect of bond vacancies on families of graphs with the property of bounded diameter as $N \to \infty$.

cond-mat.stat-mech↗
arXiv2012

Ice model and eight-vertex model on the two-dimensional Sierpinski gasket

We present the numbers of ice model and eight-vertex model configurations (with Boltzmann factors equal to one), I(n) and E(n) respectively, on the two-dimensional Sierpinski gasket SG(n) at stage $n$. For the eight-vertex model, the number of configurations is $E(n)=2^{3(3^n+1)/2}$ and the entropy per site, defined as $\lim_{v \to \infty} \ln E(n)/v$ where $v$ is the number of vertices on SG(n), is exactly equal to $\ln 2$. For the ice model, the upper and lower bounds for the entropy per site $\lim_{v \to \infty} \ln I(n)/v$ are derived in terms of the results 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 accurate. The corresponding result of ice model on the generalized two-dimensional Sierpinski gasket SG_b(n) with $b=3$ is also obtained. For the generalized vertex model on SG_3(n), the number of configurations is $2^{(8 \times 6^n +7)/5}$ and the entropy per site is equal to $\frac87 \ln 2$. The general upper and lower bounds for the entropy per site for arbitrary $b$ are conjectured.

cond-mat.stat-mech↗
arXiv2011

Asymptotic enumeration of independent sets on the Sierpinski gasket

The number of independent sets is equivalent to the partition function of the hard-core lattice gas model with nearest-neighbor exclusion and unit activity. We study the number of independent sets $m_{d,b}(n)$ on the generalized Sierpinski gasket $SG_{d,b}(n)$ at stage $n$ with dimension $d$ equal to two, three and four for $b=2$, and layer $b$ equal to three for $d=2$. The upper and lower bounds for the asymptotic growth constant, defined as $z_{SG_{d,b}}=\lim_{v \to \infty} \ln m_{d,b}(n)/v$ where $v$ is the number of vertices, on these Sierpinski gaskets are derived in terms of the results at a certain stage. The numerical values of these $z_{SG_{d,b}}$ are evaluated with more than a hundred significant figures accurate. We also conjecture the upper and lower bounds for the asymptotic growth constant $z_{SG_{d,2}}$ with general $d$.

cond-mat.stat-mech↗
arXiv2010

Acyclic orientations on the Sierpinski gasket

We study the number of acyclic orientations on the generalized two-dimensional Sierpinski gasket $SG_{2,b}(n)$ at stage $n$ with $b$ equal to two and three, and determine the asymptotic behaviors. We also derive upper bounds for the asymptotic growth constants for $SG_{2,b}$ and $d$-dimensional Sierpinski gasket $SG_d$.

math-ph↗
arXiv2009

Hamiltonian paths on the Sierpinski gasket

We derive exactly the number of Hamiltonian paths H(n) on the two dimensional Sierpinski gasket SG(n) at stage $n$, whose asymptotic behavior is given by $\frac{\sqrt{3}(2\sqrt{3})^{3^{n-1}}}{3} \times (\frac{5^2 \times 7^2 \times 17^2}{2^{12} \times 3^5 \times 13})(16)^n$. We also obtain the number of Hamiltonian paths with one end at a certain outmost vertex of SG(n), with asymptotic behavior $\frac {\sqrt{3}(2\sqrt{3})^{3^{n-1}}}{3} \times (\frac {7 \times 17}{2^4 \times 3^3})4^n$. The distribution of Hamiltonian paths on SG(n) with one end at a certain outmost vertex and the other end at an arbitrary vertex of SG(n) is investigated. We rigorously prove that the exponent for the mean $\ell$ displacement between the two end vertices of such Hamiltonian paths on SG(n) is $\ell \log 2 / \log 3$ for $\ell>0$.

cond-mat.stat-mech↗