Browsing by Author "Roeder, Roland K. W."

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    Chromatic zeros on hierarchical lattices and equidistribution on parameter space
    (EMS Press, 2021-09) Chio, Ivan; Roeder, Roland K. W.; Mathematical Sciences, School of Science
    Associated to any finite simple graph $\\Gamma$ is the chromatic polynomial $\\mathcal{P}\\Gamma(q)$ whose complex zeros are called the \_chromatic zeros of $\\Gamma$. A hierarchical lattice is a sequence of finite simple graphs ${\\Gamman}{n=0}^\\infty$ built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let $\\mun$ denote the probability measure that assigns a Dirac measure to each chromatic zero of $\\Gamma_n$. Under a mild hypothesis on the generating graph, we prove that the sequence $\\mu_n$ converges to some measure $\\mu$ as $n$ tends to infinity. We call $\\mu$ the \_limiting measure of chromatic zeros associated to ${\\Gamman}{n=0}^\\infty$. In the case of the diamond hierarchical lattice we prove that the support of $\\mu$ has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications.
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    Limiting Measure of Lee–Yang Zeros for the Cayley Tree
    (Springer, 2019-09) Chio, Ivan; He, Caleb; Ji, Anthony L.; Roeder, Roland K. W.; Mathematical Sciences, School of Science
    This paper is devoted to an in-depth study of the limiting measure of Lee–Yang zeroes for the Ising Model on the Cayley Tree. We build on previous works of Müller-Hartmann and Zittartz (Z Phys B 22:59, 1975), Müller-Hartmann (Z Phys B 27:161–168, 1977), Barata and Marchetti (J Stat Phys 88:231–268, 1997) and Barata and Goldbaum (J Stat Phys 103:857–891, 2001), to determine the support of the limiting measure, prove that the limiting measure is not absolutely continuous with respect to Lebesgue measure, and determine the pointwise dimension of the measure at Lebesgue a.e. point on the unit circle and every temperature. The latter is related to the critical exponents for the phase transitions in the model as one crosses the unit circle at Lebesgue a.e. point, providing a global version of the “phase transition of continuous order” discovered by Müller-Hartmann–Zittartz. The key techniques are from dynamical systems because there is an explicit formula for the Lee–Yang zeros of the finite Cayley Tree of level n in terms of the n-th iterate of an expanding Blaschke Product. A subtlety arises because the conjugacies between Blaschke Products at different parameter values are not absolutely continuous.
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    q-Plane Zeros of the Potts Partition Function on Diamond Hierarchical Graphs
    (AIP, 2020-07) Chang, Shu-Chiuan; Roeder, Roland K. W.; Shrock, Robert; Mathematical Sciences, School of Science
    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 Dm of the diamond hierarchical lattice, including the limit m → ∞. In this limit, we denote the continuous accumulation locus of zeros in the q-planes at fixed v = v0 as ℬ𝑞(𝑣0). We apply theorems from complex dynamics to establish the properties of ℬ𝑞(𝑣0). For v = −1 (the zero-temperature Potts antiferromagnet or, equivalently, chromatic polynomial), we prove that ℬ𝑞(−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 = q1 = 1.638 896 9…, and (v) an infinite number of points smaller than q1, converging to 32/27 from above. Similar results hold for ℬ𝑞(𝑣0) for any −1 < v < 0 (Potts antiferromagnet at nonzero temperature). The locus ℬ𝑞(𝑣0) crosses the real q-axis at only two points for any v > 0 (Potts ferromagnet). We also provide the computer-generated plots of ℬ𝑞(𝑣0) at various values of v0 in both the antiferromagnetic and ferromagnetic regimes and compare them to the numerically computed zeros of Z(D4, q, v0).
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    Some Connections Between Complex Dynamics and Statistical Mechanics
    (2020-05) Chio, Ivan; Roeder, Roland K. W.; Misiurewicz, Michal; Perez, Rodrigo A.; Yattselev, Maxim L.
    Associated to any finite simple graph $\Gamma$ is the {\em chromatic polynomial} $\P_\Gamma(q)$ whose complex zeros are called the {\em chromatic zeros} of $\Gamma$. A hierarchical lattice is a sequence of finite simple graphs $\{\Gamma_n\}_{n=0}^\infty$ built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let $\mu_n$ denote the probability measure that assigns a Dirac measure to each chromatic zero of $\Gamma_n$. Under a mild hypothesis on the generating graph, we prove that the sequence $\mu_n$ converges to some measure $\mu$ as $n$ tends to infinity. We call $\mu$ the {\em limiting measure of chromatic zeros} associated to $\{\Gamma_n\}_{n=0}^\infty$. In the case of the Diamond Hierarchical Lattice we prove that the support of $\mu$ has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree.
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    Two-point correlation functions and universality for the zeros of systems of so (n+ 1)- invariant gaussian random polynomials
    (Oxford, 2016) Bleher, Pavel M.; Homma, Yushi; Roeder, Roland K. W.; Department of Mathematical Sciences, School of Science
    We study the two-point correlation functions for the zeroes of systems of SO(n+1)-invariant Gaussian random polynomials on RPn and systems of Isom(Rn) -invariant Gaussian analytic functions. Our result reflects the same “repelling”, “neutral”, and “attracting” short-distance asymptotic behavior, depending on the dimension, as was discovered in the complex case by Bleher, Shiffman, and Zelditch. We then prove that the correlation function for the Isom(Rn)-invariant Gaussian analytic functions is “universal”, describing the scaling limit of the correlation function for the restriction of systems of the SO(k+1)-invariant Gaussian random polynomials to any n-dimensional C2 submanifold M⊂RPk. This provides a real counterpart to the universality results that were proved in the complex case by Bleher, Shiffman, and Zelditch.
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    Typical dynamics of plane rational maps with equal degrees
    (AIMS, 2016) Diller, Jeffrey; Liu, Han; Roeder, Roland K. W.; Department of Mathematical Sciences, School of Science
    Let f:CP2⇢CP2 be a rational map with algebraic and topological degrees both equal to d≥2. Little is known in general about the ergodic properties of such maps. We show here, however, that for an open set of automorphisms T:CP2→CP2, the perturbed map T∘f admits exactly two ergodic measures of maximal entropy logd, one of saddle type and one of repelling type. Neither measure is supported in an algebraic curve, and fT is 'fully two dimensional' in the sense that it does not preserve any singular holomorphic foliation of CP2. In fact, absence of an invariant foliation extends to all T outside a countable union of algebraic subsets of Aut(P2). Finally, we illustrate all of our results in a more concrete particular instance connected with a two dimensional version of the well-known quadratic Chebyshev map.
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    Values of Ramanujan's Continued Fractions Arising as Periodic Points of Algebraic Functions
    (2023-08) Akkarapakam, Sushmanth Jacob; Morton, Richard Patrick; Klimek, Slawomir D.; Roeder, Roland K. W.; Geller, William A.
    The main focus of this dissertation is to find and explain the periodic points of certain algebraic functions that are related to some modular functions, which themselves can be represented by continued fractions. Some of these continued fractions are first explored by Srinivasa Ramanujan in early 20th century. Later on, much work has been done in terms of studying the continued fractions, and proving several relations, identities, and giving different representations for them. The layout of this report is as follows. Chapter 1 has all the basic background knowledge and ingredients about algebraic number theory, class field theory, Ramanujan’s theta functions, etc. In Chapter 2, we look at the Ramanujan-Göllnitz-Gordon continued fraction that we call v(τ) and evaluate it at certain arguments in the field K = Q(√−d), with −d ≡ 1 (mod 8), in which the ideal (2) = ℘2℘′2 is a product of two prime ideals. We prove several identities related to itself and with other modular functions. Some of these are new, while some of them are known but with different proofs. These values of v(τ) are shown to generate the inertia field of ℘2 or ℘′2 in an extended ring class field over the field K. The conjugates over Q of these same values, together with 0, −1 ± √2, are shown to form the exact set of periodic points of a fixed algebraic function ˆF(x), independent of d. These are analogues of similar results for the Rogers-Ramanujan continued fraction. See [1] and [2]. This joint work with my advisor Dr. Morton, is submitted for publication to the New York Journal. In Chapters 3 and 4, we take a similar approach in studying two more continued fractions c(τ) and u(τ), the first of which is more commonly known as the Ramanujan’s cubic continued fraction. We show what fields a value of this continued fraction generates over Q, and we describe how the periodic points for described functions arise as values of these continued fractions. Then in the last chapter, we summarise all these results, give some possible directions for future research as well as mentioning some conjectures.