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The number $n_s$ of clusters (per site) of size $s$, a central quantity in percolation theory, displays at criticality an algebraic scaling behavior of the form $n_s\simeq s^{-\tau}\, A\, (1+B s^{-\Omega})$. For the Fortuin--Kasteleyn representation of the $Q$-state Potts model in two dimensions, the Fisher exponent $\tau$ is known as a function of the real parameter $0\le Q\le4$, and, for bond percolation (the $Q\rightarrow 1$ limit), the correction-to-scaling exponent is derived as $\Omega=72/91$. We theoretically derive the exact formula for the correction-to-scaling exponent $\Omega=8/[(2g+1)(2g+3)]$ as a function of the Coulomb-gas coupling strength $g$, which is related to $Q$ by $Q=2+2\cos(2 \pi g)$. Using an efficient Monte Carlo cluster algorithm, we study the O($n$) loop model on the hexagonal lattice, which is in the same universality class as the $Q=n^2$ Potts model, and has significantly suppressed finite-size corrections and critical slowing-down. The predictions of the above formula include the exact value for percolation as a special case, and agree well with the numerical estimates of $\Omega$ for both the critical and tricritical branches of the Potts model., Comment: The document contains the paper (pdflatex, 14 pages), and 6 pdf figures

We show by extensive simulations that the whole supercritical phase of the three-dimensional uniform forest model simultaneously exhibits an infinite tree and a rich variety of critical phenomena. Besides typical scalings like algebraically decaying correlation, power-law distribution of cluster sizes, and divergent correlation length, a number of anomalous behaviors emerge. The fractal dimensions for off-giant trees take different values when being measured by linear system size or gyration radius. The giant-tree size displays two-length scaling fluctuations, instead of following the central-limit theorem., Comment: 11 pages, 3 figures. Pdflatex. SciPost Physics style SciPost.cls is included. Minor differences with respect to v2. Final version

Many systems, when initially placed far from equilibrium, exhibit surprising behavior in their attempt to equilibrate. Striking examples are the Mpemba effect and the cooling-heating asymmetry. These anomalous behaviors can be exploited to shorten the time needed to cool down (or heat up) a system. Though, a strategy to design these effects in mesoscopic systems is missing. We bring forward a description that allows us to formulate such strategies, and, along the way, makes natural these paradoxical behaviors. In particular, we study the evolution of macroscopic physical observables of systems freely relaxing under the influence of one or two instantaneous thermal quenches. The two crucial ingredients in our approach are timescale separation and a nonmonotonic temperature evolution of an important state function. We argue that both are generic features near a first-order transition. Our theory is exemplified with the one-dimensional Ising model in a magnetic field using analytic results and numerical experiments., Comment: pdflatex. Main text (6pp + 5 pdf figures) + Supplemental Material (7pp + 5 pdf figures). Some minor changes w.r.t. v1. Final version

We prove the ergodicity of the Wang--Swendsen--Koteck\'y (WSK) algorithm for the zero-temperature $q$-state Potts antiferromagnet on several classes of lattices on the torus. In particular, the WSK algorithm is ergodic for $q\ge 4$ on any quadrangulation of the torus of girth $\ge 4$. It is also ergodic for $q \ge 5$ (resp. $q \ge 3$) on any Eulerian triangulation of the torus such that one sublattice consists of degree-4 vertices while the other two sublattices induce a quadrangulation of girth $\ge 4$ (resp.~a bipartite quadrangulation) of the torus. These classes include many lattices of interest in statistical mechanics., Comment: 27 pages, pdflatex, and 22 pdf figures. Corrected an error in Remark 4 after Theorem 4.4. Final version

Cooling and heating faster a system is a crucial problem in science, technology and industry. Indeed, choosing the best thermal protocol to reach a desired temperature or energy is not a trivial task. Noticeably, we find that the phase transitions may speed up thermalization in systems where there are no conserved quantities. In particular, we show that the slow growth of magnetic domains shortens the overall time that the system takes to reach a final desired state. To prove that statement, we use intensive numerical simulations of a prototypical many-body system, namely the 2D Ising model., Comment: The document contains the paper (pdflatex, 11 pages), and 12 pdf figures. Final version to be published

A Circumnuclear Disk (CND) of molecular gas occupies the central few parsecs of the Galactic Center. It is likely subject to turbulent disruptions from violent events in its surrounding environment, but the effect of such perturbations has not yet been investigated in detail. Here we perform 3D, N-body/smoothed particle hydrodynamic (SPH) simulations with an adapted general turbulence driving method to investigate the CND's structural evolution, in particular its reaction to varied scales of injected turbulence. We find that, because of shear flow in the disk, transient arcs of gas (streams) naturally arise when turbulence is driven on large scales (up to ~4 pc), as might occur when a supernova blast wave encounters the CND. Because energetic events arise naturally and often in the central parsecs of our Galaxy, this result suggests that the transient structures that characterize the CND do not imply that the CND itself is a transient structure. We also note that features similar to the density concentrations, or "clumps", detailed in literature emerge when we account for the observed orientation of the disk and for the spatial resolution of observations. As such, clumps could be an artifact of observational limitations., Comment: 19 pages, 9 figures. Accepted for publication by ApJ

The gas dynamics in the inner few kiloparsecs of barred galaxies often results in configurations that give rise to nuclear gas rings. However, the generic dynamical description of the formation of galactic nuclear rings does not take into account the effects of thermal pressure and turbulence. Here we perform 3D hydrodynamic simulations of gas in a galactic barred potential out to a radius of $2$~kpc. We include self-gravity and a large-scale turbulence-driving module. We study how the formation of gaseous nuclear rings is affected by changing the bar pattern speed and the strength of the turbulence, and compare the results to simulations with a relatively high sound speed. We use two values for the bar pattern speed that have been discussed in the literature for our Milky Way Galaxy (40 and 63 km s$^{-1}$ kpc$^{-1}$). Our results show that turbulence produces broader and smaller nuclear rings, and enhances gas migration towards the inner few pc of the galaxy, compared to simulations without turbulence., Comment: 13 pages, 3 figures, 2 tables

We study the triangular array defined by the Graham--Knuth--Patashnik recurrence $T(n,k) = (\alpha n + \beta k + \gamma)\, T(n-1,k)+(\alpha' n + \beta' k + \gamma') \, T(n-1,k-1)$ with initial condition $T(0,k) = \delta_{k0}$ and parameters $\mathbf{\mu} = (\alpha,\beta,\gamma, \alpha',\beta',\gamma')$. We show that the family of arrays $T(\mathbf{\mu})$ is invariant under a 48-element discrete group isomorphic to $S_3 \times D_4$. Our main result is to determine all parameter sets $\mathbf{\mu} \in \mathbb{C}^6$ for which the ordinary generating function $f(x,t) = \sum_{n,k=0}^\infty T(n,k) \, x^k t^n$ is given by a Stieltjes-type continued fraction in $t$ with coefficients that are polynomials in $x$. We also exhibit some special cases in which $f(x,t)$ is given by a Thron-type or Jacobi-type continued fraction in $t$ with coefficients that are polynomials in $x$., Comment: The document contains the main paper (pdflatex, 72 pages), a Supplementary Material file (pdflatex, 31 pages, 1 pdf figure) with the detailed proof of Theorem 3.1 of the main paper, and the e-jc.sty file. Version published in journal

In this paper we study the phase diagram of the five-state Potts antiferromagnet on the bisected-hexagonal lattice. This question is important since Delfino and Tartaglia recently showed that a second-order transition in a five-state Potts antiferromagnet is allowed, and the bisected-hexagonal lattice had emerged as a candidate for such a transition on numerical grounds. By using high-precision Monte Carlo simulations and two complementary analysis methods, we conclude that there is a finite-temperature first-order transition point. This one separates a paramagnetic high-temperature phase, and a low-temperature phase where five phases coexist. This phase transition is very weak in the sense that its latent heat (per edge) is two orders of magnitude smaller than that of other well-known weak first-order phase transitions., Comment: 26 pages, pdflatex. Contains 25 pdf figures. Minor changes with respect to version 1, including a slight change in the title. Final version published in journal

Turbulence is a prevalent phenomenon in the interstellar medium, and in particular, the environment at the centers of galaxies. For example, detailed observations of the Milky Way's Central Molecular Zone (CMZ) revealed that it has a complex and turbulent structure. Turbulence on galactic scales is often modeled using star formation and feedback. However, these effects do not appear to be sufficient for explaining the high-velocity dispersion observed in the CMZ, indicating that additional gas-stirring processes are likely to be operating. Here we introduce a proof-of-concept method to drive turbulence in gas that orbits under the influence of a galactic potential. Instead of relying on a particular physical mechanism, we have adopted a Fourier forcing module and have applied it using a Smoothed Particle Hydrodynamics code. To test our method, we performed simulations of a simplistic model of the CMZ. Our turbulence injection method is capable of balancing the self-gravity of the gas, which allows us to run the simulations for long timescales and thereby follow the evolution of the CMZ. Our results show that turbulence induces a flocculent spiral pattern in our model, analogous to that found in galactic-scale simulations. Furthermore, we find that our turbulence injection method induces inward migration of gas, a result consistent with previous numerical simulations. We submit that this injection method is a promising new tool to simulate turbulence in galactic centers., Comment: 20 pages, 15 figures. Accepted to AJ

Exoplanets have been observed around stars at all stages of stellar evolution, in many cases orbiting in configurations that will eventually lead to the planets being engulfed or consumed by their host stars, such as Hot Jupiters or ultra-short period planets. Furthermore, objects such as polluted white dwarfs provide strong evidence that the consumption of planets by stars is a common phenomenon. This consumption causes several significant changes in the stellar properties, such as changes to the stellar spin, luminosity, chemical composition, or mass loss processes. Here, we explore this wide variety of effects for a comprehensive range of stellar and planetary masses and stages of stellar evolution, from the main sequence over red giants to the white dwarfs. We determine that planet consumption can cause transient luminosity features that last on the order of centuries to millennia, and that the post-consumption stellar spins can often reach break-up speeds. Furthermore, stellar moss loss can be caused by this spin-up, as well as through surface grazing interactions, leading to to the formation of unusual planetary nebula shapes or collimated stellar gas ejections. Our results highlight several observable stellar features by which the presence or previous existence of a planet around a given star can be deduced. This will provide future observational campaigns with the tools to better constrain exoplanet demographics, as well as planetary formation and evolution histories., Comment: 10 pages, 5 figures, submitted to ApJ

We introduce a new representation for the rescaled Appell polynomials and use it to obtain asymptotic expansions to arbitrary order. This representation consists of a finite sum and an integral over a universal contour (i.e. independent of the particular polynomials considered within the Appell family). We illustrate our method by studying the zero attractors for rescaled Appell polynomials. We also discuss the asymptotics to arbitrary order of the rescaled Bernoulli polynomials., Comment: 18 pages, pdflatex. Contains 7 pdf figures. Minor changes in the introduction w.r.t. v2, and some references added. Final published version

A recent study using $Hubble$ $Space$ $Telescope$ observations found periodic, high-speed, collimated ejections (or "bullets") from the star V Hya. The authors of that study proposed a model associating these bullets with the periastron passage of an unseen, substellar companion in an eccentric orbit and with an orbital period of $\sim8$ yrs. Here we propose that V Hya is part of a triple system, with a substellar companion having an orbital period of $\sim8$ yrs, and a tertiary object on a much wider orbit. In this model, the more distant object causes high-eccentricity excitations on the substellar companion's orbit via the Eccentric Kozai-Lidov mechanism. These eccentricities can reach such high values that they lead to Roche-lobe crossing, producing the observed bullet ejections via a strongly enhanced accretion episode. For example, we find that a ballistic bullet ejection mechanism can be produced by a brown-dwarf-mass companion, while magnetically driven outflows are consistent with a Jovian-mass companion. Finally, we suggest that the distant companion may reside at few a hundred AU on an eccentric orbit., Comment: Accepted to MNRAS. 10 pages, 6 figures, 1 table

We study the antiferromagnetic 3-state Potts model on general (periodic) plane quadrangulations $\Gamma$. Any quadrangulation can be built from a dual pair $(G,G^*)$. Based on the duality properties of $G$, we propose a new criterion to predict the phase diagram of this model. If $\Gamma$ is of self-dual type (i.e., if $G$ is isomorphic to its dual $G^*$), the model has a zero-temperature critical point with central charge $c=1$, and it is disordered at all positive temperatures. If $\Gamma$ is of non-self-dual type (i.e., if $G$ is not isomorphic to $G^*$), three ordered phases coexist at low temperature, and the model is disordered at high temperature. In addition, there is a finite-temperature critical point (separating these two phases) which belongs to the universality class of the ferromagnetic 3-state Potts model with central charge $c=4/5$. We have checked these conjectures by studying four (resp. seven) quadrangulations of self-dual (resp. non-self-dual) type, and using three complementary high-precision techniques: Monte-Carlo simulations, transfer matrices, and critical polynomials. In all cases, we find agreement with the conjecture. We have also found that the Wang-Swendsen-Kotecky Monte Carlo algorithm does not have (resp. does have) critical slowing down at the corresponding critical point on quadrangulations of self-dual (resp. non-self-dual) type., Comment: 55 pages, pdflatex. Contains 33 pdf figures. Uses tikz package. Minor changes with respect to v1. Final version

We provide a new criterion based on graph duality to predict whether the 3-state Potts antiferromagnet on a plane quadrangulation has a zero- or finite-temperature critical point, and its universality class. The former case occurs for quadrangulations of self-dual type, and the zero-temperature critical point has central charge $c=1$. The latter case occurs for quadrangulations of non-self-dual type, and the critical point belongs to the universality class of the 3-state Potts ferromagnet. We have tested this criterion against high-precision computations on four lattices of each type, with very good agreement. We have also found that the Wang-Swendsen-Koteck\'y algorithm has no critical slowing-down in the former case, and critical slowing-down in the latter., Comment: 6 pages, LaTeX2e. Contains 4 postscript figures. Uses revtex4-1. Final journal version

We study the phase diagram of the triangular-lattice $Q$-state Potts model in the real $(Q,v)$-plane, where $v=e^J-1$ is the temperature variable. Our first goal is to provide an obviously missing feature of this diagram: the position of the antiferromagnetic critical curve. This curve turns out to possess a bifurcation point with two branches emerging from it, entailing important consequences for the global phase diagram. We have obtained accurate numerical estimates for the position of this curve by combining the transfer-matrix approach for strip graphs with toroidal boundary conditions and the recent method of critical polynomials. The second goal of this work is to study the corresponding $A_{p-1}$ RSOS model on the torus, for integer $p=4,5,\ldots,8$. We clarify its relation to the corresponding Potts model, in particular concerning the role of boundary conditions. For certain values of $p$, we identify several new critical points and regimes for the RSOS model and we initiate the study of the flows between the corresponding field theories., Comment: 56 pages (pdflatex). Contains a .bbl file, 33 figures in pdf format, and a mathematica file triangular.m. The tex source uses IOP and tikz macros (not included). Minor changes with respect to v1. Final version published in the journal

We revisit the problem of Q-colourings of the triangular lattice using a mapping onto an integrable spin-one model, which can be solved exactly using Bethe Ansatz techniques. In particular we focus on the low-energy excitations above the eigenlevel g_2, which was shown by Baxter to dominate the transfer matrix spectrum in the Fortuin-Kasteleyn (chromatic polynomial) representation for Q_0 <= Q <= 4, where Q_0 = 3.819671... We argue that g_2 and its scaling levels define a conformally invariant theory, the so-called regime IV, which provides the actual description of the (analytically continued) colouring problem within a much wider range, namely 2 < Q <= 4. The corresponding conformal field theory is identified and the exact critical exponents are derived. We discuss their implications for the phase diagram of the antiferromagnetic triangular-lattice Potts model at non-zero temperature. Finally, we relate our results to recent observations in the field of spin-one anyonic chains., Comment: 51 pages (pdflatex). Contains a .bbl file, and 9 figures in pdf format. The tex source uses IOP and tikz macros (not included). v2 contains many changes with respect to v1. Final version published in journal

We define a new family of generalized Stirling permutations that can be interpreted in terms of ordered trees and forests. We prove that the number of generalized Stirling permutations with a fixed number of ascents is given by a natural three-parameter generalization of the well-known Eulerian numbers. We give the generating function for this new class of numbers and, in the simplest cases, we find closed formulas for them and the corresponding row polynomials. By using a non-trivial involution our generalized Eulerian numbers can be mapped onto a family of generalized Ward numbers, forming a Riordan inverse pair, for which we also provide a combinatorial interpretation., Comment: 19 pages (LaTeX2e). v2: Many changes with respect to v1: old section 3 corresponds to new sections 4 and 5, and the other sections in the new version contain new combinatorial interpretations of these numbers. v3: uses e-js.sty (included). This is the journal version, with some differences with respect v2

We consider Problem 6.94 posed in the book Concrete Mathematics by Graham, Knuth, and Patashnik, and solve it by using bivariate exponential generating functions. The family of recurrence relations considered in the problem contains many cases of combinatorial interest for particular choices of the six parameters that define it. We give a complete classification of the partial differential equations satisfied by the exponential generating functions, and solve them in all cases. We also show that the recurrence relations defining the combinatorial numbers appearing in this problem display an interesting degeneracy that we study in detail. Finally, we obtain for all cases the corresponding univariate row generating polynomials., Comment: 21 pages (LaTeX2e). Major changes with respect to version 1. Section 1 (resp. 2) is an abridged version of sections 1, 2, and 3 (resp. 4) of arXiv:1307.2010v1; section 3 contains new material, section 4 is an abridged version of section 3 of arXiv:1307.5624v1, and the appendix contains a summary of section 5 of arXiv:1307.2010v1

We study the partition function Z_{G(nk,k)}(Q,v) of the Q-state Potts model on the family of (non-planar) generalized Petersen graphs G(nk,k). We study its zeros in the plane (Q,v) for 1<= k <= 7. We also consider two specializations of Z_{G(nk,k)}, namely the chromatic polynomial P_{G(nk,k)}(Q) (corresponding to v=-1), and the flow polynomial Phi_{G(nk,k)}(Q) (corresponding to v=-Q). In these two cases, we study their zeros in the complex Q-plane for 1 <= k <= 7. We pay special attention to the accumulation loci of the corresponding zeros when n -> infinity. We observe that the Berker-Kadanoff phase that is present in two-dimensional Potts models, also exists for non-planar recursive graphs. Their qualitative features are the same; but the main difference is that the role played by the Beraha numbers for planar graphs is now played by the non-negative integers for non-planar graphs. At these integer values of Q, there are massive eigenvalue cancellations, in the same way as the eigenvalue cancellations that happen at the Beraha numbers for planar graphs., Comment: 47 pages (LaTeX2e). Includes tex file, three sty files, and 40 Postscript figures. Minor changes from version 1. Final version published in Nucl. Phys. B

We exhibit infinite families of two-dimensional lattices (some of which are triangulations or quadrangulations of the plane) on which the q-state Potts antiferromagnet has a finite-temperature phase transition at arbitrarily large values of q. This unexpected result is proven rigorously by using a Peierls argument to measure the entropic advantage of sublattice long-range order. Additional numerical data are obtained using transfer matrices, Monte Carlo simulation, and a high-precision graph-theoretic method., Comment: RevTeX, 5 pages, includes 10 Postscript figures

We show how the Hintermann-Merlini-Baxter-Wu model (which is a generalization of the well-known Baxter-Wu model to a general Eulerian triangulation) can be mapped onto a particular infinite-coupling-limit of the Ashkin-Teller model. We work out some mappings among these models, also including the standard and mixed Ashkin-Teller models. Finally, we compute the phase diagram of the infinite-coupling-limit Ashkin-Teller model on the square, triangular, hexagonal, and kagome lattices., Comment: 58 pages (LaTeX2e). Includes tex file, three sty files, and 6 Postscript figures. Minor changes from v1. Final version accepted for publication

We argue that the 4-state Potts antiferromagnet has a finite-temperature phase transition on any Eulerian plane triangulation in which one sublattice consists of vertices of degree 4. We furthermore predict the universality class of this transition. We then present transfer-matrix and Monte Carlo data confirming these predictions for the cases of the union-jack and bisected hexagonal lattices., Comment: LaTeX2e/Revtex4, 5 pages

The number of nowhere zero Z_Q flows on a graph G can be shown to be a polynomial in Q, defining the flow polynomial \Phi_G(Q). According to Tutte's five-flow conjecture, \Phi_G(5) > 0 for any bridgeless G.A conjecture by Welsh that \Phi_G(Q) has no real roots for Q \in (4,\infty) was recently disproved by Haggard, Pearce and Royle. These authors conjectured the absence of roots for Q \in [5,\infty). We study the real roots of \Phi_G(Q) for a family of non-planar cubic graphs known as generalised Petersen graphs G(m,k). We show that the modified conjecture on real flow roots is also false, by exhibiting infinitely many real flow roots Q>5 within the class G(nk,k). In particular, we compute explicitly the flow polynomial of G(119,7), showing that it has real roots at Q\approx 5.0000197675 and Q\approx 5.1653424423. We moreover prove that the graph families G(6n,6) and G(7n,7) possess real flow roots that accumulate at Q=5 as n\to\infty (in the latter case from above and below); and that Q_c(7)\approx 5.2352605291 is an accumulation point of real zeros of the flow polynomials for G(7n,7) as n\to\infty., Comment: 44 pages (LaTeX2e). Includes tex file, three sty files, and a mathematica script polyG119_7.m. Many improvements from version 3, in particular Sections 3 and 4 have been mostly re-writen, and Sections 7 and 8 have been eliminated. (This material can now be found in arXiv:1303.5210.) Final version published in J. Combin. Theory B

We study the properties of the Wang-Swendsen-Kotecky cluster Monte Carlo algorithm for simulating the 3-state kagome-lattice Potts antiferromagnet at zero temperature. We prove that this algorithm is not ergodic for symmetric subsets of the kagome lattice with fully periodic boundary conditions: given an initial configuration, not all configurations are accessible via Monte Carlo steps. The same conclusion holds for single-site dynamics., Comment: Latex2e. 22 pages. Contains 11 figures using pstricks package. Uses iopart.sty. Final version accepted in journal

We study, using transfer-matrix methods, the partition-function zeros of the square-lattice q-state Potts antiferromagnet at zero temperature (= square-lattice chromatic polynomial) for the special boundary conditions that are obtained from an m x n grid with free boundary conditions by adjoining one new vertex adjacent to all the sites in the leftmost column and a second new vertex adjacent to all the sites in the rightmost column. We provide numerical evidence that the partition-function zeros are becoming dense everywhere in the complex q-plane outside the limiting curve B_\infty(sq) for this model with ordinary (e.g. free or cylindrical) boundary conditions. Despite this, the infinite-volume free energy is perfectly analytic in this region., Comment: 114 pages (LaTeX2e). Includes tex file, three sty files, and 23 Postscript figures. Also included are Mathematica files data_Eq.m, data_Neq.m,and data_Diff.m. Many changes from version 1, including several proofs of previously conjectured results. Final version to be published in J. Stat. Phys

We prove that for the class of three-colorable triangulations of a closed oriented surface, the degree of a four-coloring modulo 12 is an invariant under Kempe changes. We use this general result to prove that for all triangulations T(3L,3M) of the torus with 3<= L <= M, there are at least two Kempe equivalence classes. This result implies in particular that the Wang-Swendsen-Kotecky algorithm for the zero-temperature 4-state Potts antiferromagnet on these triangulations T(3L,3M) of the torus is not ergodic., Comment: 37 pages (LaTeX2e). Includes tex file and 3 additional style files. The tex file includes 14 figures using pstricks.sty. Minor changes. Version published in J. Phys. A

We prove that the 3-state Potts antiferromagnet on the diced lattice (dual of the kagome lattice) has entropically-driven long-range order at low temperatures (including zero). We then present Monte Carlo simulations, using a cluster algorithm, of the 3-state and 4-state models. The 3-state model has a phase transition to the high-temperature disordered phase at v = e^J - 1 = -0.860599 +- 0.000004 that appears to be in the universality class of the 3-state Potts ferromagnet. The 4-state model is disordered throughout the physical region, including at zero temperature., Comment: 4 pages, LaTeX2e/revtex4, includes 4 figures. Version 3 (published in PRL) contains several small improvements in response to referees' comments

We derive some new structural results for the transfer matrix of square-lattice Potts models with free and cylindrical boundary conditions. In particular, we obtain explicit closed-form expressions for the dominant (at large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as the solution of a special one-dimensional polymer model. We also obtain the large-q expansion of the bulk and surface (resp. corner) free energies for the zero-temperature antiferromagnet (= chromatic polynomial) through order q^{-47} (resp. q^{-46}). Finally, we compute chromatic roots for strips of widths 9 <= m <= 12 with free boundary conditions and locate roughly the limiting curves., Comment: 111 pages (LaTeX2e). Includes tex file, three sty files, and 19 Postscript figures. Also included are Mathematica files data_CYL.m and data_FREE.m. Many changes from version 1: new material on series expansions and their analysis, and several proofs of previously conjectured results. Final version to be published in J. Stat. Phys