arXiv subjects
Publications and source records attributed to A. J. Guttmann.
We have dramatically extended the zero field susceptibility series at both high and low temperature of the Ising model on the triangular and honeycomb lattices, and used these data and newly available further terms for the square lattice to calculate a number of terms in the scaling function expansion around both the ferromagnetic and, for the square and honeycomb lattices, the antiferromagnetic critical point.
We study the class of non-holonomic power series with integer coefficients that reduce, modulo primes, or powers of primes, to algebraic functions. In particular we try to determine whether the susceptibility of the square-lattice Ising model belongs to this class, and more broadly whether the susceptibility is a solution of a differentially algebraic equation. Initial results on Tutte's non-linear ordinary differential equation (ODE) and other simple quadratic non-linear ODEs suggest that a large set of differentially algebraic power series solutions with integer coefficients might reduce to algebraic functions modulo primes, or powers of primes. Here we give several examples of series with integer coefficients and non-zero radius of convergence that reduce to algebraic functions modulo (almost) every prime (or power of a prime). These examples satisfy differentially algebraic equations with the encoding polynomial occasionally possessing quite high degree (and thus difficult to identify even with long series). Additionally, we have extended both the high- and low-temperature Ising square-lattice susceptibility series to 5043 coefficients. We find that even this long series is insufficient to determine whether it reduces to algebraic functions modulo $3$, $5$, etc. This negative result is in contrast to the comparatively easy confirmation that the corresponding series reduce to algebraic functions modulo powers of $2$.
We study the full susceptibility of the Ising model modulo powers of primes. We find exact functional equations for the full susceptibility modulo these primes. Revisiting some lesser-known results on discrete finite automata, we show that these results can be seen as a consequence of the fact that, modulo 2^r, one cannot distinguish the full susceptibility from some simple diagonals of rational functions which reduce to algebraic functions modulo 2^r, and, consequently, satisfy exact functional equations modulo 2^r. We sketch a possible physical interpretation of these functional equations modulo 2^r as reductions of a master functional equation corresponding to infinite order symmetries such as the isogenies of elliptic curves. One relevant example is the Landen transformation which can be seen as an exact generator of the Ising model renormalization group. We underline the importance of studying a new class of functions corresponding to ratios of diagonals of rational functions: they reduce to algebraic functions modulo powers of primes and they may have solutions with natural boundaries.
We obtain in exact arithmetic the order 24 linear differential operator $L_{24}$ and right hand side $E^{(5)}$ of the inhomogeneous equation$L_{24}(Φ^{(5)}) = E^{(5)}$, where $Φ^{(5)} =\tildeχ^{(5)}-\tildeχ^{(3)}/2+\tildeχ^{(1)}/120$ is a linear combination of $n$-particle contributions to the susceptibility of the square lattice Ising model. In Bostan, et al. (J. Phys. A: Math. Theor. {\bf 42}, 275209 (2009)) the operator $L_{24}$ (modulo a prime) was shown to factorize into $L_{12}^{(\rm left)} \cdot L_{12}^{(\rm right)}$; here we prove that no further factorization of the order 12 operator $L_{12}^{(\rm left)}$ is possible. We use the exact ODE to obtain the behaviour of $\tildeχ^{(5)}$ at the ferromagnetic critical point and to obtain a limited number of analytic continuations of $\tildeχ^{(5)}$ beyond the principal disk defined by its high temperature series. Contrary to a speculation in Boukraa, et al (J. Phys. A: Math. Theor. {\bf 41} 455202 (2008)), we find that $\tildeχ^{(5)}$ is singular at $w=1/2$ on an infinite number of branches.
We consider the Fuchsian linear differential equation obtained (modulo a prime) for $\tildeχ^{(5)}$, the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination of $\tildeχ^{(1)}$ and $\tildeχ^{(3)}$ can be removed from $\tildeχ^{(5)}$ and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth order linear differential operator occurs as the left-most factor of the "depleted" differential operator and it is shown to be equivalent to the symmetric fourth power of $L_E$, the linear differential operator corresponding to the elliptic integral $E$. This result generalizes what we have found for the lower order terms $\tildeχ^{(3)}$ and $\tildeχ^{(4)}$. We conjecture that a linear differential operator equivalent to a symmetric $(n-1)$-th power of $L_E$ occurs as a left-most factor in the minimal order linear differential operators for all $\tildeχ^{(n)}$'s.
We calculate very long low- and high-temperature series for the susceptibility $χ$ of the square lattice Ising model as well as very long series for the five-particle contribution $χ^{(5)}$ and six-particle contribution $χ^{(6)}$. These calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150000 CPU hours on computer clusters. For $χ^{(5)}$ 10000 terms of the series are calculated {\it modulo} a single prime, and have been used to find the linear ODE satisfied by $χ^{(5)}$ {\it modulo} a prime. A diff-Padé analysis of 2000 terms series for $χ^{(5)}$ and $χ^{(6)}$ confirms to a very high degree of confidence previous conjectures about the location and strength of the singularities of the $n$-particle components of the susceptibility, up to a small set of ``additional'' singularities. We find the presence of singularities at $w=1/2$ for the linear ODE of $χ^{(5)}$, and $w^2= 1/8$ for the ODE of $χ^{(6)}$, which are {\it not} singularities of the ``physical'' $χ^{(5)}$ and $χ^{(6)},$ that is to say the series-solutions of the ODE's which are analytic at $w =0$. Furthermore, analysis of the long series for $χ^{(5)}$ (and $χ^{(6)}$) combined with the corresponding long series for the full susceptibility $χ$ yields previously conjectured singularities in some $χ^{(n)}$, $n \ge 7$. We also present a mechanism of resummation of the logarithmic singularities of the $χ^{(n)}$ leading to the known power-law critical behaviour occurring in the full $χ$, and perform a power spectrum analysis giving strong arguments in favor of the existence of a natural boundary for the full susceptibility $χ$.
Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their `concavity index', $m$. Such polygons are called \emph{$m$-convex} polygons and are characterised by having up to $m$ indentations in their perimeter. We first describe how we conjectured the (isotropic) generating function for the case $m=2$ using a numerical procedure based on series expansions. We then proceed to prove this result for the more general case of the full anisotropic generating function, in which steps in the $x$ and $y$ direction are distinguished. In so doing, we develop tools that would allow for the case $m > 2$ to be studied. %In our proof we use a `divide and conquer' approach, factorising 2-convex %polygons by extending a line along the base of its indents. We then use %the inclusion-exclusion principle, the Hadamard product and extensions to %known methods to derive the generating functions for each case.
Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their `concavity index', $m$. Such polygons are called \emph{$m$-convex} polygons and are characterised by having up to $m$ indentations in the side. We use a `divide and conquer' approach, factorising 2-convex polygons by extending a line along the base of its indents. We then use the inclusion-exclusion principle, the Hadamard product and extensions to known methods to derive the generating functions for each case.
We study a restricted class of self-avoiding walks (SAW) which start at the origin (0, 0), end at $(L, L)$, and are entirely contained in the square $[0, L] \times [0, L]$ on the square lattice ${\mathbb Z}^2$. The number of distinct walks is known to grow as $λ^{L^2+o(L^2)}$. We estimate $λ= 1.744550 \pm 0.000005$ as well as obtaining strict upper and lower bounds, $1.628 < λ< 1.782.$ We give exact results for the number of SAW of length $2L + 2K$ for $K = 0, 1, 2$ and asymptotic results for $K = o(L^{1/3})$. We also consider the model in which a weight or {\em fugacity} $x$ is associated with each step of the walk. This gives rise to a canonical model of a phase transition. For $x < 1/μ$ the average length of a SAW grows as $L$, while for $x > 1/μ$ it grows as $L^2$. Here $μ$ is the growth constant of unconstrained SAW in ${\mathbb Z}^2$. For $x = 1/μ$ we provide numerical evidence, but no proof, that the average walk length grows as $L^{4/3}$. We also consider Hamiltonian walks under the same restriction. They are known to grow as $τ^{L^2+o(L^2)}$ on the same $L \times L$ lattice. We give precise estimates for $τ$ as well as upper and lower bounds, and prove that $τ< λ.$
We analyse new exact enumeration data for self-avoiding polygons, counted by perimeter and area on the square, triangular and hexagonal lattices. In extending earlier analyses, we focus on the perimeter moments in the vicinity of the bicritical point. We also consider the shape of the critical curve near the bicritical point, which describes the crossover to the branched polymer phase. Our recently conjectured expression for the scaling function of rooted self-avoiding polygons is further supported. For (unrooted) self-avoiding polygons, the analysis reveals the presence of an additional additive term with a new universal amplitude. We conjecture the exact value of this amplitude.
Poland-Scheraga models were introduced to describe the DNA denaturation transition. We give a rigorous and refined discussion of a family of these models. We derive possible scaling functions in the neighborhood of the phase transition point and review common examples. We introduce a self-avoiding Poland-Scheraga model displaying a first order phase transition in two and three dimensions. We also discuss exactly solvable directed examples. This complements recent suggestions as to how the Poland-Scheraga class might be extended in order to display a first order transition, which is observed experimentally.
We enumerate self-avoiding walks and polygons, counted by perimeter, on the quasiperiodic rhombic Penrose and Ammann-Beenker tilings, thereby considerably extending previous results. In contrast to similar problems on regular lattices, these numbers depend on the chosen start vertex. We compare different ways of counting and demonstrate that suitable averaging improves converge to the asymptotic regime. This leads to improved estimates for critical points and exponents, which support the conjecture that self-avoiding walks on quasiperiodic tilings belong to the same universality class as self-avoiding walks on the square lattice. For polygons, the obtained e numeration data does not allow to draw decisive conclusions about the exponent.
For the study of Ising models of general spin S on the square lattice, we have combined our recently extended high-temperature expansions with the low-temperature expansions derived some time ago by Enting, Guttmann and Jensen. We have computed for the first time various critical parameters and improved the estimates of others. Moreover the properties of hyperscaling and of universality (spin S independence) of exponents and of various dimensionless amplitude combinations have been verified accurately. Assuming the validity of the lattice-lattice scaling, from our estimates of critical amplitudes for the square lattice we have also obtained estimates of the corresponding amplitudes for the spin S Ising model on the triangular, honeycomb, and kagome` lattices.
Exactly solvable models of planar polygons, weighted by perimeter and area, have deepened our understanding of the critical behaviour of polygon models in recent years. Based on these results, we derive a conjecture for the exact form of the critical scaling function for planar self-avoiding polygons. The validity of this conjecture was recently tested numerically using exact enumeration data for small values of the perimeter on the square and triangular lattices. We have substantially extended these enumerations and also enumerated polygons on the hexagonal lattice. We also performed Monte-Carlo simulations of the model on the square lattice. Our analysis supports the conjecture that the scaling function is given by the logarithm of an Airy function.
The perimeter and area generating functions of exactly solvable polygon models satisfy q-functional equations, where q is the area variable. The behaviour in the vicinity of the point where the perimeter generating function diverges can often be described by a scaling function. We develop the method of q-linear approximants in order to extract the approximate scaling behaviour of polygon models when an exact solution is not known. We test the validity of our method by approximating exactly solvable q-linear polygon models. This leads to scaling functions for a number of q-linear polygon models, notably generalized rectangles, Ferrers diagrams, and stacks.
We analyze new data for self-avoiding polygons, on the square and triangular lattices, enumerated by both perimeter and area, providing evidence that the scaling function is the logarithm of an Airy function. The results imply universal amplitude combinations for all area moments and suggest that rooted self-avoiding polygons may satisfy a $q$-algebraic functional equation.
We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N^6) to determine its first N coefficients. As a result, we have generated and analyzed series with more than 300 terms in both the high- and low-temperature regime. We quantify the effect of irrelevant variables to the scaling-amplitude functions. In particular, we find and quantify the breakdown of simple scaling, in the absence of irrelevant scaling fields, arising first at order |T-T_c|^{9/4}, though high-low temperature symmetry is still preserved. At terms of order |T-T_c|^{17/4} and beyond, this symmetry is no longer present. The short-distance terms are shown to have the form (T-T_c)^p(log|T-T_c|)^q with p \ge q^2. Conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments.
We report computations of the short-distance and the long-distance (scaling) contributions to the square-lattice Ising susceptibility in zero field close to T_c. Both computations rely on the use of nonlinear partial difference equations for the correlation functions. By summing the correlation functions, we give an algorithm of complexity O(N^6) for the determination of the first N series coefficients. Consequently, we have generated and analysed series of length several hundred terms, generated in about 100 hours on an obsolete workstation. In terms of a temperature variable, τ, linear in T/T_c-1, the short-distance terms are shown to have the form τ^p(ln|τ|)^q with p>=q^2. To O(τ^14) the long-distance part divided by the leading τ^{-7/4} singularity contains only integer powers of τ. The presence of irrelevant variables in the scaling function is clearly evident, with contributions of distinct character at leading orders |τ|^{9/4} and |τ|^{17/4} being identified.