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Anthony J. Guttmann

Publications and source records attributed to Anthony J. Guttmann.

36 records · Page 2Linked to original sources
arXiv2012

An integral arising from the chiral sl(n) Potts model

We show that the integral $J(t) = (1/π^3) \int_0^π\int_0^π\int_0^πdx dy dz \log(t - \cos{x} - \cos{y} - \cos{z} + \cos{x}\cos{y}\cos{z})$, can be expressed in terms of ${_5F_4}$ hypergeometric functions. The integral arises in the solution by Baxter and Bazhanov of the free-energy of the $sl(n)$ Potts model, which includes the term $J(2)$. Our result immediately gives the logarithmic Mahler measure of the Laurent polynomial $k - (x+1/x) - (y+1/y) - (z+1/z) + 1/4(x+1/x) (y+1/y) (z+1/z)$ in terms of the same hypergeometric functions.

math-ph↗
arXiv2012

Off-critical parafermions and the winding angle distribution of the O($n$) model

Using an off-critical deformation of the identity of Duminil-Copin and Smirnov, we prove a relationship between half-plane surface critical exponents $γ_1$ and $γ_{11}$ as well as wedge critical exponents $γ_2(α)$ and $γ_{21}(α)$ and the exponent characterising the winding angle distribution of the O($n$) model in the half-plane, or more generally in a wedge of wedge-angle $α.$ We assume only the existence of these exponents and, for some values of $n,$ the conjectured value of the critical point. If we assume their values as predicted by conformal field theory, one gets complete agreement with the conjectured winding angle distribution, as obtained by CFT and Coulomb gas arguments. We also prove the exponent inequality $γ_1-γ_{11} \ge 1,$ and its extension $γ_2(α)-γ_{21}(α) \ge 1$ for the edge exponents. We provide conjectured values for all exponents for $n \in [-2,2).$

math-ph↗
arXiv2011

Two-dimensional self-avoiding walks and polymer adsorption: Critical fugacity estimates

Recently Beaton, de Gier and Guttmann proved a conjecture of Batchelor and Yung that the critical fugacity of self-avoiding walks interacting with (alternate) sites on the surface of the honeycomb lattice is $1+\sqrt{2}$. A key identity used in that proof depends on the existence of a parafermionic observable for self-avoiding walks interacting with a surface on the honeycomb lattice. Despite the absence of a corresponding observable for SAW on the square and triangular lattices, we show that in the limit of large lattices, some of the consequences observed for the honeycomb lattice persist irrespective of lattice. This permits the accurate estimation of the critical fugacity for the corresponding problem for the square and triangular lattices. We consider both edge and site weighting, and results of unprecedented precision are achieved. We also \emph{prove} the corresponding result fo the edge-weighted case for the honeycomb lattice.

math-ph↗
arXiv2010

Form factor expansions in the 2D Ising model and Painlevé VI

We derive a Toda-type recurrence relation, in both high and low temperature regimes, for the $λ$ - extended diagonal correlation functions $C(N,N;λ)$ of the two-dimensional Ising model, using an earlier connection between diagonal form factor expansions and tau-functions within Painlevé VI (PVI) theory, originally discovered by Jimbo and Miwa. This greatly simplifies the calculation of the diagonal correlation functions, particularly their $λ$-extended counterparts. We also conjecture a closed form expression for the simplest off-diagonal case $C^{\pm}(0,1;λ)$ where a connection to PVI theory is not known. Combined with the results for diagonal correlations these give all the initial conditions required for the $ł$-extended version of quadratic difference equations for the correlation functions discovered by McCoy, Perk and Wu. The results obtained here should provide a further potential algorithmic improvement in the $ł$-extended case, and facilitate other developments.

math-ph↗
arXiv2009

Polyominoes with nearly convex columns: An undirected model

Column-convex polyominoes were introduced in 1950's by Temperley, a mathematical physicist working on "lattice gases". By now, column-convex polyominoes are a popular and well-understood model. There exist several generalizations of column-convex polyominoes; an example is a model called multi-directed animals. In this paper, we introduce a new sequence of supersets of column-convex polyominoes. Our model (we call it level m column-subconvex polyominoes) is defined in a simple way. We focus on the case when cells are hexagons and we compute the area generating functions for the levels one and two. Both of those generating functions are complicated q-series, whereas the area generating function of column-convex polyominoes is a rational function. The growth constants of level one and level two column-subconvex polyominoes are 4.319139 and 4.509480, respectively. For comparison, the growth constants of column-convex polyominoes, multi-directed animals and all polyominoes are 3.863131, 4.587894 and 5.183148, respectively.

math.CO↗
arXiv2008

Prudent walks and polygons

We have produced extended series for two-dimensional prudent polygons, based on a transfer matrix algorithm of complexity O$(n^5),$ for a series of length $n.$ We have extended the definition to three dimensions and produced series expansions for both prudent walks and polygons in three dimensions. For prudent polygons in two dimensions we find the growth constant to be smaller than that for the corresponding walks, and by considering three distinct classes of polygons, we find that the growth constant for polygons varies with class, while for walks it does not. We give the critical exponent for both walks and polygons. In the three-dimensional case we estimate the growth constant for both walks and polygons and also estimate the usual critical exponents $γ,$ $ν$ and $α.$

cond-mat.stat-mech↗
arXiv2008

Area distribution and scaling function for punctured polygons

Punctured polygons are polygons with internal holes which are also polygons. The external and internal polygons are of the same type, and they are mutually as well as self-avoiding. Based on an assumption about the limiting area distribution for unpunctured polygons, we rigorously analyse the effect of a finite number of punctures on the limiting area distribution in a uniform ensemble, where punctured polygons with equal perimeter have the same probability of occurrence. Our analysis leads to conjectures about the scaling behaviour of the models. We also analyse exact enumeration data. For staircase polygons with punctures of fixed size, this yields explicit expressions for the generating functions of the first few area moments. For staircase polygons with punctures of arbitrary size, a careful numerical analysis yields very accurate estimates for the area moments. Interestingly, we find that the leading correction term for each area moment is proportional to the corresponding area moment with one less puncture. We finally analyse corresponding quantities for punctured self-avoiding polygons and find agreement with the conjectured formulas to at least 3-4 significant digits.

math.CO↗
arXiv2007

Modeling force-induced bio-polymer unfolding

We study the conformations of polymer chains in a poor solvent, with and without bending rigidity, by means of a simple statistical mechanics model. This model can be exactly solved for chains of length up to N=55 using exact enumeration techniques. We analyze in details the differences between the constant force and constant distance ensembles for large but finite N. At low temperatures, and in the constant force ensemble, the force-extension curve shows multiple plateaus (intermediate states), in contrast with the abrupt transition to an extended state prevailing in the $N \to \infty$ limit. In the constant distance ensemble, the same curve provides a unified response to pulling and compressing forces, and agrees qualitatively with recent experimental results. We identify a cross-over length, proportional to $N$, below which the critical force of unfolding decreases with temperature, while above, it increases wiyh temperature. Finally, the force-extension curve for stiff chains exhibits "saw-tooth" like behavior, as observed in protein unfolding experiments.

cond-mat.stat-mech↗
arXiv2007

Role of conformational entropy in force-induced bio-polymer unfolding

A statistical mechanical description of flexible and semi-flexible polymer chains in a poor solvent is developed in the constant force and constant distance ensembles. We predict the existence of many intermediate states at low temperatures stabilized by the force. A unified response to pulling and compressing forces has been obtained in the constant distance ensemble. We show the signature of a cross-over length which increases linearly with the chain length. Below this cross-over length, the critical force of unfolding decreases with temperature, while above, it increases with temperature. For stiff chains, we report for the first time "saw-tooth" like behavior in the force-extension curves which has been seen earlier in the case of protein unfolding.

cond-mat.soft↗
arXiv2006

The perimeter generating function of punctured staircase polygons

Using a simple transfer matrix approach we have derived very long series expansions for the perimeter generating function of punctured staircase polygons (staircase polygons with a single internal staircase hole). We find that all the terms in the generating function can be reproduced from a linear Fuchsian differential equation of order 8. We perform an analysis of the properties of the differential equation.

cond-mat.stat-mech↗
arXiv2005

Correction-to-scaling exponents for two-dimensional self-avoiding walks

We study the correction-to-scaling exponents for the two-dimensional self-avoiding walk, using a combination of series-extrapolation and Monte Carlo methods. We enumerate all self-avoiding walks up to 59 steps on the square lattice, and up to 40 steps on the triangular lattice, measuring the mean-square end-to-end distance, the mean-square radius of gyration and the mean-square distance of a monomer from the endpoints. The complete endpoint distribution is also calculated for self-avoiding walks up to 32 steps (square) and up to 22 steps (triangular). We also generate self-avoiding walks on the square lattice by Monte Carlo, using the pivot algorithm, obtaining the mean-square radii to ~0.01% accuracy up to N = 4000. We give compelling evidence that the first non-analytic correction term for two-dimensional self-avoiding walks is Delta_1 = 3/2. We compute several moments of the endpoint distribution function, finding good agreement with the field-theoretic predictions. Finally, we study a particular invariant ratio that can be shown, by conformal-field-theory arguments, to vanish asymptotically, and we find the cancellation of the leading analytic correction.

cond-mat.stat-mech↗
arXiv2004

Self-avoiding walks and trails on the 3.12 lattice

We find the generating function of self-avoiding walks and trails on a semi-regular lattice called the $3.12^2$ lattice in terms of the generating functions of simple graphs, such as self-avoiding walks, polygons and tadpole graphs on the hexagonal lattice. Since the growth constant for these graphs is known on the hexagonal lattice we can find the growth constant for both walks and trails on the $3.12^2$ lattice. A result of Watson then allows us to find the generating function and growth constant of neighbour-avoiding walks on the covering lattice of the $3.12^2$ lattice which is tetra-valent. A mapping into walks on the covering lattice allows us to obtain improved bounds on the growth constant for a range of lattices.

cond-mat.stat-mech↗
arXiv2002

Vicious walkers, friendly walkers and Young tableaux III: Between two walls

We derive exact and asymptotic results for the number of star and watermelon configurations of vicious walkers confined to lie between two impenetrable walls, as well as for the analogous problem for $\infty$-friendly walkers. Our proofs make use of results from symmetric function theory and the theory of basic hypergeometric series.

cond-mat.stat-mech↗
arXiv2000

Vicious walkers, friendly walkers and Young tableaux II: With a wall

We derive new results for the number of star and watermelon configurations of vicious walkers in the presence of an impenetrable wall by showing that these follow from standard results in the theory of Young tableaux, and combinatorial descriptions of symmetric functions. For the problem of $n$-friendly walkers, we derive exact asymptotics for the number of stars and watermelons both in the absence of a wall and in the presence of a wall.

cond-mat.stat-mech↗
arXiv1995

Series expansions of the percolation probability on the directed triangular lattice

We have derived long series expansions of the percolation probability for site, bond and site-bond percolation on the directed triangular lattice. For the bond problem we have extended the series from order 12 to 51 and for the site problem from order 12 to 35. For the site-bond problem, which has not been studied before, we have derived the series to order 32. Our estimates of the critical exponent $β$ are in full agreement with results for similar problems on the square lattice, confirming expectations of universality. For the critical probability and exponent we find in the site case: $q_c = 0.4043528 \pm 0.0000010$ and $β= 0.27645 \pm 0.00010$; in the bond case: $q_c = 0.52198\pm 0.00001$ and $β= 0.2769\pm 0.0010$; and in the site-bond case: $q_c = 0.264173 \pm 0.000003$ and $β= 0.2766 \pm 0.0003$. In addition we have obtained accurate estimates for the critical amplitudes. In all cases we find that the leading correction to scaling term is analytic, i.e., the confluent exponent $Δ= 1$.

cond-mat↗
arXiv1995

Series expansions of the percolation probability for directed square and honeycomb lattices

We have derived long series expansions of the percolation probability for site and bond percolation on directed square and honeycomb lattices. For the square bond problem we have extended the series from 41 terms to 54, for the square site problem from 16 terms to 37, and for the honeycomb bond problem from 13 terms to 36. Analysis of the series clearly shows that the critical exponent $β$ is the same for all the problems confirming expectations of universality. For the critical probability and exponent we find in the square bond case, $q_c = 0.3552994\pm 0.0000010$, $β= 0.27643\pm 0.00010$, in the square site case $q_c = 0.294515 \pm 0.000005$, $β= 0.2763 \pm 0.0003$, and in the honeycomb bond case $q_c = 0.177143 \pm 0.000002$, $β= 0.2763 \pm 0.0002$. In addition we have obtained accurate estimates for the critical amplitudes. In all cases we find that the leading correction to scaling term is analytic, i.e., the confluent exponent $Δ= 1$.

cond-mat↗
arXiv1993

Universal Amplitude Combinations for Self-Avoiding Walks, Polygons and Trails

We give exact relations for a number of amplitude combinations that occur in the study of self-avoiding walks, polygons and lattice trails. In particular, we elucidate the lattice-dependent factors which occur in those combinations which are otherwise universal, show how these are modified for oriented lattices, and give new results for amplitude ratios involving even moments of the area of polygons. We also survey numerical results for a wide range of amplitudes on a number of oriented and regular lattices, and provide some new ones.

cond-mat↗