We describe a novel algorithm for random sampling of freely reduced words equal to the identity in a finitely presented group. The algorithm is based on Metropolis Monte Carlo sampling. The algorithm samples from a stretched Boltzmann distribution be
gin{align*}pi(w) &= (|w|+1)^{alpha} beta^{|w|} cdot Z^{-1} end{align*} where $|w|$ is the length of a word $w$, $alpha$ and $beta$ are parameters of the algorithm, and $Z$ is a normalising constant. It follows that words of the same length are sampled with the same probability. The distribution can be expressed in terms of the cogrowth series of the group, which then allows us to relate statistical properties of words sampled by the algorithm to the cogrowth of the group, and hence its amenability. We have implemented the algorithm and applied it to several group presentations including the Baumslag-Solitar groups, some free products studied by Kouksov, a finitely presented amenable group that is not subexponentially amenable (based on the basilica group), and Richard Thompsons group $F$.
We compute the cogrowth series for Baumslag-Solitar groups $mathrm{BS}(N,N) = < a,b | a^N b = b a^N > $, which we show to be D-finite. It follows that their cogrowth rates are algebraic numbers.
We propose a numerical method for studying the cogrowth of finitely presented groups. To validate our numerical results we compare them against the corresponding data from groups whose cogrowth series are known exactly. Further, we add to the set of
such groups by finding the cogrowth series for Baumslag-Solitar groups $mathrm{BS}(N,N) = < a,b | a^N b = b a^N >$ and prove that their cogrowth rates are algebraic numbers.
In this paper the number and lengths of minimal length lattice knots confined to slabs of width $L$, is determined. Our data on minimal length verify the results by Sharein et.al. (2011) for the similar problem, expect in a single case, where an impr
ovement is found. From our data we construct two models of grafted knotted ring polymers squeezed between hard walls, or by an external force. In each model, we determine the entropic forces arising when the lattice polygon is squeezed by externally applied forces. The profile of forces and compressibility of several knot types are presented and compared, and in addition, the total work done on the lattice knots when it is squeezed to a minimal state is determined.
An implementation of BFACF-style algorithms on knotted polygons in the simple cubic, face centered cubic and body centered cubic lattice is used to estimate the statistics and writhe of minimal length knotted polygons in each of the lattices. Data ar
e collected and analysed on minimal length knotted polygons, their entropy, and their lattice curvature and writhe.
Let $p_n$ denote the number of self-avoiding polygons of length $n$ on a regular three-dimensional lattice, and let $p_n(K)$ be the number which have knot type $K$. The probability that a random polygon of length $n$ has knot type $K$ is $p_n(K)/p_n$
and is known to decay exponentially with length. Little is known rigorously about the asymptotics of $p_n(K)$, but there is substantial numerical evidence that $p_n(K)$ grows as $p_n(K) simeq , C_K , mu_emptyset^n , n^{alpha-3+N_K}$, as $n to infty$, where $N_K$ is the number of prime components of the knot type $K$. It is believed that the entropic exponent, $alpha$, is universal, while the exponential growth rate, $mu_emptyset$, is independent of the knot type but varies with the lattice. The amplitude, $C_K$, depends on both the lattice and the knot type. The above asymptotic form implies that the relative probability of a random polygon of length $n$ having prime knot type $K$ over prime knot type $L$ is $frac{p_n(K)/p_n}{p_n(L)/p_n} = frac{p_n(K)}{p_n(L)} simeq [ frac{C_K}{C_L} ]$. In the thermodynamic limit this probability ratio becomes an amplitude ratio; it should be universal and depend only on the knot types $K$ and $L$. In this letter we examine the universality of these probability ratios for polygons in the simple cubic, face-centered cubic, and body-centered cubic lattices. Our results support the hypothesis that these are universal quantities. For example, we estimate that a long random polygon is approximately 28 times more likely to be a trefoil than be a figure-eight, independent of the underlying lattice, giving an estimate of the intrinsic entropy associated with knot types in closed curves.
In this paper the elementary moves of the BFACF-algorithm for lattice polygons are generalised to elementary moves of BFACF-style algorithms for lattice polygons in the body-centred (BCC) and face-centred (FCC) cubic lattices. We prove that the ergod
icity classes of these new elementary moves coincide with the knot types of unrooted polygons in the BCC and FCC lattices and so expand a similar result for the cubic lattice. Implementations of these algorithms for knotted polygons using the GAS algorithm produce estimates of the minimal length of knotted polygons in the BCC and FCC lattices.
We show that the classical Rosenbluth method for sampling self-avoiding walks can be extended to a general algorithm for sampling many families of objects, including self-avoiding polygons. The implementation relies on an elementary move which is a g
eneralisation of kinetic growth; rather than only appending edges to the endpoint, edges may be inserted at any vertex providing the resulting objects still lie within the same family. We implement this method using pruning and enrichment to sample self-avoiding walks and polygons. The algorithm can be further extended by mixing it with length-preserving moves, such pivots and crank-shaft moves.
The exact solution of directed self-avoiding walks confined to a slit of finite width and interacting with the walls of the slit via an attractive potential has been calculated recently. The walks can be considered to model the polymer-induced steric
stabilisation and sensitised floculation of colloidal dispersions. The large width asymptotics led to a phase diagram different to that of a polymer attached to, and attracted to, a single wall. The question that arises is: can one interpolate between the single wall and two wall cases? In this paper we calculate the exact scaling functions for the partition function by considering the two variable asymptotics of the partition function for simultaneous large length and large width. Consequently, we find the scaling functions for the force induced by the polymer on the walls. We find that these scaling functions are given by elliptic theta-functions. In some parts of the phase diagram there is more a complex crossover between the single wall and two wall cases and we elucidate how this happens.
We investigate the existence and location of the surface phase known as the Surface-Attached Globule (SAG) conjectured previously to exist in lattice models of three-dimensional polymers when they are attached to a wall that has a short range potenti
al. The bulk phase, where the attractive intra-polymer interactions are strong enough to cause a collapse of the polymer into a liquid-like globule and the wall either has weak attractive or repulsive interactions, is usually denoted Desorbed-Collapsed or DC. Recently this DC phase was conjectured to harbour two surface phases separated by a boundary where the bulk free energy is analytic while the surface free energy is singular. The surface phase for more attractive values of the wall interaction is the SAG phase. We discuss more fully the properties of this proposed surface phase and provide Monte Carlo evidence for self-avoiding walks up to length 256 that this surface phase most likely does exist. Importantly, we discuss alternatives for the surface phase boundary. In particular, we conclude that this boundary may lie along the zero wall interaction line and the bulk phase boundaries rather than any new phase boundary curve.
