Lattice paths: vicious walkers and friendly walkers

https://doi.org/10.1016/S0378-3758(01)00158-6Get rights and content

Abstract

We introduce a model of friendly walkers which generalises the well-known vicious walker model. Friendly walkers refers to a model in which any number P of directed lattice paths, starting at adjacent lattice sites, simultaneously proceed in one of the allowed lattice directions. In the case of n-friendly walkers the paths may stay together for n vertices. The previously considered case of vicious walkers corresponds to the case n=0. The Gessel–Viennot theorem applies only to vicious walkers, and not to the cases n>0. The connection between this model and the m-vertex models of Statistical Mechanics is described. For planar configurations, we solve the two-walker case for all n. Conjectured solutions for the three-walker case with n=1 are also obtained. Numerical studies lead to the conjectured asymptotic behaviour for all n, for an arbitrary number of walkers P and in arbitrary spatial dimension d⩾2.

Introduction

In an earlier paper (Essam and Guttmann, 1995) the problem of vicious random walkers on a d-dimensional directed lattice was considered. “Vicious walkers” describes the situation in which two or more walkers arriving at the same lattice site annihilate one another. Accordingly, the only allowed configurations are those in which contacts are forbidden. Alternatively expressed as a static rather than a dynamic problem, vicious walkers are mutually self-avoiding networks of directed lattice walks—that is directed lattice paths—which in turn model directed polymer networks.

The problem of vicious walkers was introduced to the mathematical physics literature by Fisher (1984) who also discussed a number of physical applications. The general model is one of P random walkers on a d-dimensional lattice, who at regular time intervals simultaneously take one step with equal probability in the direction of one of the allowed lattice vectors. In the combinatorics literature, an equivalent problem for any planar graph was treated by Gessel and Viennot (1985) though some of their results had been anticipated in other settings (Karlin and McGregor, 1959; Lindström, 1973).

In considering lattice path problems in which the Gessel–Viennot formulation does not apply, Viennot (private communication) suggested a model where the mutual avoidance constraint is relaxed to the extent that two paths may share a site, and may even stay together for just one step, but must then diverge—though they may subsequently touch. We now call this model “the 2-friendly-walk model” since the number of consecutive sites the walkers are allowed to share, here two, distinguishes this model from vicious walkers, which are not allowed to share any consecutive sites. It also distinguishes it from the so-called osculating walkers, which form directed lattice paths that can share one, but no more than one, consecutive sites. This nomenclature also leads us to make further natural generalisations, given below.

Interestingly, each of the vicious, osculating and 2-friendly-walk models can be formulated as lattice statistical mechanical vertex models, which are models of ferroelectric materials (see Guttmann et al., 1998). In particular the vicious, osculating and 2-friendly-walk models can be mapped to five, six and ten vertex models, respectively, as shown in Fig. 1.

We now make the natural generalisation of the vicious, osculating and 2-friendly-walk models to models where the walks are allowed to share up to n consecutive sites. These models are then referred to as the “n-friendly-walk models”. It is important to note that in any of the models considered in this paper, no more than two walkers are allowed to share any site. In our naming scheme the vicious and osculating walker models become the 0- and 1-friendly-walk models respectively. As the parameter n increases, from n=0 corresponding to vicious walkers, n=1 corresponding to osculating walkers, the walkers may be thought of as becoming increasingly friendly. Hence we shall refer to the n-friendly-walk models other than vicious walkers as partially-friendly walkers, the degree of friendliness being labelled by n.

In this paper we shall consider the n-friendly-walk model on the square, and more generally on the directed d-dimensional body-centered hyper-cubic lattices. To define the n-friendly-walk model on the directed square lattice rotated through 45°, so that the unit vectors on the lattice are (i+j)/2 and (ij)/2, let us consider P lattice paths labelled as p=1,…,P of length L (see Fig. 2) which start at l=0 at vertical positions zp,0 having intermediate positions zp,l after l steps and finishing at l=L at zp,L. These P walks never cross so that they are always ordered with zp,lzp′,l for p<p′. The conditions for the n-friendly-walk model can then be expressed as follows:

(1) All paths must fulfill the non-crossing condition, that is zp,lzp+1,l for 1⩽p⩽P−1,0⩽l⩽L.

(2) They must satisfy the n-consecutive-site or n-friendly condition. That is two walkers may stay together for at most n consecutive sites. For 0⩽l⩽L−n,1⩽p⩽P−1, there exists at least one k∈[0,n] such that zp,l+kzp+1,l+k.

(3) One must also exclude three walkers occupying the same site: the no-three-walker condition. This is achieved by requiring zp,l<zp+2,l for 1⩽pP−2 and 0⩽lL.

Note that for mn any m-friendly-walk configuration is also a valid n-friendly-walk configuration, so by increasing n one increases the size of the configuration space of the model.

We consider in this paper one further model, which for convenience we shall call the ∞-friendly-walk model. In this model two adjacent walks may share any number of sites; that is, we simply remove the condition (2) above from our definition of n-friendly-walks. Intriguingly, the ∞-friendly-walk model, in contrast to the n-friendly-walk models with n⩾3, though in common with the n-friendly-walk models with n=0,1,2, as described above, can be formulated as a vertex model, in this case with fourteen vertices. The four vertices that occur in addition to those of the 2-friendly-walk model are shown in Fig. 3 We point out that the ∞-friendly-walk model differs by virtue of the “no-three-walker condition” (3) from a model where walks are directed and stay ordered vertically at each time step but have any number of walks occupying the same site. This latter model, in which only condition (1) above is obeyed, may simply be called the “non-crossing walker model”. We also note in passing that by allowing more than two walkers at one site but keeping the restriction of two walkers per bond, one obtains a walk model that can be mapped to a nineteen vertex model.

Two standard topologies of interest for networks of lattice paths are that of a star and a watermelon. Again consider a directed square lattice rotated 45°. Both configurations consist of P chains of length L which start at (0,0),(0,2),(0,4),…,(0,2P−2). Watermelon configurations end at (L,k),(L,k+2),(L,k+4),…,(L,k+2P−2). For stars, the end-points of the chain all have l-coordinate equal to L, but the z-coordinates are unconstrained, apart from the ordering imposed by the ordering of the walks. Thus if the end-points are (L,z1,L),(L,z2,L),…,(L,zP,L), then −Lz1,Lz2,L⩽⋯⩽zP,L⩽2P−2+L, with zp,l denoting the y-coordinate of the pth walker after the lth step. Note that for vicious walkers these inequalities would be strict (apart from the first and the last). We shall use SP,model and WP,model to denote the generating functions for stars and watermelons respectively where the subscript “model” is given in this paper by either the integer n to refer to the n-friendly-walk model or the symbol ∞ to refer to the ∞-friendly-walk model. In particular we will consider generating functions for both isotropic and anisotropic models on the directed, rotated square lattice. In the first case we denote them SP,model(t) and WP,model(t), where t is the variable associated with the length of the walks. HenceSP,model(t)=L=0sP,model,LtLandWP,model(t)=L=0wP,model,LtL,where sP,model,L and wP,model,L are the numbers of valid star and watermelon configurations respectively of P walks of length L in the model described by the associated subscript. In the anisotropic case we distinguish between NE and SE steps. In that case we write the generating functions as SP,model(x,y) and WP,model(x,y) where x(y) is associated with the number of NE (SE) steps in the graphs. HenceSP,model(x,y)=j,k=0sP,model,j,kxjykandWP,model(x,y)=j,k=0wP,model,j,kxjyk,where sP,model,j,k and wP,model,j,k are the numbers of valid star and watermelon configurations respectively of P walks with j NE steps and k SE steps in total in the model described by the associated subscript. Note that we associate the variables in the anisotropic generating function with the steps in the graph and in the isotropic generating function we associate the variable with the length of the walks. Hence sP,model,j,k≠0 and wP,model,j,k≠0 only if j+k=0(modP). This implies that one can obtain the isotropic generating function from the anisotropic one with the mappings xt1/P and yt1/P.

This paper considers the calculation of the generating functions of isotropic and anisotropic cases of the n-friendly-walk models, for all n, and the ∞-friendly-walk model on the directed square lattice. We concentrate on the cases n=0,1,2 and ∞-friendly. We also later summarise the known results from higher dimensional analogues on directed d-dimensional hyper-cubic lattices. Firstly we briefly mention previous work in this area.

Essam and Guttmann (1995) obtained recurrence relations, and the corresponding differential equations, for certain stars and watermelons, both on the directed square lattice and on the general d-dimensional directed hyper-cubic lattice. In two dimensions, the Gessel–Viennot determinant for watermelons was evaluated by standard techniques, while in the case of stars the results obtained were conjectural. More recently Guttmann et al. (1998) showed that the conjectured result for stars follows from the q→1 limit of the Bender–Knuth theorem, the first published proof of which appears in Gordon (1983). Further, the connections between the celebrated 6-vertex model of statistical mechanics, and these vicious walker problems were discussed. Connections between the vicious walker problem and Young tableaux and integer partitions were also presented. In a subsequent paper (Guttmann et al., 2000) analogous results for stars and watermelons adjacent to an impenetrable wall were developed.

We begin our investigation in the following section by dealing with stars and watermelons with two walks (P=2) in the n-friendly-walk models, for all n∈[0,1,2,3,…) and in the ∞-friendly-walk model. We solve every case for the anisotropic generating functions by decomposing the graphs involved into parts whose generating functions are known.

In Section 3 we briefly summarise the computational method, based upon the series analysis technique of differential approximants, that we utilise in later sections to conjecture exact solutions as well as the location and nature of the singularities of the generating functions of interest.

Section 4 deals with three walkers, firstly with the isotropic cases and then with the anisotropic cases of our models. In both cases we analyse stars and watermelons in the vicious, osculating and 2-friendly-walk models. For the isotropic model we add to the known results for vicious walkers an analytical generating function for osculating stars and a differential equation for osculating watermelons. Further we use the techniques described in Section 3 to guess the exponents for the 2-friendly-walk model. In the second part, where we deal with the anisotropic model, we were not able to find a closed form solution for the three walker models. Therefore we look at generating functions of subsets of the configurations which when summed give the anisotropic generating function. We then try to find patterns in the generating functions of the subclasses so as to perhaps conjecture the exact generating function. We explain how far we have progressed in this endeavour.

In Section 5 we give the dominant exponents for models with more than three walkers obtained from numerical analysis. For four osculating walkers we were able to identify a singularity other than the dominant one and we conjecture that any exact solution would be constrained to contain both these singularities.

Section 6 builds on the work in Essam and Guttmann (1995) and concludes that all n-friendly-walk models in three or more dimensions behave essentially like free walkers, and hence we deduce their exponents.

In Section 7 we introduce the concept of inversion relations, which are equations that relate the generating function to its analytic continuation, since they may provide a route to an exact solution for some of these problems. We were able to find inversion relations that hold for vicious walker watermelons with an arbitrary number of walkers even though we do not have the explicit anisotropic generating functions for three or more walkers.

Section snippets

Two walkers

In this section we consider the n-friendly-walk models with two walkers and solve for the generating functions for stars and watermelons for all n. We first derive the generating functions of watermelons by decomposing the graphs into sections whose generating functions are known. By distinguishing between NE and SE steps, we give the 2-variable or anisotropic generating functions for the four cases of special interest explicitly. Secondly, we extend the derivation to stars and give their

Analysis of numerical data

In situations where we cannot derive exact results, we have written computer programs to generate the first N terms of the star and watermelon generating functions, typically N≈100. We use the techniques of series analysis, reviewed in Guttmann (1989), to obtain estimates of the location of singularities, and their associated exponents and in favourable circumstances to conjecture a recurrence relation. We base our numerical analysis on the method of differential approximants which we will

Three walkers

In this section we deal with stars and watermelons with three walkers only. Of the n-friendly-walk models we consider the vicious walker (n=0), osculating walker (n=1), 2-friendly-walk case both isotropically and anisotropically and further the isotropic ∞-friendly-walk model.

First we extend isotropic results given for vicious walkers in Essam and Guttmann (1995) to osculating walkers and analyse the isotropic 2-friendly-walk and ∞-friendly-walk series. In the second part we analyse the

More than three walkers

We have no exact results for more than three walkers, although one could carry out a similar analysis of the anisotropic series generating functions, though this is unlikely to be fruitful. We did however calculate the first terms of the anisotropic generating functions of vicious walker watermelons with more than three walkers. We used these to determine an inversion relation which is discussed below in Section 7. Further, we calculated the first few terms in the isotropic generating functions

Three and more dimensions

We know from earlier analysis (Essam and Guttmann, 1995) that three is the so called upper critical dimension for the directed walker networks. That is to say, above three dimensions the freedom to move afforded by the increased spatial dimensionality is such that the constraints imposed by the different models is not felt. (At the critical dimension the dominant singularity is usually modified by a confluent logarithmic term. Essam and Guttmann (1995) saw that this was the case for vicious

Inversion relations

To provide further insight into these models, we introduce the concept of inversion relations. This concept applies to generating functions of two or more variables (Baxter, 1982; Bousquet-Mélou et al., 1999; Maillard, 1982). An inversion relation is a relation of the formG(x1,…,xn)+f1(x1,…,xn)G(g1(x1,…,xn),…,gn(x1,…,xn))=f2(x1,…,xn),usually with relative simple functions f1, f2 and gi. Typically one of the gi is of the form gi=ĝi/xi, (hence the name inversion relation) with ĝi a constant or

Conclusion

We have introduced a family of lattice path problems, and solved a number of them. Those that were not solved have been studied numerically, and their singularity structure conjectured. Higher dimensional analogues have also been studied. A combination of exact and numerical study allows us to conjecture—and in some instances prove—the asymptotic behaviour of star and watermelon generating functions for any number of walkers (P), for all models (n), and all lattice dimensions (d). We find for

Acknowledgements

AJG gratefully acknowledges financial support from the Australian Research Council. MV would like to thank Will Orrick for providing the Mathematica program solver, and fruitful discussions and to thank the DAAD for support with a doctoral stipend in the framework of the “gemeinsames Hochschulsonderprogramm III von Bund und Ländern”. We would particularly like to thank Aleks Owczarek for many suggestions to improve the manuscript, and Xavier Viennot for posing the 2-friendly-walk model, which

References (19)

There are more references available in the full text version of this article.

Cited by (0)

View full text