# Moments of vicious walkers and Möbius graph expansions

###### Abstract

A system of Brownian motions in one-dimension all started from the origin and conditioned never to collide with each other in a given finite time-interval is studied. The spatial distribution of such vicious walkers can be described by using the repulsive eigenvalue-statistics of random Hermitian matrices and it was shown that the present vicious walker model exhibits a transition from the Gaussian unitary ensemble (GUE) statistics to the Gaussian orthogonal ensemble (GOE) statistics as the time is going on from 0 to . In the present paper, we characterize this GUE-to-GOE transition by presenting the graphical expansion formula for the moments of positions of vicious walkers. In the GUE limit , only the ribbon graphs contribute and the problem is reduced to the classification of orientable surfaces by genus. Following the time evolution of the vicious walkers, however, the graphs with twisted ribbons, called Möbius graphs, increase their contribution to our expansion formula, and we have to deal with the topology of non-orientable surfaces. Application of the recent exact result of dynamical correlation functions yields closed expressions for the coefficients in the Möbius expansion using the Stirling numbers of the first kind.

###### pacs:

05.40.-a, 02.50.Ey, 02.10.Ox## I Introduction

Statistics of a set of one-dimensional random walks conditioned never to collide in a given time interval, say , has its own importance in statistical physics, since, if we set , it realizes the one-dimensional Fermi statistics deG68 , and with it is used to analyze the models for wetting and melting phenomena Fis84 . We will refer to such non-colliding random walks and the continuum counterpart, non-colliding Brownian motions, simply as vicious walks following the terminology used by M. Fisher Fis84 . For the pioneering work on vicious walker models, see HF84 ; Fold ; AME91 ; MB93 ; EG95 . A generic setting of vicious walk problems is discussed in CK03 . Recently the interest on the vicious walks in mathematical physics is renewed and growing very rapidly, for close relationships of the vicious walk problem with the study of ensembles of Young tableaux and the symmetric functions GOV98 ; KGV00 ; KGV02 , the theories of orthogonal polynomials and random matrices Bai00 ; NF02 , and some topics of representation theory and probability theory Gra99 ; OY02 ; KT02a have been clarified.

In an earlier paper KT02b , the continuum limit of non-colliding random walks on a lattice was taken by letting the temporal and spatial units go to zero with the relation and a system of non-colliding Brownian motions was derived. Since each random walk tends to be a Brownian motion in this diffusion scaling limit, such a construction of non-colliding Brownian motion is plausible, and indeed mathematical rigor can be established as a functional central limit theorem of vicious walks KT02a . The important fact is that the repulsive interaction among the obtained Brownian particles is no longer contact interaction as in the original vicious random walks on a lattice but is long ranged. The origin of this long-ranged interaction is the restriction of allowed configurations by the non-colliding condition, so that we can say it is an example of entropy-origin effective force. Moreover, the following setting was considered in our vicious walks; (i) even after taking the continuum limit, still we let the time interval, in which the non-colliding condition is imposed, be finite , and (ii) all the Brownian particles are assumed to start from an origin and the time interval of non-colliding is .

In this setting the process is temporally inhomogeneous. At the very early stage the repulsive interaction should be strong, since the Brownian motions will be restricted so that they will not collide for a long time period up to time in the future. As the time going on, the strength of repulsion is decreasing as is the remaining time until , and attains its minimum at the final time , at which there is no more restriction of motion in the future . It was Dyson’s idea that such systems of Brownian motions with long-ranged repulsion could have the equilibrium states, which can be described by the distribution functions of eigenvalues of random matrices in the ensembles appropriately specified by symmetry of the system Dys62 . The previous paper KT02b showed that the spatial distribution of vicious walkers at realizes the eigenvalue-statistics of the Gaussian unitary ensemble (GUE), one at does that of the Gaussian orthogonal ensemble (GOE), and the GUE to GOE transition is observed in the intermediate time, which is equivalent with the transition studied in the two-matrix model by Pandey and Mehta PM83 ; Meh91 .

In the present paper, we will characterize this transition in distribution by calculating the moments of the positions of particles as functions of time ,

(1) |

where denotes the position of -th vicious walker and the average at time . In addition to such an interest of statistical physics, we can put emphasis that our present study possesses another importance as an interesting application of the graphical expansion theory. In high energy physics, graphical expansions for the matrix models of SU() gauge theory were studied using ribbon graphs representing propagators and it was shown that the dominant graphs for large are the planar ones and the leading corrections come from the graphs embedded in a torus, which are depressed by a factor with respect to the planar graphs tHo74 ; BIPZ78 . On the other hand, it was clarified that in the graphical expansion of SO() gauge theory the leading corrections only are depressed by a factor for large with respect to the dominant planar graphs, in which propagators can be represented by twisted ribbons Cic82 ; BN91 . In the gauge theory, the presence of non-Gaussian interaction terms of cubic or higher power is crucial and it leads to triangulations of random surfaces. Although the Gaussian matrix models associated with the present vicious walker model are not related with the triangulation problem of random surfaces and only give purely enumerative problems of surfaces, the proper structures of large expansions in SU() and SO() gauge theories are also found in the GUE and GOE models, respectively. We can show that, if in our vicious walker model, the moments of the walker positions (1) can be calculated by the graphical expansion method of GUE using the ribbon graphs and the results are given in the form of power series in the inverse of the square of matrix size ,

Here the coefficients are the numbers of orientable surfaces of genus made from -gon by some specified procedure HZ86 (see also Zvo97 and Sec.5.5 in Meh91 ). On the other hand, for the GOE we have to take into account the non-orientable surfaces as well as the orientable ones and the expansion is in the form of power series in . In other words, in order to generate necessary surfaces, we need to use twisted ribbons, whose type of graphs is now called of Möbius graphs Sil97 ; MW02 . For the moments (1) of our vicious walkers, we will perform the Möbius graph expansion in the present paper. Now the weights of the graphs are depending on the time ; in the limit all the weights on the twisted ribbons are zero, but they are growing as time going on, and at twisted ribbons are equally weighted as untwisted ones. This gives another characterization of the temporally inhomogeneity of the process and the GUE-to-GOE transition.

Quite recently Nagao, Tanemura and one of the present authors applied the method of skew orthogonal polynomials and quaternion determinants developed for the multimatrix models NF99 ; FNH99 ; Nag01 to the vicious walk problem and derived the quaternion determinantal expressions for dynamical correlation functions NKT03 . Using this result, we will present an expression of the coefficients in our expansion of moments using the Stirling number of the first kind.

The paper is organized as follows. In Sec.II we briefly review the previous results reported in KT02b and give the precise definition of the moments which we will study. Graphical representations are demonstrated in Sec.III. Application of the result of NKT03 is given in Sec.IV to give the expression for the coefficients of expansion and some concluding remarks are given in Sec.V. Appendices A and B are prepared to derive the expression of the density function used in Sec.IV and the -expansions of the one-point Green function discussed in Sec.V, respectively.

## Ii Vicious Walks and Matrix Model

We study a continuum model of vicious walks, the non-colliding Brownian motions in the time interval , constructed in KT02a ; KT02b as the diffusion scaling limit of vicious random walks on a lattice. First we briefly review our previous results. The configuration space of the present vicious walkers is , where is a set of all real numbers. The probability density of vicious walkers at time with the initial condition that all walkers start from the origin is denoted by . It was given as

(2) |

with for , where is the Gamma function, and

By using the Harish-Chandra/Itzykson-Zuber integral formula HC57 ; IZ80 ; Meh81 , we will see that is proportional to the integral

(3) |

with

where is the diagonal matrix with , and the integrals and are taken over the groups of unitary matrices and real symmetric matrices , respectively. The proportional coefficient is determined so that the probability density is normalized. On the other hand, the integral over in (3) can be regarded as the convolution of the Gaussian distribution of complex Hermitian matrices with variance and that of real symmetric matrices with variance , and thus we have the expression

(4) |

with

(5) |

where and in the following we use the abbreviations and for the real and the imaginary part of the complex number , respectively, i.e. for with . Remark that, if we set

(6) |

and , is equal to the probability density of the two-matrix model of Pandey and Mehta with the parameter PM83 ; Meh91 . Corresponding to changing the parameter from 1 to 0 in the Pandey-Mehta two-matrix model, a GUE-to-GOE transition occurs in the time development of particle distribution in our vicious walks.

Now we define the quantity, which we will study in the present paper; the moment of particle positions in the vicious walks. Since the distribution (2) of is symmetric about the origin , all of the odd moments vanish. The even moments are defined and denoted as follows,

(7) | |||||

for .

## Iii Graphical Expansions

### iii.1 Wick formula

First we notice that (4) is invariant under any permutation of . Then (7) is written as

(8) |

Next we introduce the integration measure for the complex Hermitian matrices

Since , if are the eigenvalues of and (e.g. see Meh91 ), and for with any unitary matrix , (8) with (5) becomes

(9) |

where for functions of the elements of with

(10) | |||||

Note that

where the sum is taken over all combinations of indices , and that with the Hermitian condition , the integrand in (9) is a polynomial of the independent random variables . Since the probability density (10) is a product of independent Gaussian integration-kernels, we can apply the Wick formula with the variances

(11) |

for , where is Kronecker’s delta.

We can readily prove that (11) is equivalent with

(12) |

where is given by (6) and

(13) |

The Wick formula for (9) is thus

(14) | |||||

with the identification , where the first sum is taken over all combinations of indices , and the second one over the set of permutations of with the restriction

The total number of the terms in the second summation is .

### iii.2 An example: the fourth moment

In this section, by performing calculation of the fourth moment , we will demonstrate how to obtain graphical expansions from the Wick formula (14) with the variance (12). We start from the Wick formula for ,

which has terms in the summand of . Substitution of (12) and binomial expansion give the terms in the form,

with

Graphically, we prepare a square with four vertices labeled in a cyclic order for each term as shown in Fig.1 and connect the vertices and by a line for each Kronecker’s delta . We then regard these lines connecting vertices as the hems of ribbons connecting the two edges of the square. For example, the two lines connecting and in the term are considered as the two hems of a ribbon, say , connecting the edges and of the square, while the lines and are as those of a ribbon, say , connecting and . There are two ways to connect two distinct edges by a ribbon, by untwisting as and by twisting as in the above example, respectively. For each twisted ribbon, we put a factor .

Next we take the summation over in each term. Again consider the term for example. Under the restrictions on indices specified by the Kronecker deltas, , only two indices, say and , can be chosen arbitrary from . Then the summation over all possible choices of indices gives for this term. The contribution of is thus . We list up the contributions of all terms in Fig.1, and the sum of them gives

(15) | |||||

This shows that the 12 terms are classified into 6 equivalence classes, , , , , and , with respect to the contribution to the moment, and Fig.1 implies that all graphs for the terms in an equivalence class are topologically equivalent.

### iii.3 General formula

For the general -th moment, , , we have Wick couplings in the formula (14), each term of which is the products of the variances . By applying (12) and expanding in , we will have the terms, . As demonstrated in the above section, one-to-one correspondence is established between terms and graphs each of which consists of a -gon with its edges connected by ribbons to each other. For each graph corresponding to , let be the number of twisted ribbons in the ribbons and be the “free indices” remaining after the identification of indices under the Kronecker delta conditions. Then the contribution from the term is given by . As mentioned at the end of the previous section, we consider the equivalence classes of the terms having the same contribution to the -th moment, and let each equivalence class be represented by a graph . We let and be the numbers of free indices and of twisted ribbons. Moreover, we denote the number of elements in the equivalence class by . In other words, is the number of ways to generate graphs, which are topologically equivalent with , using a -gon and ribbons by gluing edges of -gons by ribbons. Define be the collection of all graphs generated by the present procedure. Then we have

(16) |

Each graph having no twisted ribbons, , defines a way of drawing a graph on an orientable surface , called a map, and each map specifies the surface on which the graph is drawn (see, for example, Zvo97 ; Oko99 ). In general, the specified orientable surface has “holes” or “handles” and their number is called the genus . The genus is related with , (the number of distinct edges; the original sides of polygon were glued together in pairs by ribbons) and (the number of faces) through the Euler characteristic,

(17) |

Then the contribution from all graphs having no twisted ribbons is expressed as

(18) | |||||

where is the indicator; if the condition is satisfied and otherwise, and .

In the similar way, the graphs having twisted ribbons, , are considered to define non-orientable surfaces . The genus for non-orientable surface may be defined by the Euler characteristics as Thu97

instead of (17). Then all the contribution to the moment from such non-orientable surface graphs is

(19) |

where . The moment (16) is then given by the summation

(20) |

It should be noted that defined by (13) is a monotonically increasing function of and changes its value from to 1 as the time passes from to . The above formula (19) shows the fact that the contribution from Möbius graphs with twisted ribbons is growing in time and at twisted ribbons contribute with the same weights as untwisted ribbons (the GOE case).

## Iv Calculation by Density Function

Set

for . It is easy to confirm that is invariant under any permutation of and that of , and (2) is equal to , if . Then the density function at time is defined as

(21) |

and the -th moment is given by

(22) |

Let be the -th Hermitian polynomial, satisfying the orthogonality with . As shown in Appendix A, the general formula for the dynamical correlation functions of vicious walks reported in NKT03 ; KNT03 gives the expression

(23) |

Substituting (23) into (22) and replacing the integral variable by give

In the second equality, we used the Christoffel-Darboux formula (see page 193 in Bat53 )

and the relation

Now we apply the integration formula for the triple of Hermitian polynomials (see page 290 in Bat54 )

Then we arrive at , with

(25) |

Here we do not repeat the explanation how to characterize the quantity in (18) by using the formula (IV), and only mention that it is obtained as the solution of the recurrence relation

(26) |

with the boundary conditions

In order to express the expansion in for (25), here we introduce the number defined as the coefficients of the expansion

for . It is known that is the number of elements in the set of all permutations of , which are products of disjoint cycles. These numbers are called the Stirling numbers of the first kind AS65 . For example, there are distinct permutations of . We denote a permutation simply by . We regard as a cycle , as a product of two cycles and , and the identity transformation as that of three cycles and . In this example, we see (for there are two elements and with in ), () and (other three permutations). For convenience, we will assume here that if , or or . Then we have expression (19) from (25) with the coefficients

(27) |