# Random vicious walks and random matrices

###### Abstract

Lock step walker model is a one-dimensional integer lattice walker model in discrete time. Suppose that initially there are infinitely many walkers on the non-negative even integer sites. At each tick of time, each walker moves either to its left or to its right with equal probability. The only constraint is that no two walkers can occupy the same site at the same time. It is proved that in the large time limit, a certain conditional probability of the displacement of the leftmost walker is identical to the limiting distribution of the properly scaled largest eigenvalue of a random GOE matrix (GOE Tracy-Widom distribution). The proof is based on the bijection between path configurations and semistandard Young tableaux established recently by Guttmann, Owczarek and Viennot. Statistics of semistandard Young tableaux is analyzed using the Hankel determinant expression for the probability from the work of Rains and the author. The asymptotics of the Hankel determinant is obtained by applying the Deift-Zhou steepest-descent method to the Riemann-Hilbert problem for the related orthogonal polynomials.

## 1 Introduction

In [12], two types of random vicious
walkers models, *random turn walker model* and
*lock step walker model*, are considered.
In these models, walkers are on one-dimensional integer lattice,
and time is discrete.
For their applications and earlier results, see, for example,
[12, 15, 16, 17, 18, 7, 8, 14] and references therein.
In this paper, we present results on lock step model showing a relation
to random matrix theory.
For random turn walker model, see [14, 5] and
discussions following Theorem 1.1 below.

At time , infinitely many walkers are located at the sites . We label the walkers by from the left to the right. In the lock step model, at each time , all the particles move either to their right or to their left with equal probability. The only constraint is that no two particles can occupy the same site at the same time. This is why the walkers are called “vicious”. One typical path configuration is shown in Figure 1.

This model can also be thought of as a certain
totally asymmetric exclusion process in discrete time.
Initially there are infinitely many particles at .
A particle is called *left-movable* if its left site is vacant.
Particles are called
*successors* of a particle at a certain time
if they are next to each other
in the order of the indices.
At each (discrete) time step, a left-movable particle
either moves to its left site *together* with
arbitrarily taken number of its successors,
or stays at the same site with equal probability.
It is easy to see that this process is equivalent to the above
lock step model ; right move of lock step corresponds staying
at the same site in the exclusion process.
Figure 2 represent an exclusion process
equivalent to the lock step model in Figure 1.

Suppose that during time steps, total left moves are made by all the particles. In the example of Figure 1, and . We denote by the set of all path configurations during time steps with total left moves. Then each configuration has equal probability, over the cardinal of . Hence our probability space is with uniform probability given by . We denote by the number of left moves made by the particle in a path . We are interested in the limiting statistics of the random variables as .

To state the main result, we need a definition. Let be the solution of the differential equation

(1.1) |

where is the Airy function.
The above equation is called Painlevé II equation.
It is known that there is unique global solution to (1.1)
(see, e.g. [2] and references in it).
Define the function, called the
*GOE Tracy-Widom distribution function*, by

(1.2) |

This is indeed a distribution function, and the decay rate is given by

(1.3) | |||||

(1.4) |

for some (see, e.g. (2.11)-(2.14) of [4]). In [29], Tracy and Widom proved that is the limiting distribution of the properly centered and scaled largest eigenvalue of a random matrix taken from a Gaussian orthogonal ensemble. The subscription in is a general convention : there are also GUE and GSE Tracy-Widom distribution functions and [28, 29]. One can find general discussion for random matrices in [23, 9].

Now the main theorem is

###### Theorem 1.1.

For fixed , let

(1.5) |

Let be the GOE Tracy-Widom distribution function. Under the condition that in time steps total left moves are made, the (conditional) probability distribution of the number of left moves made by the leftmost particle satisfies

(1.6) |

Also all the moments of the scaled random variable converge to the corresponding moments of the limiting random variable.

In other words, in the large limit, the (conditional) fluctuation of the displacement of the leftmost particle in lock step model is identical to the fluctuation of the largest eigenvalue of a random GOE matrix. Naturally we expect that the particle corresponds to the eigenvalue of random GOE matrix.

It is interesting to compare the above result with
the results for random turn walker model.
Initially there are infinitely many walkers
at the position .
We again call a walker *left-movable* if its left site
is vacant.
At each time, we select *one* walker at random among
left-movable walkers, and move it to its left site.
Hence there is one and only one movement at each time and
all the movements are to the left.
An example of random turn walker path configuration is in
Figure 3.

Let be the displacement of the walker after time step. It is shown in [5] that for , we have

(1.7) |

where is the limiting distribution of the (scaled) largest eigenvalue of a random GOE matrix. Especially we have . On the contrary, if we assume that the walkers move to their left in the first time steps, and then move to their right in the next time steps so that at the end walkers come back to their original positions, then we obtain the GUE Tracy-Widom distribution in the limit [14]. Indeed in this case, a lot more are known. The general row statistics and also the correlation functions converge to the corresponding quantities of random GUE matrix in the limit [14].

The first step to prove the above theorem is to map the path statistics into tableaux statistics following [18]. By definition, a semistandard Young tableau (SSYT) is an array of positive integers top and left adjusted as in Figure 4 so that the numbers in each row increase weakly and the numbers in each column increase strictly. A reference for tableaux is [26], and we freely use the notations in it. In [18], Guttmann, Owczarek and Viennot established a simple bijection between path configurations of lock step model and the set of SSYT : for a path configuration, we write down the time steps at which the particle made movement to its left on the column. Hence the top row is the array of time steps the particles made first movement to their left, the second row is the array of time steps the particles made their second movement to their left, and so on. If we draw boxes around each number, the result is a SSYT. See figure 4 for the tableau corresponding to the path configuration of Figure 1.

This map is a bijection between and the set of SSYT of size with fillings taken from . Moreover, under this bijection, is equal to the number of boxes in the column of the corresponding SSYT. Therefore the statistics of is identical to the column statistics of random tableaux.

After the work of Guttmann, Owczarek and Viennot,
Forrester in [14] observed that similar bijection
can be established between path configuration of random turn model
and the set of standard Young tableaux (SYT).
The above result (1.7) is obtained based on this
bijection and the recent work [4] on statistics of
the first row of random SYT.
Also as mentioned above, if we assume that the walkers move
to their left in the first time steps, and then move to
their right in the next time steps so that at the end
walkers come back to their original positions,
the limiting fluctuation is not GOE type, but GUE type.
This difference comes from the fact that in this case,
the path configuration is in bijection with the *pairs* of SYT.
The statistics of pairs of SYT is more well studied than that of single SYT
[2, 1, 24, 6, 19], and we have stronger results
mentioned earlier.

This paper is organized as follows. In Section 2, using the result of [3], we express the generating function for the probability of the first column of random tableaux in terms of a Hankel determinant. It is a general fact that Hankel determinant is related to orthogonal polynomials on the unit circle. The asymptotics of orthogonal polynomials is obtained via Riemann-Hilbert problem and summarized separately in Section 3. We can obtain the limiting statistics of the first column from the knowledge of Hankel determinant asymptotics. This work occupies the second half of Section 2. The proof of Theorem 1.1 is given at the end of Section 2.

Acknowledgments. The author would like to thank Percy Deift for his interest and encouragement.

## 2 Proof

Let be the number of semistandard Young tableaux of shape with fillings taken from , and let be the number of rows of (parts of , or the length of the first column). From the bijection of path configurations and tableaux, the number of path satisfying is equal to

(2.1) |

In our analysis (see also [3, 4]), it turns out that in addition to the number of rows, the number of odd columns plays an important role in describing a tableau. For a partition , we define to be the transpose of , to be the number of odd row in , and to be the size of . Let be the number of semistandard Young tableaux of size with odd columns with at most columns with fillings taken from :

(2.2) |

We use the notation for the sum above without restriction on . Now we define a generating function

(2.3) |

where the sum in the first expression is taken over all the partitions satisfying .

The starting point of our analysis is the following result of [3].

###### Lemma 2.1.

###### Remark.

Note that the right hand side of (2.4) does not depend on .

###### Proof.

This proof is in [3] in a slightly different form. Let , , be the Schur function, and define with by

(2.6) |

In (5.65) of [3], it is proved that

(2.7) |

which is an identity as a formal power series in . But the combinatorial definition of the Schur function is (see, e.g. Chapter 7.10 of [26])

(2.8) |

where the sum is over all semistandard Young tableaux of shape , and is the number of parts of equal to (type of ). Since , if we take the special case where the first elements are and the rest are , then becomes

(2.9) |

Hence for this special choice of (2.7) is now

(2.10) |

Now using Weyl’s integration formula for (see, e.g. [25]), the expectation in (2.10) becomes

(2.11) |

Standard Vandermonde determinant manipulations yield

(2.12) |

which again after change of variables , is equal to

(2.13) |

where , is the Chebyshev polynomial of the second kind. Note that . Hence elementary row and column operations yield Lemma 2.1. ∎ ,

Using this expression, we first obtain the asymptotic result for the generating function. The limit is insensitive to since so is .

###### Proposition 2.2.

Let and be fixed satisfying . For each and , define by

(2.14) |

where and are defined in (1.5) Then there exits a positive constant such that for , there are constants , independent of , and which may depend on so that

(2.15) |

Also we have

(2.16) | |||||

(2.17) |

###### Proof.

It is enough to consider the limit for since from the definition (2.3), is monotone increasing in . First we related the determinant in (2.4) with certain quantities of orthogonal polynomials on the circle.

Let be the monic orthogonal polynomial with respect to the weight on the interval , and let be the norm of :

(2.18) |

It is a well known result of orthogonal polynomial theory (see, e.g. [27]) that , where with

(2.19) |

which is equal to (recall (2.5)). Hence . Since the Szegö strong limit theorem for Hankel determinants (see, e.g. [20]) implies that for fixed , we have

(2.20) |

Now we use the relation between orthogonal polynomials on the unit circle and those on the interval . Let be the monic orthogonal polynomials on the unit circle with respect to the weight

(2.21) |

and let be the norm of :

(2.22) |

There is a simple relation between orthogonal polynomials on the unit circle and orthogonal polynomials on the interval (see the forth equation of (11.5.2) in [27]) :

(2.23) |

Especially comparing the coefficient of the leading term , we have the relation

(2.24) |

But we also have (see (11.3.6) in [27]). Hence (2.24) is equal to

(2.25) |

Therefore (2.20) becomes

(2.26) |

This argument appeared in Corollary 2.7 of [3]. Now using Proposition 3.2, computations similar to Lemma 7.1 of [2] (also Corollary 7.2 and Corollary 7.6 of [4]) yield the result. ∎

Interpreting the notation as the sum without constraints on in (2.2), the number of path configuration in is equal to

(2.27) |

and the probability of interest that for is equal to

(2.28) |

where

(2.29) |

For fixed , as , the Szegö strong limit theorem for Hankel determinants (see, e.g. [20]) implies that (2.4) becomes . Thus we have the identity

(2.30) |

By taking Taylor expansion of the right hand side in and , we obtain

(2.31) |

There is a more direct way to see this. See the remark after Lemma 2.6 below. Now from (2.27), the total number of paths in is equal to

(2.32) |

Now it is straightforward to obtain the following result on the number of all paths.

###### Lemma 2.3.

Let and let

(2.33) |

As , we have

(2.34) |

where the term vanishes as , and is defined by

(2.35) |

which is of order from (2.33). Moreover, the main contribution to the sum comes from for some ; precisely, there is a constant such that for any , we have

(2.36) |

###### Proof.

From (2.32), we have

(2.37) |

The ratio of is

(2.38) |

One can directly check that under the condition (2.33), the above ratio is decreasing in , and becomes closest to at

(2.39) |

Hence is unimodal: it is increasing for and is decreasing for . Now consider the neighborhood of of size for some fixed . For in , set

(2.40) |

For any , Stirling’s formula yields

(2.41) |

Using (2.33), (2.40) and (2.41), we have for in ,

(2.42) |

and

(2.43) |

Thus we have

(2.44) |

Let

(2.45) |

where denotes the set of satisfying (2.40) and denotes the rest of the range of . From (2.44), the first sum over is equal to the right hand side of (2.34). Also from the unimodality, in is less than or equal to the largest of and where . The number of summand in is of order . Hence using (2.44) again for , if we take , for large , we obtain

(2.46) |

which establishes the proof. ∎

###### Lemma 2.4.

For any , there are constants such that for all ,

(2.48) |

where

(2.49) | |||||

(2.50) |

The proof follows by using the following Lemma twice for and indices together with the Lemma 2.6. (Recall the (2.31)).

###### Lemma 2.5.

For a sequence , we define the following generating function

(2.51) |

For each , there are constants such that for any sequence satisfying (i) and (ii) ,

(2.52) |

where

(2.53) | |||||

(2.54) |

###### Proof.

This proof is parallel to that of the de-Poissonization lemma in [21]. We have

(2.55) |

Stirling’s formula yields for ,

(2.56) |

with some constant . Thus we have

(2.57) |

One can directly check the following estimates of :

(2.58) | |||||

(2.59) |

We take a constant satisfying . From (2.58) and the condition (ii), we have

(2.60) |

with a new constant . Similarly,

(2.61) |

Also using (2.59), we have

(2.62) |

for some constants . Thus we have

(2.63) |

with a possibly different constant . Now from the monotonicity condition (i), we have

(2.64) |

and

(2.65) |

using the equality (2.63) for the second equality. Thus we obtained the desired result. ∎

To use the above Lemma to , we need monotonicity in . It is more convenient now to view semistandard Young tableaux (SSYT) as generalized permutations. A two-rowed array

(2.66) |

is called a generalized permutation if either or , . Suppose the elements in the upper row of come from and the elements in the bottom row come from . One can represent a generalized permutation as a matrix where is the number of times when occurs in . For example, the generalized permutation

(2.67) |

corresponds to

(2.68) |

In the proof of the Lemma below, we regard a generalized permutation as a square board with stacks of balls in each position . We denote by the length of the longest strictly decreasing subsequence of . In the example (2.67), .

Let be the set of matrices which is symmetric , and satisfies and . This is a certain subset of the set of generalized permutations. The celebrated Robinson-Schensted-Knuth correspondence [22] establishes a bijection between and the set of SSYT of size with odd columns with fillings taken from . Moreover, under this bijection, for (viewed as a generalized permutation) is equal to the number of rows of the corresponding SSYT.

With this preliminary, we can prove the following.

###### Lemma 2.6 (monotonicity).

For any , we have

(2.69) |

###### Proof.

We first consider the second inequality. From (2.29), we need to show that

(2.70) |

By the definition (2.2) and the Robinson-Schensted-Knuth correspondence, is equal to the number of satisfying .

Consider all possible distinct (strict) upper triangular parts of elements in . It is equal to putting identical balls into boxes ; distinct ways. Hence we have a disjoint union where each consists of with same upper triangular part, and elements in and have different upper triangular parts when . Similarly, where has the upper triangular part same as that of .

Now for each , we generate elements in as follows. For , assign identical such that and for . (One can think this as adding a new ball in an array of balls ; there are ways.) Since , there result (many identical) elements of . Note that under this assignment,

(2.71) |

Now fix . Since there are diagonal entries, there are exactly (many identical) elements of from which is generated under the above assignment. (Considering each entry as a ball, each balls on the diagonal can be a newly added one.) Thus we have the identity . Furthermore, from the remark regarding (2.71), we have , where is the subset of satisfying . Therefore the second inequality in the Lemma is obtained.

The first inequality follows from a similar argument. ∎

###### Remark.

As mentioned before, using the generalized permutation interpretation of SSYT, we can see (2.31) directly. From non-negative integer matrix representation of generalized permutations, is the number of matrices with non-negative integer entries such that and