In this paper, we investigate the behavior of the unique rightmost species-1 particle in a two-species ASEP. We start with an introduction, where we first present the model itself, which was presented by Frank Spitzer in the paper [4]. The main result of the thesis is an integral formula for the probability distribution of a unique particle of type 1, which at time t=0 is on the far right in the ASEP of type 2.
In the paper [3] Gunter Sch¨utz constructed a solution to the (single species) TASEP using two different ways: the constructive Bethe Ansatz technique, and as a determinant of a certain matrix. He also obtained the solutions for the ASEP by Bethe Ansatz in the number of particles N =1 and N =2. Then, in the paper [5], Craig Tracy and Harold Widom obtained the solutions to the forward equation for the ASEP with general N.
Subsequently, in another paper [6], Craig Tracy and Harold Widom derived the form of the solutions to the forward equation for the multi-species ASEP with general N . Moreover, in the TASEP case he expressed the solution as the determinant of a certain matrix.
What is a multi-species ASEP?
Finding transition probability in a single-species ASEP
N=1
Then, it is not difficult to see that the integrand satisfies the initial condition: lett=0; then, there are 2 cases: osex=y, orx≠y.
N=2
The reason for this extension is that we will need an expression u(x, x;t) that accepts an input that does not belong to the state space of our Markov process. Then it is not difficult to check that this expression satisfies the initial condition: lett=0; then there are 2 cases: x=y or x≠y. Therefore, our construction actually satisfies the initial condition P(x1, x2; 0) =δxy=δx1y1⋅δx2y2 and is therefore our desired solution.
General N
The constant ( 1 . 2πi)N is unwieldy and has no meaning; therefore we will further include each constant 2πi1 in differentialdξi,i=1, .., N.
Finding transition probability in a multi-species ASEP
N = 2
As is now clear, what we have is a system of coupled linear differential equations, and therefore they cannot be solved separately from each other. And so, as noted above, we must solve these equations together; and therefore it is not possible to approach the problem as in single-species ASEP. First of all, note that we do not limit ourselves to the cases where species are different: we also include the cases where all particles belong to the same species (especially the matrix entries (1,1) and (2,2) )) , which reduces to single-species ASEP in these cases.
Now, given this matrix, we want to construct the matrix forward differential equation; for this we need to consider cases again, but now we use matrices of functions instead of ordinary functions. Now, as before, we will combine everything into a single matrix equation with some boundary conditions.
General N
Permutations, inversions, and subsets
In this chapter, we experiment with special cases to find out what we are looking for: the probability that the rightmost particle η belongs to type-1 at time xat, given that all other particles belong to type-2: PY(η( t) =x). Our goal in this chapter is to discover the main patterns of our general particle system by studying specific cases and lay the groundwork for future work. However, it is possible to deform the contour into a circle with a radius greater than 1, because the only singularity of the integrand is at the origin.
As before, we need to increase our c2 contour, but this time without approximating the poly function (p−qξp+qξ2)(ξ1−1). And in order for this inequality to make sense, we need to have this ∣ξp.
N = 3
As before, we need to break each integral into a sum of integrals, and treat each integral separately. Reduce the contourc1 and increase the contourc3 to C3 as in the third integral; then reduce the contourc2 as in the second integral. Decrease contourc2 and enlarge contourc3 to C3 as in the third integral, and shrink.
Enlarge contourc1 to C1 large enough and shrink contourc2 small enough to avoid the singularities. This time there is no need to deform anything; just choose contours small enough to avoid singularities. These cancellations seem somewhat miraculous; However, it becomes much clearer if we didn't do any evaluations in the first place: consider, for example, the last cancellation in the variables ξ1, ξ3.
It turns out that there is no need to fear these singularities: Tracy and Widom in [5] gave an argument that the residue due to the denominator p+qξαξβ−ξα (which is given by ξa = p . 1−qξβ ) is O (1) at infinity in the variable ξβ, so it vanishes when integrated over the variable ξβ.
Some simplifications for further work
The first thing we need to do is remove the Qβα terms by writing: Qβα=Sβα−pTβα. This simplification implies that we must always remove Qβα terms, then take the summations over the columns of the [Aσ] tableau with Tβα rearranged in the next column.
General formula conjectured
The formula suggests that integrals that contain the variable ξN behave differently from those that do not. Indeed, to calculate integrals of lower dimensions, we will need to evaluate [Aσ] in variables not present in the integral. The coefficients of [Aσ] in the variable ξm, m However, the coefficients of [Aσ] in the variableξN contain the terms TN α and the unique QN α, and therefore this case requires special attention. This is a departure from the standard notation: it is customary to write [n]q for the q bracket. The Q brackets are not compound, but rather occur naturally in mathematics, as in the following sentence (which we will use later):. First of all, since mapσ is order-preserving in the first∣Sc∣variables, it follows that there are no inversions among the first∣Sc∣variables. Next, since the position of 1 ismden from the right, it follows that we have (m−1)pTN α-factors, a single QN α-factor, and (∣S∣ −m) SN α-factors. TN α. We now make the following important observation: the coefficient[A○σ] is not zero initially if and only ifN ∈Sc. We say that the coefficient before the sum of the integrands over the kth column in the table [A○σ] is given byCN,k= (−1)N−k[N−1. We conclude the chapter with the study of lower-dimensional integrals, which leads us to the final form of the integral formula. There is nothing special about taking the residual on the value with only the highest index; it's just the convention we adhere to. First of all, we note sen≤N−k, because the maximum number of variables in the geometric series whenk is the position of 1 from the right is N−k. Then, from the form of the denominator we see that if there is a residue at ξcn = 1. Therefore, almost every integrand gives rise to an N−1-dimensional residue; the only exception is the integrands with the coefficient [Aσ which do not require any contour deformation, and therefore give no residuals at all. Finally, we notice that the sum in the brackets is nothing but the sum we had for the N-times integral, but over the variablesξs1, .., ξsk. Now we notice that each sum separately is nothing but the coefficient for the N-fold integrand in the Tracy-Widom integral formula for the kth rightmost particle (equivalent to ∣S∣ −k+1th leftmost particle), 1≤k≤ ∣ S∣.
