On the real line initially there are infinite number of particles on the positive half line, each having one of

We consider one-dimensional dynamical model of the boundary between two phases (particles and antiparticles, bears and bulls) where the boundary moves due to reaction (annihilation, transaction) of pairs of particles of different phases.

Assume that at time

The main result of the paper is the explicit formula for the asymptotic velocity of the boundary as the function of

Main technical tool of the proof may seem surprising (and may be of its own interest), we reduce this infinite particle problem to the study of a special random walk of one particle in the orthant

The organization of the paper is the following. In Section

At time

We assume that all

We denote this infinite particle process

We define the boundary

For any pair

The asymptotic velocity of the boundary exists and is equal to

Note that the definition of

Now, we will explain this result in more detail. As

if

if

Item

Normally, the minimal difference between consecutive prices (a tick) is very small. Moreover, one customer can have many units of the commodity. That is why it is natural to consider the scaled densities:

This scaling suggests a curious interpretation of the model—the simplest model of one instrument (e.g., a stock) market. Particle initially at

There are models of the market having similar type (but very different from ours, see [

The case

More complicated situation was considered in [

The phase transition appears already in case when

Then, the asymptotic velocity is the function:

Assume that the speed

These numbers should be equal (balance equations), and after dividing by

One could think that on this way one can get rigorous proof. However, it is not so easy. We develop here different techniques, that gives much more information about the process than simple balance equations.

One can consider the phase boundary as a special kind of server where the customers (particles) arrive in pairs and are immediately served. However, the situation is more involved than in standard queuing theory, because the server moves, and correlation between its movement and arrivals is sufficiently complicated. That is why this analogy does not help much. However, we describe the crucial correspondence between random walks in

Denote

Denote

Now, we describe the trajectories

Note that the increments of the coordinates

We call the process

Two following two conditions are equivalent:

the process

All other cases of boundary movement correspond to nonergodic random walks. Even more, we will see that in all other cases the process

To understand the corresponding random walk dynamics introduce a new family of processes.

Let

Define the following set of “appropriate” faces

It holds that

The proof will be given in Appendix

One can define a family

Similarly to

Consider a face

Introduce the plane:

Let

This vector

Let

The set

This lemma will be proved in Section

We define now the piece-wise constant

It follows that the trajectories

We call the ergodic face

The following theorem, together with Theorem

This theorem will be proved in Section

If

If

Now, we come back to our infinite particle process

If the process

Remind that the collisions can be presented as follows. If

We will need below the following new notation, (

Let the process

the speed of the boundary

Consider the faces

The following lemma is quite similar to Lemma

If the process

Introduce the following notation:

Due to (

For any process

Let

the boundary velocity for the process

Note that

For any ergodic face

This is quite similar to Lemma 2.2, page 143 of [

It follows from (

Note that by condition (

Intuitive interpretation of this formula is the following. For example, the inequality

The implication

The system (

If the assumption

For ergodic face

Note that the matrix

The theorem is proved.

For any nonergodic face

Introduce the face

So, there can be two possible cases.

If

If

We show that

By formula (

To finish the proof of Lemma

So

Let

The first goal of this subsection is to study trajectories

According to the definition of

There exists a monotone sequence of faces:

The proof of Proposition

First, we will present here some algorithm for constructing the sequence

Notation

(i) If

(ii) If

(i) If

(ii) If

Find

If

If

If

If

If

If

If

If

If

If

If

If

If the algorithm did not stop at the steps (

where

We need not only the final group, corresponding to the face along which the trajectory escapes to infinity, but also the whole chain:

Let us remark that in the algorithm we excluded cases where some of

The next lemma is needed for the proof of the Theorem

In fact, if

To prove assertion

Consider the case

Thus, the inequality (

The

This gives the proof.

Let

Remind that the face

Let the chain (

Remind the notation

Consider now the case when the set

This ends the proof.

Assume the above algorithm produces the chain of groups (

Let

If the Case

(

Thus, there exists a time moment

Important remark is that the sequence of times

Consider the following modification of the algorithm: in Cases (2a) and (

From the above, it follows that any trajectory

For any initial point

Consider an arbitrary trajectory

Two cases are possible for the transition

Note the following fact. Take, for example, some

Let the chain (

The fourth assertion of theorem 3 is a corollary of Proposition

Theorem

If associated random walk

Let the process

The following assertion is an obvious analog of the proposition 1.4.3 of [

For any

Let

Let the face

For shortness, denote

one has

We prove the first three items. The others are quite similar. Using (

The Lemma is proved.

Let

A. D. Manita was supported by the Russian Foundation of Basic Research (Grants 12-01-00897 and 11-01-90421).