Approximation for extinction probability of the contact process based on the Gr\"obner basis

TL;DR

This paper introduces a novel method utilizing Gr"obner basis techniques to approximate the extinction probability of the one-dimensional contact process.

Contribution

The paper presents a new approach for approximating extinction probabilities using algebraic methods, specifically Gr"obner bases, in the context of the contact process.

Findings

Provides a series of approximations for extinction probability.

Demonstrates effectiveness of algebraic methods in stochastic process analysis.

Offers a potentially more accurate or efficient computational technique.

Abstract

In this note we give a new method for getting a series of approximations for the extinction probability of the one-dimensional contact process by using the Gr\"obner basis.

Equations

$$\Omega f(\eta )=\sum _{x\in {\mathbb{Z}}^d}c(x,\eta )[f({\eta }^x)-f(\eta )],$$

$$c(x,\eta )=(1-\eta (x))\times \lambda \sum _{y:\mid y-x\mid =1}\eta (y)+\eta (x),$$

$$001\to 011$$

$$100\to 110$$

$$101\to 111$$

$$1\to 0$$

$$\displaystyle\lambda\sum_{x\in A}\sum_{y:\mid y-x\mid=1}\Bigl{[}\nu_{\lambda}(A\cup\{y\})-\nu_{\lambda}(A)\Bigr{]}+\sum_{x\in A}\Bigl{[}\nu_{\lambda}(A\setminus\{x\})-\nu_{\lambda}(A)\Bigr{]}=0.$$

$${\nu }_{\lambda }(\circ )={\nu }_{\lambda }(\{0\}),{\nu }_{\lambda }(\circ \circ )={\nu }_{\lambda }(\{0,1\}),{\nu }_{\lambda }(\circ \times \circ )={\nu }_{\lambda }(\{0,2\}),\dots .$$

$$2\lambda {\nu }_{\lambda }(\circ \circ )-(2\lambda +1){\nu }_{\lambda }(\circ )+1=0,$$

$$\lambda {\nu }_{\lambda }(\circ \circ \circ )-(\lambda +1){\nu }_{\lambda }(\circ \circ )+{\nu }_{\lambda }(\circ )=0,$$

$$2\lambda {\nu }_{\lambda }(\circ \circ \circ \circ )+{\nu }_{\lambda }(\circ \times \circ )-(2\lambda +3){\nu }_{\lambda }(\circ \circ \circ )+2{\nu }_{\lambda }(\circ \circ )=0,$$

$$\lambda {\nu }_{\lambda }(\circ \circ \times \circ )-(2\lambda +1){\nu }_{\lambda }(\circ \times \circ )+\lambda {\nu }_{\lambda }(\circ \circ \circ )+{\nu }_{\lambda }(\circ )=0.$$

$$I_1=⟨2\lambda y-2\lambda x-x+1,y-x^2⟩\subset \mathbb{R}[\lambda ,x,y].$$

$$G_1=\{(x-1)(2\lambda x-1),y-x^2\}$$

$${\rho }_{\lambda }^{(1)}=1-{\nu }_{\lambda }^{(1)}(\circ )=\frac{2\lambda -1}{2\lambda },$$

$$I_2=⟨2\lambda y-2\lambda x-x+1,\lambda z-\lambda y-y+x,xz-y^2⟩\subset \mathbb{R}[\lambda ,x,y,z].$$

$$G_2=\{(x-1)((2\lambda -1)x-1),1+2\lambda (y-x)-x,$$

$$-y-yx+2x^2,-z-y(2+y)+4x^2\}$$

$${\rho }_{\lambda }^{(2)}=\frac{2(\lambda -1)}{2\lambda -1},$$

$$I_2^{\prime }=⟨2\lambda y-2\lambda x-x+1,\lambda z-\lambda y-y+x,y-x^2,z-x^3⟩\subset \mathbb{R}[\lambda ,x,y,z],$$

$$G_2^{\prime }=\{z-1,y-1,x-1\}$$

$$I_3=⟨2\lambda y-2\lambda x-x+1,\lambda z-\lambda y-y+x,$$

$$2\lambda s+w-(2\lambda +3)z+2y,\lambda u-(2\lambda +1)w+\lambda z+x,$$

$$ys-z^2,xu-yw⟩\subset \mathbb{R}[\lambda ,x,y,z,w,s,u].$$

$$G_3=\{(x-1)((12{\lambda }^3-5\lambda -1)x^2-2\lambda (2\lambda +3)x-\lambda +1),\dots \}$$

$${\rho }_{\lambda }^{(3)}=\frac{4\lambda (3{\lambda }^2-\lambda -3)}{12{\lambda }^3-2{\lambda }^2-8\lambda -1+\sqrt{D}},$$

Full text

Approximation for extinction probability of the contact process based on the Gröbner basis

Norio Konno

Department of Applied Mathematics

Yokohama National University

**Abstract. In this note we give a new method for getting a series of approximations for the extinction probability of the one-dimensional contact process by using the Gröbner basis. **

1 Introduction

Let $X=\{0,1\}^{\mathbb{Z}^{d}}$ denote a configuration space, where $\mathbb{Z}^{d}$ is the $d$-dimensional integer lattices. The contact process $\{\eta_{t}:t\geq 0\}$ is an $X$-valued continuous-time Markov process. The model was introduced by Harris in 1974 [1] and is considered as a simple model for the spread of a disease with the infection rate $\lambda$. In this setting, an individual at $x\in\mathbb{Z}^{d}$ for a configuration $\eta\in X$ is infected if $\eta(x)=1$ and healthy if $\eta(x)=0$. The formal generator is given by

[TABLE]

where $\eta^{x}\in X$ is defined by $\eta^{x}(y)=\eta(y)\>(y\neq x),$ and $\>\eta^{x}(x)=1-\eta(x)$. Here for each $x\in\mathbb{Z}^{d}$ and $\eta\in X,$ the transition rate is

[TABLE]

with $|x|=|x_{1}|+\cdots+|x_{d}|.$ In particular, the one-dimensional contact process is

[TABLE]

Let $Y=\{A\subset\mathbb{Z}^{d}:|A|<\infty\}$, where $|A|$ is the number of elements in $A$. Let $\xi^{A}_{t}(\subset\mathbb{Z}^{d})$ denote the state at time $t$ of the contact process with $\xi^{A}_{0}=A$. There is a one-to-one correspondence between $\xi^{A}_{t}(\subset\mathbb{Z}^{d})$ and $\eta_{t}\in X$ such that $x\in\xi^{A}_{t}$ if and only if $\eta_{t}(x)=1$. For any $A\in Y$, we define the extinction probability of $A$ by $\lim_{t\to\infty}P(\xi^{A}_{t}=\emptyset).$ Define $\nu_{\lambda}(A)=\nu_{\lambda}\{\eta\>:\>\eta(x)=0\>$ for any $\>x\in A\}$, where $\nu_{\lambda}$ is an invariant measure of the process starting from a configuration: $\eta(x)=1\>(x\in\mathbb{Z}^{d})$ and is called the upper invariant measure. In other words, let $\delta_{1}S(t)$ denote the probability measure at time $t$ for initial probability measure $\delta_{i}$ which is the pointmass $\eta\equiv i\>(i=0,1).$ Then $\nu_{\lambda}=\lim_{t\to\infty}\delta_{1}S(t)$. Then self-duality of the process implies that $\nu_{\lambda}(A)=\lim_{t\to\infty}P(\xi^{A}_{t}=\emptyset)$. The correlation identities for $\nu_{\lambda}(A)$ can be obtained as follows:

Theorem 1.1

For any $A\in Y$,

[TABLE]

From now on we consider the one-dimensional case. We introduce the following notation:

[TABLE]

By Theorem 1.1, we obtain

Corollary 1.2

[TABLE]

The detailed discussion concerning results in this section can be seen in Konno [2, 3]. If we regard $\lambda,\nu_{\lambda}(\circ),\nu_{\lambda}(\circ\circ),\nu_{\lambda}(\circ\circ\circ),\ldots$ as variables, then the left hand sides of the correlation identities by Theorem 1.1 are polynomials of degree at most two. In the next section, we give a new procedure for getting a series of approximations for extinction probabilities based on the Gröbner basis by using Corollary 1.2. As for the Gröbner basis, see [4], for example.

2 Our results

Put $x=\nu_{\lambda}(\circ),\>y=\nu_{\lambda}(\circ\circ),\>z=\nu_{\lambda}(\circ\circ\circ),\>w=\nu_{\lambda}(\circ\times\circ),\>s=\nu_{\lambda}(\circ\circ\circ\circ),\>u=\nu_{\lambda}(\circ\circ\times\circ)$. Let $\prec$ denote the lexicographic order with $\lambda\prec x\prec y\prec w\prec z\prec u\prec s.$ For $m=1,2,3$, let $I_{m}$ be the ideals of a polynomial ring $\mathbb{R}[x_{1},x_{2},\ldots,x_{n(m)}]$ over $\mathbb{R}$ as defined below. Here $x_{1}=\lambda,x_{2}=x,x_{3}=y,x_{4}=z,x_{5}=w,x_{6}=s,x_{7}=u$ and $n(1)=3,n(2)=4,n(3)=7.$

2.1 First approximation

We consider the following ideal based on Corollary 1.2 (1):

[TABLE]

Here $y-x^{2}$ corresponds to the first (or mean-field) approximation: $\nu_{\lambda}^{(1)}(\circ\circ)=(\nu_{\lambda}^{(1)}(\circ))^{2}$. Then

[TABLE]

is the reduced Gröbner basis for $I_{1}$ with respect to $\prec$. Therefore the solution except a trivial one $x(=y)=1$ is $x=\nu_{\lambda}^{(1)}(\circ)=1/(2\lambda)$. Remark that the trivial solution means that the invariant measure is $\delta_{0}$. From this, we obtain the first approximation of the density of the particle, $\rho_{\lambda}=E_{\nu_{\lambda}}(\eta(x))$, as follows:

[TABLE]

for any $\lambda\geq 1/2$. This result gives the first lower bound $\lambda_{c}^{(1)}$ of the critical value $\lambda_{c}$ of the one-dimensional contact process, that is, $\lambda_{c}^{(1)}=1/2\leq\lambda_{c}.$ However it should be noted that the inequality is not proved in our approach. The estimated value of $\lambda_{c}$ is about 1.649.

2.2 Second approximation

Consider the following ideal based on Corollary 1.2 (1) and (2):

[TABLE]

Here $xz-y^{2}$ corresponds to the second (or pair) approximation: $\nu_{\lambda}^{(2)}(\circ)\nu_{\lambda}^{(2)}(\circ\circ\circ)=(\nu_{\lambda}^{(2)}(\circ\circ))^{2}$. Then

[TABLE]

is the reduced Gröbner basis for $I_{2}$ with respect to $\prec$. Therefore the solution except a trivial one $x(=y=z)=1$ is $x=\nu_{\lambda}^{(2)}(\circ)=1/(2\lambda-1)$. As in a similar way of the first approxaimation, we get the second approximation of the density of the particle:

[TABLE]

for any $\lambda\geq 1$. This result implies the second lower bound $\lambda_{c}^{(2)}=1$. We should remark that if we take

[TABLE]

then we have

[TABLE]

is the reduced Gröbner basis for $I_{2}^{\prime}$ with respect to $\prec$. Here $y-x^{2}$ and $z-x^{3}$ correspond to an approximation: $\nu_{\lambda}^{(2^{\prime})}(\circ\circ)=(\nu_{\lambda}^{(2^{\prime})}(\circ))^{2}$ and $\nu_{\lambda}^{(2^{\prime})}(\circ\circ\circ)=(\nu_{\lambda}^{(2^{\prime})}(\circ))^{3}$, respectively. Then we have only trivial solution: $x=y=z=1.$

2.3 Third approximation

Consider the following ideal based on Corollary 1.2 (1)–(4):

[TABLE]

Here $ys-z^{2}$ and $xu-yw$ correspond to the third approximation: $\nu_{\lambda}^{(3)}(\circ\circ)\nu_{\lambda}^{(3)}(\circ\circ\circ\circ)=(\nu_{\lambda}^{(3)}(\circ\circ\circ))^{2}$ and $\nu_{\lambda}^{(3)}(\circ)\nu_{\lambda}^{(3)}(\circ\circ\times\circ)=\nu_{\lambda}^{(3)}(\circ\circ)\nu_{\lambda}^{(3)}(\circ\times\circ)$, respectively. Then

[TABLE]

is the reduced Gröbner basis for $I_{3}$ with respect to $\prec$. Therefore the solution except a trivial one $x=1$ is $x=\nu_{\lambda}^{(3)}(\circ)=(\lambda(2\lambda+3)+\sqrt{D})/(12\lambda^{3}-5\lambda-1)$, where $D=16\lambda^{4}+4\lambda^{2}+4\lambda+1$. Then we obtain the third approximation of the density of the particle:

[TABLE]

for any $\lambda\geq(1+\sqrt{37})/6$. This result corresponds to the third lower bound $\lambda_{c}^{(3)}=(1+\sqrt{37})/6\approx 1.180$.

3 Summary

We obtain the first, second, and third approximations for the extinction probability, the density of the particle, and the lower bound of the one-dimensional contact process by using the Gröbner basis with respect to a suitable term order. These results coincide with results given by the Harris lemma (more precisely, the Katori-Konno method, see [3]) or the BFKL inequality [5] (see also [3]). As we saw, the generators of $I_{m}$ in Section 2 have degree at most two in $x_{1},x_{2},\ldots$, such as $2\lambda y-2\lambda x-x+1,\>ys-z^{2}$ in the case of $I_{3}$. We expect that this property will lead to get the higher order approximations of the process (and other interacting particle systems having a similar property) effectively.

Acknowledgment. The author thanks Takeshi Kajiwara for valuable discussions and comments.