%0 Journal Article %1 cox1995hybrid %A Cox, J. T. %A Durrett, R. %D 1995 %J Bernoulli %K clines hybrid_zone particle_systems random_walk spatial_structure voter_model %N 4 %P 343--370 %T Hybrid zones and voter model interfaces %U http://dx.doi.org/10.2307/3318488 %V 1 %X In this paper, the authors study the dynamics of hybrid zones in the absence of selection: They consider a process $(ŋ_t)_t0$ with state space $(0,0),(0,1),(1,0),(1,1)^Z^d1,2$, where $(ŋ_t(x,1),ŋ_t(x,2))$ is a pair of genes which represent the state of the individual at site $xınZ^d$ at time $t$. It evolves according to a translation-invariant, irreducible, symmetric probability transition $p(·,·)$ on $Z^d$ with finite second moments, in the following way. At rate 1, each individual is replaced by a new one: it chooses each of its two parents with probability $p(·,·)$, then picks at random one of the parents' two genes (all these choices being independent); the two genes obtained are the individual's new state. This description corresponds to a voter model see T. M. Liggett, Interacting particle systems, Springer, New York, 1985; MR0776231 (86e:60089). Taking for initial configuration $(0,0)$ on sites $xınZ^d$ with $x_10$, and $(1,1)$ if $x_1<0$ ($x_1$ is the first coordinate of $x$), the authors look at the evolution of the interface between 0's and 1's, which gives rise to a hybrid zone. They show that, in dimension $d>1$, the width of the hybrid zone grows as $t$. In dimension 1, assuming that $p(·,·)$ has finite third moments, they prove that the interface process is an irreducible, positive recurrent Markov chain, with an equilibrium distribution of infinite mean (except in the nearest neighbor case where nothing moves). The first step of the proofs transforms a particle system's problem into a random walk problem, because the voter model is in duality with coalescing random walks. The key tools are therefore refined properties of random walks (in particular, about the recurrent potential kernel), and these theorems are interesting on their own. Finally, the authors illustrate their belief that tight interfaces are common in one-dimensional particle systems: They explain conjectures for the contact process see T. M. Liggett, op. cit., for a model of oxidation of carbon monoxide on a catalyst surface (defined by R. M. Ziff, E. Gulari and Y. Barshad Phy. Rev. Lett. 56 (1986), no. 24, 2553--2556), mention results on simple exclusion see P. A. Ferrari, C. Kipnis and E. Saada, Ann. Probab. 19 (1991), no. 1, 226--244; MR1085334 (92b:60099), and on stochastic PDEs see R. Tribe, Stochastics Stochastics Rep. 56 (1996), no. 3-4, 317--340; C. Mueller and R. B. Sowers, J. Funct. Anal. 128 (1995), no. 2, 439--498; MR1319963 (97a:60083) MR1396765 .

@article{cox1995hybrid, abstract = {In this paper, the authors study the dynamics of hybrid zones in the absence of selection: They consider a process $(ŋ_t)_{t\geq 0}$ with state space ${(0,0),(0,1),(1,0),(1,1)}^{\bold Z^d\times {1,2}}$, where $(ŋ_t(x,1),ŋ_t(x,2))$ is a pair of genes which represent the state of the individual at site $x\in\bold Z^d$ at time $t$. It evolves according to a translation-invariant, irreducible, symmetric probability transition $p(·,·)$ on $\bold Z^d$ with finite second moments, in the following way. At rate 1, each individual is replaced by a new one: it chooses each of its two parents with probability $p(·,·)$, then picks at random one of the parents' two genes (all these choices being independent); the two genes obtained are the individual's new state. This description corresponds to a voter model [see T. M. Liggett, Interacting particle systems, Springer, New York, 1985; MR0776231 (86e:60089)]. Taking for initial configuration $(0,0)$ on sites $x\in\bold Z^d$ with $x_1\geq 0$, and $(1,1)$ if $x_1<0$ ($x_1$ is the first coordinate of $x$), the authors look at the evolution of the interface between 0's and 1's, which gives rise to a hybrid zone. They show that, in dimension $d>1$, the width of the hybrid zone grows as $\sqrt t$. In dimension 1, assuming that $p(·,·)$ has finite third moments, they prove that the interface process is an irreducible, positive recurrent Markov chain, with an equilibrium distribution of infinite mean (except in the nearest neighbor case where nothing moves). The first step of the proofs transforms a particle system's problem into a random walk problem, because the voter model is in duality with coalescing random walks. The key tools are therefore refined properties of random walks (in particular, about the recurrent potential kernel), and these theorems are interesting on their own. Finally, the authors illustrate their belief that tight interfaces are common in one-dimensional particle systems: They explain conjectures for the contact process [see T. M. Liggett, op. cit.], for a model of oxidation of carbon monoxide on a catalyst surface (defined by R. M. Ziff, E. Gulari and Y. Barshad [Phy. Rev. Lett. 56 (1986), no. 24, 2553--2556]), mention results on simple exclusion [see P. A. Ferrari, C. Kipnis and E. Saada, Ann. Probab. 19 (1991), no. 1, 226--244; MR1085334 (92b:60099)], and on stochastic PDEs [see R. Tribe, Stochastics Stochastics Rep. 56 (1996), no. 3-4, 317--340; C. Mueller and R. B. Sowers, J. Funct. Anal. 128 (1995), no. 2, 439--498; MR1319963 (97a:60083) MR1396765 ].}, added-at = {2009-01-29T21:13:33.000+0100}, author = {Cox, J. T. and Durrett, R.}, biburl = {https://www.bibsonomy.org/bibtex/2d9dd98b278afe26ba73f2dd4cc0c5d0e/peter.ralph}, fjournal = {Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability}, interhash = {2e54df652cde9169bbe4326fed6eb832}, intrahash = {d9dd98b278afe26ba73f2dd4cc0c5d0e}, issn = {1350-7265}, journal = {Bernoulli}, keywords = {clines hybrid_zone particle_systems random_walk spatial_structure voter_model}, mrclass = {60K35 (60J15 82B24 82C22)}, mrnumber = {MR1369166 (97k:60270)}, mrreviewer = {Ellen Saada}, number = 4, pages = {343--370}, timestamp = {2016-04-26T07:24:12.000+0200}, title = {Hybrid zones and voter model interfaces}, url = {http://dx.doi.org/10.2307/3318488}, volume = 1, year = 1995 }