Mathematics Colloquia and Seminars

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I will discuss law of large numbers and central limit theorem for a class of growth models that includes a well studied particle system known as simple exclusion process. Given a function $v:{Bbb Z}^d o {Bbb N}$, we define the space of configurations to be the set of height functions $h:{Bbb Z}^d o {Bbb Z}$ such that $h(i)-h(j)le v(i-j)$. A $v$-exclusion process is a Markov process with the following stochastic rules: At a site $i$ the height $h(i)$ increases (respect. decreases) by one unit with rate $lambda^+$ (respect. $lambda^-$) provided that the resulting height function is still in the configuration space, otherwise the increase (respect. decrease) is suppressed. Under some conditions on $v$, the rescaled height function $u^{epsilon}(x,t)=epsilon h([frac xepsilon],frac tepsilon)$ converges to a deterministic function $u(x,t)$ that satisfies a Hamilton-Jacobi equation of the form $u_t+H(u_x)=0$. If the growth rates change with space, the corresponding equation is of the form $u_t+H(x,u,u_x)=0$. Such a Hamilton-Jacobi eqution is not well-posed in general. However, the above growth model leads to a variational formula for the limit $u$. When $v(i)=i^+$, the $v$-exclusion model is the celebrated simple exclusion process. In this case, I will discuss a central limit theorem for the convergence of $u^epsilon$ to $u$.