Speaker: Jean-Dominique Deuschel

Title: The Random Line model

Abstract: Let $t\in \mathbb{R}_+\to X(t)=(X_1(t),X_2(t))\in \mathbb{Z}^2$ be the variable speed  random walk in random environment (RWRE)  with horizontal jump rate $\omega(x,x\pm e_1)=H(x_2)>0$and vertical jump rate $\omega(x,x\pm e_2)=V(x_1)>0,$ where the jump rates are independent with fat tails 

$$ 

\mathbb{P}(H(x_2)>L) = c_H L^{-\alpha_H(1 + o(1))}, \quad \mathbb{P}(V(x_1)>L) = c_V L^{-\alpha_V(1 + o(1))}.

$$

 This RWRE is both a conductance model with symmetric jump rate and a balanced random walk. In particular the coordinate $X_1(t)$ and $X_2(t)$ are local martingales with quadratic variation

$$

<X_1>(t)=\int_0^t H(X_2(s))\,ds, \quad <X_2>(t)=\int_0^t V(X_1(s))\,ds.

$$

        In case $\alpha_H>1$ and $\alpha_V>1$, both $\mathbb{E}[V(X_1(s))]<\infty$ and $\mathbb{E}[H(X_2(s))]<\infty,$ so that the rescaled walks $X^T(t)=T^{-1/2}X(Tt)$ converge in law to 2 independent Brownian motions $(\beta_1(t),\beta_2(t))$. 

        In case $\alpha_H<1$ and $\alpha_V>1$  then $\mathbb{E}[V(X_1(s))]<\infty$   but $\mathbb{E}[H(X_2(s))]=\infty$, the second diffusively rescaled coordinate  $X^T_2$  and its rescaled local time converge to a Brownian motion $\beta_2$ with local time $L(\beta_2)$ whereas the first coordinate super diffusively rescaled

$$

X^T_1(t)=T^{-\delta_H/2}X_1(Tt),\quad \delta_H =\frac12+\frac{1}{\alpha_H}>1,

$$

converges to a time changed independent Brownian motion $\beta_1(\Delta(t))$, where

$$

\Delta(t)=\int_{\mathbb{R}}L(\beta_2)(t,y)\, d H(y)

$$

is the self similar process introduced by Kesten-Spitzer in 1979 in context of random walk in random scenery. 

        Moreover we show that the first coordinate of the corresponding constant speed random walk converges to an independent Brownian motion while the second coordinate converges to a Fontes-Isopi-Newman diffusion as is known in $1-d$ trapping model of Ben Arous-Cerny.The more challenging case where both $\alpha_H<1$ and $\alpha_V<1$, remains open. In this setting we show  a non-explosion criteria and assuming that the Kesten-Spitzer convergence takes place we can derive the corresponding scaling parameters.