Random Walks with Local Memory

Abstract

We prove a quenched invariance principle for a class of random walks in random environment on ${\mathbb {Z}}^d$, where the walker alters its own environment. The environment consists of an outgoing edge from each vertex. The walker updates the edge e at its current location to a new random edge $e'$ (whose law depends on e) and then steps to the other endpoint of $e'$. We show that a native environment for these walks (i.e., an environment that is stationary in time from the perspective of the walker) consists of the wired uniform spanning forest oriented toward the walker, plus an independent outgoing edge from the walker.

Appendix A: Random Walks with Hidden Local Memory

In this section we present a more general version of random walk with local memory inspired by hidden Markov chains. We refer to [2] for a more detailed discussion on hidden Markov chains.

For each $x \in V$, a hidden mechanism at x is a Markov chain $M_x$ with finite state space $S_x$ and probability transition function $p_x(\cdot ,\cdot )$. A jump rule is a map $f_x:S_x \rightarrow {\mathcal {P}}({N}(x))$ from $S_x$ to the set of probability distributions on the set of neighbors of x. A hidden state configuration is a map $\kappa : V \rightarrow \sqcup _{x \in V} S_x$ such that $\kappa (x) \in S_x$ for all $x \in V$.

Definition A.1

(Random walk with hidden local memory) A random walk with hidden local memory, or RWHLM for short, is a sequence $(X_n, \rho _n, \kappa _n)_{n \ge 0}$ satisfying the following transition rules:

• (i) $\kappa _{n+1}(x):={\left\{ \begin{array}{ll} K_n &{} \text {if } x=X_n;\\ \kappa _{n}(x) &{} \text {if } x\ne X_n. \end{array}\right. }$;

• (ii) $\rho _{n+1}(x):={\left\{ \begin{array}{ll} Y_n &{} \text {if } x=X_n;\\ \rho _{n}(x) &{} \text {if } x\ne X_n, \end{array}\right. }$

• (iii) $X_{n+1}:=Y_n$,

where $K_n$ is a random element of $S_{X_n}$ sampled from $p_{X_n}(\kappa _{n}(X_n), \cdot )$ independent of the past, and $Y_n$ is a random neighbor of x sampled from $f_{X_n}(K_n)$ independent of the past.

Described in words, at each time step (i) the walker first updates the hidden state of its current location using the given hidden mechanism. Then, (ii) the walker updates the rotor of its current location by sampling the new rotor from the probability distribution corresponding to the new hidden state. Finally, (iii) the walker travels to the vertex specified by the new rotor.

Example A.2

(Hidden triangular walk) Let G be the triangular lattice. For each ${\mathbf {x}}\in V$, the hidden mechanism at ${\mathbf {x}}\in V$ has the following state space and transition probability:

$$\begin{aligned} S_{{\mathbf {x}}}:=\{ s_1,s_2,s_3 \};\qquad p_{\mathbf {x}}:=\begin{bmatrix} 0 &{} \frac{1}{2} &{} \frac{1}{2}\\ 0 &{} 0 &{} 1\\ 1 &{}0 &{}0 \end{bmatrix}. \end{aligned}$$

That is, $s_1$ transitions to either $s_2$ or $s_3$ with equal probability, $s_2$ transitions to $s_3$ with probability 1, and $s_3$ transitions to $s_1$ with probability 1.

We now describe the jump rule $f_{\mathbf {x}}$. Let $N_1 \sqcup N_2$ be the partition of the neighbors ${N}({\mathbf {x}})$ of ${\mathbf {x}}$ given by:

$$\begin{aligned} N_1:= {\mathbf {x}}+\left\{ \begin{pmatrix} 1\\ 0 \end{pmatrix}, \frac{1}{2}\begin{pmatrix} -{1} \\ {\sqrt{3}} \end{pmatrix} , \frac{1}{2}\begin{pmatrix} -1 \\ {-\sqrt{3}} \end{pmatrix}\right\} ; \quad N_2:={\mathbf {x}}+ \left\{ \begin{pmatrix} -1\\ 0 \end{pmatrix},\frac{1}{2}\begin{pmatrix} 1 \\ {\sqrt{3}} \end{pmatrix}, \frac{1}{2}\begin{pmatrix} 1\\ -\sqrt{3} \end{pmatrix}\right\} . \end{aligned}$$

The distribution $f_{\mathbf {x}}(s_1)$ is then given by the uniform distribution on $N_1$, while $f_{\mathbf {x}}(s_2)$ and $f_{\mathbf {x}}(s_3)$ are the uniform distribution on $N_2$.

Without knowing the hidden states, an outside observer will not be able to predict the future dynamics of this RWHLM even while knowing the past and present location of the walker and rotor configuration, as illustrated in Figure 8.

Fig. 8

Two instances of a two-step hidden triangular walk with the same walker’s trajectory and rotor configurations. The number at the origin records the hidden state of the origin. The pictures at the right side illustrate the future hidden state of the origin and the arrows point to (possible) future locations of the walker

Note that a non-hidden RWLM is a special case of RWHLM, with $S_x$ ($x \in V$) being the set of neighbors of x and with $f_x(y)$ $(y \in {N}(x))$ being the probability distribution concentrated on y. On the other hand, every RWHLM on a simple graph G can be emulated by a non-hidden RWLM on a larger graph (with multiple edges) in the following manner.

Let $G^\times$ be the undirected graph with vertex set V(G) and with an edge incident to x and y in $G^\times$ for each $\{x,y\}\in E(G)$ and each hidden state $s \in S_x$ of the RWHLM. Such an edge is labeled e(x, y, s).

For any $x \in V(G^\times )$, the mechanism of this RWLM on x is the Markov chain with state space the set of edges incident to x in $G^\times$ (instead of the set of neighbors of x), and with probability transition function

$$\begin{aligned} p_x^\times (e({x,y,s}),e({x,y',s'})) := p_{x}(s,s')\, (f_x(s'))(y'), \end{aligned}$$

where $p_x$ and $f_x$ are the probability transition function and the jump rule for the RWHLM, respectively.

This RWLM on $G^\times$ emulates the RWHLM on G in the following sense. Let $(X_n,\rho _n,\kappa _n)_{n \ge 0}$ be an RWHLM on G. Start an RWLM $(X_n^\times , \rho _n^\times )_{n \ge 0}$ on $G^\times$ with the following initial configuration:

$$\begin{aligned} X_0^\times&:= X_0; \qquad \rho ^\times _0(x):= e({x, \rho _0(x), \kappa _0(x))} \quad (x \in V). \end{aligned}$$

Then $(X_n, \rho _n)_{n \ge 0}$ is equal in distribution to $(X_n^\times , h(\rho _n^\times ))_{n \ge 0}$, where $h(\rho _n^\times )$ is the rotor configuration of G given by $h(\rho _n^\times )(x):=y$ if $\rho _n^\times (x)=e({x,y,s})$ for some $s \in S_x$.

As a consequence of this reduction, we can convert the hidden triangular walk from Example A.2 to a non-hidden random walk with local memory, and then apply a version of Proposition 1.4 for non-simple graphs to conclude that the scaling limit of this hidden triangular walk is a Brownian motion in ${\mathbb {R}}^2$.