The Annals of Probability

On self-attracting $d$-dimensional random walks

Erwin Bolthausen and Uwe Schmock

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Let $\{X_t\}_{t \geq 0}$ be a symmetric, nearest-neighbor random walk on $\mathbb{Z}^d$ with exponential holding times of expectation $1/d$, starting at the origin. For a potential $V: \mathbb{Z}^d \to [0, \infty)$ with finite and nonempty support, define transformed path measures by $d \hat{\mathbb{P}}_T \equiv \exp (T^{-1} \int_0^T \int_0^T V(X_s - X_t) ds dt) d \mathbb{P}/Z_T$ for $T > 0$, where $Z_T$ is the normalizing constant. If $d = 1$ or if the self-attraction is sufficiently strong, then $||X_t||_{\infty}$ has an exponential moment under $\mathbb{P}_T$ which is uniformly bounded for $T > 0$ and $t \in [0, T]$. We also prove that ${X_t}_{t \geq 0}$ under suitable subsequences of ${\hat{\mathbb{P}}_T}_{T > 0}$ behaves for large $T$ asymptotically like a mixture of space-inhomogeneous ergodic random walks. For special cases like a sufficiently strong Dirac-type interaction, we even prove convergence of the transformed path measures and the law of $X_T$ as well as of the law of the empirical measure $L_T$ under ${\hat{\mathbb{P}}_T}_{T > 0}$.

Article information

Ann. Probab., Volume 25, Number 2 (1997), 531-572.

First available in Project Euclid: 18 June 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60F10: Large deviations 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

$d$-dimensional random walk attractive interaction large deviations weak convergence maximum entropy principle Dirac-type interaction


Bolthausen, Erwin; Schmock, Uwe. On self-attracting $d$-dimensional random walks. Ann. Probab. 25 (1997), no. 2, 531--572. doi:10.1214/aop/1024404411.

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