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April 1997 On self-attracting $d$-dimensional random walks
Erwin Bolthausen, Uwe Schmock
Ann. Probab. 25(2): 531-572 (April 1997). DOI: 10.1214/aop/1024404411


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 $\hat{\mathbb{P}}_T$ 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 $\{\hat{\mathbb{P}}_T\}_{T > 0}$.


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Erwin Bolthausen. Uwe Schmock. "On self-attracting $d$-dimensional random walks." Ann. Probab. 25 (2) 531 - 572, April 1997.


Published: April 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0873.60008
MathSciNet: MR1434118
Digital Object Identifier: 10.1214/aop/1024404411

Primary: 60F05
Secondary: 60F10 , 60K35

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

Rights: Copyright © 1997 Institute of Mathematical Statistics

Vol.25 • No. 2 • April 1997
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