Large deviations and interacting random walks. (English) Zbl 1045.60026

Bolthausen, Erwin

Bolthausen, Erwin et al., Lectures on probability theory and statistics. Ecole d’été de probabilités de Saint-Flour XXIX - 1999, Saint-Flour, France, July 8–24, 1999. Berlin: Springer (ISBN 3-540-43736-3). Lect. Notes Math. 1781, 1-124 (2002).

In these lecture notes the author presents three closely related topics on random walks with self-interactions or with interactions with a wall. All topics have version for Brownian motions, but not in all cases both versions have been proved.
The first part of the lectures originates from an outstanding open problem in probability theory. This is the determination of the mean end to end distance of a standard self-avoiding random walk on the \(d\)-dimensional lattice \( \mathbb{Z}^d \) for \( d=2,3,4\). The author discusses mainly the results for the very weakly interactive case for dimension \( d= 3\). Here all paths in the set \( \Omega_n \) of paths \( \omega \) of length \( n \) receive positive weight, but the ones with many interactions are “punished”. This is given by the Gibbsian formalism as the following transformed path measure \[ \widehat{P}_{n,\beta}(\omega)=\exp\left.\left[-\frac{\beta}{2}\sum\limits_{i,j=1}^n 1_{\{\omega_i=\omega_j\}}\right]\right/Z_{n,\beta}. \] This can also be written as a sum over the whole lattice of the square of the discrete local time. This is called Domb-Joyce model. The corresponding set-up for Brownian motions leads to difficulties because evidently even the expectation under the Wiener measure is divergent for \( d\geq 2 \). Thus different techniques may apply here. One is the so-called gap regularization, where one replaces the \( \delta\) function by some smoothed version, with some parameter \( \varepsilon >0 \) and integrating over time with gap \( \varepsilon \) between the two time scales. Removing this regularization is possible, shown in Theorems 1.1 and 1.2. Next the skeleton inequalities and boundedness properties are presented as the main techniques for the proof.
The second part is devoted to random walks with self-attracting path interactions which are all closely related to large deviation theory. For technical reasons continuous time Markov processes with discrete state space are considered. The interaction is given by the following transformed path measure where the corresponding interaction (attraction) is decaying with time: \[ \widehat P_{T,\beta}(d\omega)=\exp\left[\frac{\beta}{T}\int\limits_0^Tds\int\limits_0^Tdt 1_{\{\omega_t=\omega_s\}}\right]P(d\omega)/Z_{T,\beta}. \] For \( d=1 \) and in all other dimensions if \( \beta \) is large enough, the path measure is localized. But for \( d\geq 2 \) collapse transition occurs, where the measure behaves diffusively if \( \beta \) is small. A related model for Brownian motion is the Wiener sausage in such a way that large volumes are suppressed. After providing basic large deviation theory tools the author discusses the maximum entropy principle first for simple examples and then for the transformed path measures. In two further subsections the diffusive and collapsed phase is discussed. The last two subsections of this part deal with Wiener sausage, first the large deviation for the volume and the droplet construction and then second some moderate deviation results.
In the last third part of the notes wetting transitions for one-dimensional random walks are discussed. For example one works with the tied down random walk, i.e., \( \Omega_{n,0}=\{\omega\in\Omega_n:\omega_{2n}=0\} \). The attraction to the “wall” \( (0,\ldots,0) \) is again given by a transformed path measure for some coupling parameter \( \beta > 0 \) as \[ \widehat{P}_{2n,\beta}(\omega)=\frac{1}{Z_{2n,\beta}}\exp\left[\beta\sum\limits_{i=1}^{2n-1}1_{\{\omega_i=0\}}\right]P^0_{2n}(\omega). \] Localization is then proved for this model. In another model the interaction of the random walk (the “hetero-polymer”) with the wall is produced by a random environment.
For the entire collection see [Zbl 0996.00039].

Reviewer: Stefan Adams (Berlin)

MSC:

60F10	Large deviations
82B41	Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
60J25	Continuous-time Markov processes on general state spaces
60J55	Local time and additive functionals

Keywords:

random walks; Brownian motion; Edwards’ model; local time; large deviation; polymer measure; self-attracting random walks; pinning-depinning transition