Geodesic
Canonical ensemble of particles in a self-avoiding random walk
Teoretičeskaâ i matematičeskaâ fizika, Tome 191 (2017) no. 1, pp. 100-115 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider an ensemble of particles not interacting with each other and randomly walking in the $d$-dimensional Euclidean space $\mathbb R^d$. The individual moves of each particle are governed by the same distribution, but after the completion of each such move of a particle, its position in the medium is "marked" as a region in the form of a ball of diameter $r_0$, which is not available for subsequent visits by this particle. As a result, we obtain the corresponding ensemble in $\mathbb R^d$ of marked trajectories in each of which the distance between the centers of any pair of these balls is greater than $r_0$. We describe a method for computing the asymptotic form of the probability density $W_n(\mathbf r)$ of the distance $r$ between the centers of the initial and final balls of a trajectory consisting of $n$ individual moves of a particle of the ensemble. The number $n$, the trajectory modulus, is a random variable in this model in addition to the distance $r$. This makes it necessary to determine the distribution of $n$, for which we use the canonical distribution obtained from the most probable distribution of particles in the ensemble over the moduli of their trajectories. Averaging the density $W_n(\mathbf r)$ over the canonical distribution of the modulus $n$ allows finding the asymptotic behavior of the probability density of the distance $r$ between the ends of the paths of the canonical ensemble of particles in a self-avoiding random walk in $\mathbb R^d$ for $2\le d<4$.
Mots-clés : canonical ensemble, constitutive equation
Keywords: self-avoiding random walk, renormalization group, saddle-point method, asymptotic distribution. 
@article{TMF_2017_191_1_a6,
     author = {V. I. Alkhimov},
     title = {Canonical ensemble of particles in a~self-avoiding random walk},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {100--115},
     year = {2017},
     volume = {191},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2017_191_1_a6/}
}
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V. I. Alkhimov. Canonical ensemble of particles in a self-avoiding random walk. Teoretičeskaâ i matematičeskaâ fizika, Tome 191 (2017) no. 1, pp. 100-115. http://geodesic.mathdoc.fr/item/TMF_2017_191_1_a6/

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