Probabilistic analysis of directed polymers in a random environment: a review. (English) Zbl 1114.82017

Comets, Francis; Shiga, Tokuzo; Yoshida, Nobuo

Funaki, Tadahisa (ed.) et al., Stochastic analysis on large scale interacting systems. Papers from the 11th international conference, Hayama, Japan, July 17–26, 2002 and the international conference on stochastic analysis and statistical mechanics, Kyoto, Japan, July 29–30, 2002. Tokyo: Mathematical Society of Japan (ISBN 4-931469-24-8/hbk). Advanced Studies in Pure Mathematics 39, 115-142 (2004).

The model of directed polymers in random environment has appeared as a mathematical interpretation of certain physical models with random impurities. This model is defined on the set \({\mathbb N} \times{\mathbb Z}^d\) as follows. At each \((n,x) \in {\mathbb N} \times{\mathbb Z}^d\) there exists a random \(\eta (x,n)\in{\mathbb R}\), which models environment. All \(\eta(n,x)\)’s are i.i.d; the probabilistic law of each \(\eta(n,x)\) is defined by a measure \(Q\). Let \(P\) be the probability distribution of a simple random walk on \({\mathbb Z}^d\), which starts at \(n=0\), \(x = 0\). Then the polymer measure at a given \(n\in {\mathbb N}\) and inverse temperature \(\beta>0\) is \[ \mu_n (d \omega) = \exp \left(\beta \sum_{j=1}^n \eta (j, \omega_j) \right)P(d \omega)/ Z_n. \] It is a random probability measure on the space of random walks \(\Omega = \{(\omega_n)_{n\geq 0}\mid \omega_n \in {\mathbb Z}^d\}\). The article under reviewing gives a survey of the results for the above model in the case where the distribution of \(\eta(n,x)\)’s is either Bernoulli \[ Q(\eta (n,x) = -1) = p>0, \quad Q(\eta (n,x) = +1) = 1- p>0; \] or standard Gaussian.
For the entire collection see [Zbl 1050.60001].

Reviewer: Yuri Kozitsky (Lublin)

MSC:

82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60K35	Interacting random processes; statistical mechanics type models; percolation theory
60K37	Processes in random environments
82D60	Statistical mechanical studies of polymers

Keywords:

random walk; weak disorder; strong disorder; Bernoulli environment; diffusive behavior