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)