# zbMATH — the first resource for mathematics

Interacting partially directed self avoiding walk. From phase transition to the geometry of the collapsed phase. (English) Zbl 1360.60173
The self-avoiding random walk on $$\mathbb Z^2$$ studied in this paper models a homopolymer in a poor solvent, composed of $$L$$-monomers (= of length $$L$$). The random walk takes only unitary steps upward, downward and to the right. The paper considers the case of uniform probability law on the set $$W_L$$ of all $$L$$-paths. To each self-touching between monomers, an energy reward $$\beta\geq 0$$ is assigned and with every trajectory $$w$$ of the random walk, a Hamiltonian $$H_L(w)$$ is associated, so that a probability law on the set of $$L$$-polymers is induced. Denote by $$Z_{L,\beta}$$ the partition function of the law. Consequently, the authors define the free energy $$f$$ per step: $f(\beta)= \lim_{L\to \infty}\frac{Z_{L,\beta}}{L}.$ The collapse is defined as the loss of analyticity of $$f$$ at the critical point $\beta_c:=\inf\{\beta\geq 0: f(\beta)=\beta\}$ (note that $$f(\beta)\geq \beta,$$ by the definition of $$f$$). The authors prove that the phase transition is of second order with the critical exponent $$3/2$$ and with the first order asymptotics at the critical point $$\beta_c$$ given by $f(\beta_c-\varepsilon)-\beta\sim \gamma \varepsilon^{3/2}, \;\;\varepsilon\to 0^+,$ where $$\gamma$$ is computed explicitly. The convergence of the region occupied by the properly scaled path to the deterministic limiting shape is also proved.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G50 Sums of independent random variables; random walks 82B26 Phase transitions (general) in equilibrium statistical mechanics 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics 82D60 Statistical mechanical studies of polymers
Full Text: