Interacting partially directed self avoiding walk. From phase transition to the geometry of the collapsed phase (Q341482)

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: NEWLINE\[NEWLINEf(\beta)= \lim_{L\to \infty}\frac{Z_{L,\beta}}{L}.NEWLINE\]NEWLINE The collapse is defined as the loss of analyticity of \(f\) at the critical point NEWLINE\[NEWLINE\beta_c:=\inf\{\beta\geq 0: f(\beta)=\beta\} NEWLINE\]NEWLINE (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 NEWLINE\[NEWLINEf(\beta_c-\varepsilon)-\beta\sim \gamma \varepsilon^{3/2}, \;\;\varepsilon\to 0^+,NEWLINE\]NEWLINE where \(\gamma \) is computed explicitly. The convergence of the region occupied by the properly scaled path to the deterministic limiting shape is also proved.