The rounding of the phase transition for disordered pinning with stretched exponential tails

The presence of frozen-in or quenched disorder in a system can often modify the nature of its phase transition. A particular instance of this phenomenon is the so-called rounding effect: it has been shown in many cases that the free-energy curve of the disordered system at its critical point is smoother than that of the homogenous one. In particular some disordered systems do not allow first-order transitions. We study this phenomenon for the pinning of a renewal with stretched-exponential tails on a defect line (the distribution $K$ of the renewal increments satisfies $K(n) \sim c_K\exp(-n^{\alpha}),$ $\alpha\in (0,1)$) which has a first order transition when disorder is not present. We show that the critical behavior of the disordered system depends on the value of $\alpha$: when $\alpha>1/2$ the transition remains first order, whereas the free-energy diagram is smoothed for $\alpha\le 1/2$. Furthermore we show that the rounding effect is getting stronger when $\alpha$ diminishes.