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 homogeneous 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) ~ cK exp(-n.), . . (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 .: when . > ½ the transition remains of first order, whereas the free energy diagram is smoothed for . . ½. Furthermore we show that the rounding effect is getting stronger when . diminishes.