We study the inflated phase of two-dimensional lattice polygons, both convex and column-convex, with fixed area A and variable perimeter, when a weight μ t exp[−Jb] is associated with a polygon with perimeter t and b bends. The mean perimeter is calculated as a function of the fugacity μ and the bending rigidity J. In the limit μ → 0, the mean perimeter has the asymptotic behaviour t /4 √ A 1 − K(J)/(ln μ) 2 + O(μ/ ln μ). The constant K(J) is found to be the same for both kinds of polygons, suggesting that self-avoiding polygons may also exhibit the same asymptotic behaviour.

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Horizontal directional drilling (HDD) is the method expected to prevail for laying lifelines that avoids the need to make an open cut. However, the location of the drill-bit underground must be known at all times when conducting this method. As conventional electromagnetic wave-based methods are known to have several problems, herein we suggest a new method based on the propagation of elastic waves in the ground. In this study, "Vibro" is used as the generation method of elastic waves. This is a feature loaded on the heavy machineries used for HDD. Differences of arrival time are obtained from first peak times of the waveforms measured by the geophones set up on the ground. The possibility to actualize this method is examined by application of an approximate three-dimensional (3D) location method to the differences. As the result of this study, there is the possibility to actualize this method but accuracy improvement is challenges for the future.