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We consider a model of semi-flexible interacting self-avoiding trails(sISAT's) on a lattice, where the walks are constrained to visit each latticeedge at most once. Such models have been studied as an alternative to theself-attracting self-avoiding walks (SASAW) to investigate the collapsetransition of polymers, with the attractive interactions being on site, asopposed to nearest-neighbor interactions in SASAW. The grand-canonical versionof the sISAT model is solved on a four-coordinated Bethe lattice, and fourphases appear: non-polymerized (NP), regular polymerized (P), dense polymerized(DP) and anisotropic nematic (AN), the last one present in the phase diagramonly for sufficiently stiff chains. The last two phases are dense, in the sensethat all lattice sites are visited once in AN phase and twice in DP phase. Ingeneral, critical NP-P and DP-P transition surfaces meet with a NP-DPcoexistence surface at a line of bicritical points. The region in which the ANphase is stable is limited by a discontinuous critical transition to the Pphase, and we study this somewhat unusual transition in some detail. In thelimit of rods, where the chains are totally rigid, the P phase is absent andthe three coexistence lines (NP-AN, AN-DP, and NP-DP) meet at a triple point,which is the endpoint of the bicritical line.

Grand-canonical solution of semi-flexible self-avoiding trails on the Bethe lattice