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We study the Polyakov loop correlator in the (2+1) dimensional Z_2 gaugemodel. An algorithm that we have presented recently, allows us to reach highprecision results for a large range of distances and temperatures, giving usthe opportunity to test predictions of the effective Nambu-Goto string model.Here we focus on the regime of low temperatures and small distances. Incontrast to the high temperature, large distance regime, we find that ournumerical results are not well described by the two loop-prediction of theNambu-Goto model. In addition we compare our data with those for the SU(2) andSU(3) gauge models in (2+1) dimensions obtained by other authors. We generalizethe result of L\"uscher and Weisz for a boundary term in the interquarkpotential to the finite temperature case.Comment: 38 pages, 7 figures, version accepted for publication in JHE

Short distance behaviour of the effective string