Some physical results in four dimensional large N gauge theories on a periodic torus are summarized.

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6.PhasetransitionintheWilsonloopoperator

Non-abeliangaugetheoriesinthecon nedphasearestronglyinteractingatlargedistancesandweaklyinteractingatshortdistances.Ifthisisseenasaphasetransitioninsomeobservable,onecouldusetheuniversalbehaviorofthistransitiontoconnectthelowenergyphysicsofQCDtothehighenergyphysicsofQCD.TheWilsonloopoperatorasafunctionofitssizeisthemostlikelycandidatetostudythistransitionwithinthe0c-phaseoflargeNQCD13.SuchatransitionexistsintwodimensionallargeNQCD14anditisclaimedthatthetransitioninfourdimensionalQCDisinthesameuniversalityclass.ThistransitionisreferredtoastheDurhuus-OlesenphasetransitioninFig1.

asafunctionofitssizeinlargeNQCD.ThedistributionoftheeigenvaluesofthesmearedWilsonloopoperatorarecomparedtotheDurhuus-Olesendistributions.Re-sultsareshownforloopsofsizesl/lc=0.740,0.660,0.560,0.503.Theymatchwithk=4.03,2.30,1.41,1.15wherekisthedimensionlessareaintwodimensionlargeNQCD.k=2isthecriticalarea.

Wilsonloopoperatorssu erfromaperimeterdivergenceinfourdimen-sionsandonehastoeliminateitwhende ningthisoperatorifonewereto