2022/10/18 by Zargar, Masoud
#Differential Geometry (math.DG) #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2210.09547
Each signature \underlineλ(n)=(λ1(n),…,λn(n)), where λ1(n)≥…≥λn(n) are integers, gives an irreducible representation π\underlineλ(n):U(n)\rightarrowGL(V\underlineλ(n)) of the unitary group U(n). Suppose X is a finite-area cusped hyperbolic surface, χ is a random surface representation in Hom(π1(X),U(n)) equipped with a Haar unitary probability measure, and (\underlineλ(n))n=1∞ is a sequence of signatures. Let |\underlineλ(n)|:=∑i|λi(n)|. We show that there is an absolute constant c>0 such that if 0≠ |\underlineλ(n)|≤ c(log n)/(loglog n) for sufficiently large n, then the Laplacians Δχ,\underlineλ(n) acting on sections of the flat unitary bundles associated to the surface representations π1(X)\xrightarrowχ U(n)\xrightarrowπ\underlineλ(n)GL(V\underlineλ(n)) have the property that for every ε>0 ℙ[χ:\infSpec(Δχ,\underlineλ(n))≥(1)/(4)-ε]\xrightarrown→∞1, where Spec(Δχ,\underlineλ(n)) is the spectrum of Δχ,\underlineλ(n). A special case of this is that flat unitary bundles associated to χ:π1(X)→ U(n) asymptotically almost surely as n→∞ have least eigenvalue at least (1)/(4)-ε, irrespective of the spectral gap of X itself. This is proved using the Hide--Magee method. Using the spectral theorem above and proving a probabilistic prime geodesic theorem, we also obtain a probabilistic equidistribution theorem for the images under χ of geodesics of lengths dependent on the rank n.