Abstract
We obtain large N asymptotics for the random matrix partition function$$Z_N(V)=\int_{\mathbb{R}^N} \prod_{i < j}(x_ix_j)^2\prod_{j=1}^Ne^{NV(x_j)}dx_j,$$ZN(V)=∫<sup>RN</sup>∏i<j<sup>(xixj)2</sup>∏j=1N<sup>eNV(xj)</sup>dxj,in the case where V is a polynomial such that the random matrix eigenvalues accumulate on two disjoint intervals (the twocut case). We compute leading and subleading terms in the asymptotic expansion for log Z<inf>N</inf>(V), up to terms that are small as $${N \to \infty}$$N→∞. Our approach is based on the explicit computation of the first terms in the asymptotic expansion for a quartic symmetric potential V. Afterwards, we use deformation theory of the partition function and of the associated equilibrium measure to generalize our results to general twocut potentials V. The asymptotic expansion of log Z<inf>N</inf>(V) as $${N \to \infty}$$N→∞ contains terms that depend analytically on the potential V and that have already appeared in the literature. In addition, our method allows us to compute the Vindependent terms of the asymptotic expansion of log Z<inf>N</inf>(V) which, to the best of our knowledge, had not appeared before in the literature. We use rigorous orthogonal polynomial and Riemann–Hilbert techniques, which had to this point only been successful to compute asymptotics for the partition function in the onecut case.
Original language  English 

Pages (fromto)  513587 
Number of pages  75 
Journal  Communications in Mathematical Physics 
Volume  339 
Issue number  2 
DOIs  
Publication status  Published  25 Oct 2015 
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Professor Tamara Grava
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