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Abstract
We investigate the matrix model with weight
w(x):=exp(−z22x2+tx−x22)and unitary symmetry. In particular we study the double scaling limit as N→∞ and (N−−√t,Nz2)→(u1,u2), where N is the matrix dimension and the parameters (u_{1}, u_{2}) remain finite. Using the DeiftZhou steepest descent method, we compute the asymptotics of the partition function when z and t are of order O(N−1/2). In this regime we discover a phase transition in the (z, N)plane
characterised by the Painlevé III equation. This is the first time that
Painlevé III appears in studies of double scaling limits in Random
Matrix Theory and is associated to the emergence of an essential
singularity in the weighting function. The asymptotics of the partition
function is expressed in terms of a particular solution of the Painlevé
III equation. We derive explicitly the initial conditions in the limit Nz2→u2 of this solution.
Original language  English 

Pages (fromto)  13171364 
Number of pages  47 
Journal  Communications in Mathematical Physics 
Volume  333 
Issue number  3 
Early online date  31 May 2014 
DOIs  
Publication status  Published  Feb 2015 
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Professor Francesco Mezzadri
 Probability, Analysis and Dynamics
 School of Mathematics  Professor of Mathematical Physics
 Applied Mathematics
 Mathematical Physics
Person: Academic , Member