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We study Fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points. The kernels are natural higher order analogues of the Airy kernel and are built out of functions associated with the Painleve I hierarchy. The Fredholm determinants related to those kernels are higher order generalizations of the Tracy-Widom distribution. We give an explicit expression for the determinants in terms of a distinguished smooth solution to the Painleve II hierarchy. In addition we compute large gap asymptotics for the Fredholm determinants.
We determine completely the Tracy-Widom distribution for Dysons beta-ensemble with beta=6. The problem of the Tracy-Widom distribution of beta-ensemble for general beta>0 has been reduced to find out a bounded solution of the Bloemendal-Virag equatio
We study the Cauchy problem for the Korteweg-de Vries (KdV) hierarchy in the small dispersion limit where $eto 0$. For negative analytic initial data with a single negative hump, we prove that for small times, the solution is approximated by the solu
We consider TASEP in continuous time with non-random initial conditions and arbitrary fixed density of particles rho. We show GOE Tracy-Widom universality of the one-point fluctuations of the associated height function. The result phrased in last pas
We present four infinite families of mutually commuting difference operators which include the deformed elliptic Ruijsenaars operators. The trigonometric limit of this kind of operators was previously introduced by Feigin and Silantyev. They provide
Iorgov, Lisovyy, and Teschner established a connection between isomonodromic deformation of linear differential equations and Liouville conformal field theory at $c=1$. In this paper we present a $q$ analog of their construction. We show that the gen