we start with two of the most well-studied random matrix ensembles. the limiting eigenvalue distribution of one of which is uniform on the unit disk, and the other of which is a semicircular distribution on the real line. these two distributions have a simple relation: 2 times the real part of the uniform measure on the disk gives you the semicircular distribution. in the first part of the talk i will speak about the "heat flow conjecture" which states that this simple relation can be accomplished in the matrix level by applying the heat operator to the characteristic polynomial of one of the random matrix. then i will move to the case where we apply the heat operator to a random polynomial which has roots distributed approximately uniform on the disk. in this random polynomial case, we can prove a version of the heat flow conjecture if we replace the characteristic polynomial by the random polynomial in the statement of the conjecture. finally, i will speak about the case of heat flow on the plane gaussian analytic function (gaf). these are joint work with brian hall and joint work with brian hall, jonas jalowy, and zakhar kabluchko.