Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  investigate a two-dimensional statistical model of   N     N        charged particles interacting via logarithmic repulsion in the presence of an oppositely charged compact region   K     K        whose charge density is determined by its equilibrium potential at an inverse temperature corresponding to   β=2     β  =  2       . When the charge on the region,   s     s       , is greater than   N     N       , the particles accumulate in a neighborhood of the boundary of   K     K       , and form a determinantal point process on the complex plane.  investigate the scaling limit, as   N→∞     N  →  ∞       , of the associated kernel in the neighborhood of a point on the boundary under the assumption that the boundary is sufficiently smooth.  find that the limiting kernel depends on the limiting value of   N/s     N  /  s       , and prove universality for these kernels. That is,  show that, the scaled kernel in a neighborhood of a point   ζ∈∂K     ζ  ∈  ∂  K        can be succinctly expressed in terms of the scaled kernel for the closed unit disk, and the exterior conformal map which carries the complement of   K     K        to the complement of the closed unit disk. When   N/s→0     N  /  s  →  0         recover the universal kernel discovered by Lubinsky (2010) in   [13]  .
['Christopher D. Sinclair', 'Maxim Yattselev']
I am writing a research paper on a idea Universality for ensembles of matrices with potential theoretic weights on domains with smooth boundary
[{"from": "human", "value": "I am writing a research paper on a idea Universality for ensembles of matrices with potential theoretic weights on domains with smooth boundary"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  investigate a two-dimensional statistical model of   N     N        charged particles interacting via logarithmic repulsion in the presence of an oppositely charged compact region   K     K        whose charge density is determined by its equilibrium potential at an inverse temperature corresponding to   \u03b2=2     \u03b2  =  2       . When the charge on the region,   s     s       , is greater than   N     N       , the particles accumulate in a neighborhood of the boundary of   K     K       , and form a determinantal point process on the complex plane.  investigate the scaling limit, as   N\u2192\u221e     N  \u2192  \u221e       , of the associated kernel in the neighborhood of a point on the boundary under the assumption that the boundary is sufficiently smooth.  find that the limiting kernel depends on the limiting value of   N/s     N  /  s       , and prove universality for these kernels. That is,  show that, the scaled kernel in a neighborhood of a point   \u03b6\u2208\u2202K     \u03b6  \u2208  \u2202  K        can be succinctly expressed in terms of the scaled kernel for the closed unit disk, and the exterior conformal map which carries the complement of   K     K        to the complement of the closed unit disk. When   N/s\u21920     N  /  s  \u2192  0         recover the universal kernel discovered by Lubinsky (2010) in   [13]  ."}]