LARGE DEVIATIONS FOR RANDOM HIVES AND THE SPECTRUM OF THE SUM OF TWO RANDOM MATRICES

• 成果类型: Article
• 署名作者: Narayanan, Hariharan; Sheffield, Scott
• 署名单位: Tata Institute of Fundamental Research (TIFR); Massachusetts Institute of Technology (MIT)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/24-AOP1687
• 发表日期: 2024
• 页码: 1093-1152

摘要

Suppose a, beta are Lipschitz, strongly concave functions from [0, 1] to R and y is a concave function from [0, 1] to R such that a(0) = y(0) = 0, a(1)=8(0) = 0 and 8(1)=y(1) = 0. For an nxn Hermitian matrix W, let spec(W) denote the vector in R whose coordinates are the eigenvalues of W listed in nonincreasing order. Let lambda=aa, mu=-beta on (0, 1) and v=ay, at all points of (0, 1], where a is the left derivative. Let lambda(i);= n2(a)-a()), for i = [n], and similarly, () == n2(B()-B(2)) and vn()=2()-(-)). Let X. Y be independent random Hermitian matrices from unitarily in- variant distributions with spectra lambda n, n, respectively. We define norm || - || 7 to correspond in a certain way to the sup norm of an antiderivative. We prove that the following limit exists: limn -> log P|||spec(Xn+Yn) - vn]]] < n2e]/n(2) We interpret this limit in terms of the surface tension o of continuum limits of the discrete hives defined by Knutson and Tao. We provide matching large deviation upper and lower bounds for the spec- trum of the sum of two random matrices X and Y, in terms of the surface tension o mentioned above. We also prove large deviation principles for random hives with a and B that are c2, where the rate function can be interpreted in terms of the max- imizer of a functional that is the sum of a term related to the free energy of hives associated with a, beta and y and a quantity related to logarithms of Vandermonde determinants associated with y.