On the permanent of a random symmetric matrix

Matthew Kwan ^[1] ; Lisa Sauermann ^[2]
1. [1] Stanford University
  
  Stanford University
  
  Estados Unidos
2. [2] Institute for Advanced Study
  
  Institute for Advanced Study
  
  Estados Unidos
Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 28, Nº. 1, 2022
Idioma: inglés
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Resumen
- Let Mn denote a random symmetric n×n matrix, whose entries on and above the diagonal are i.i.d. Rademacher random variables (taking values ±1 with probability 1/2 each). Resolving a conjecture of Vu, we prove that the permanent of Mn has magnitude nn/2+o(n) with probability 1−o(1). Our result can also be extended to more general models of random matrices. In our proof, we build on and extend some techniques introduced by Tao and Vu, studying the evolution of permanents of submatrices in a random matrix process.