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Combinatorial theory of permutation-invariant random matrices I: partitions, geometry and renormalization | | Franck Gabriel
; | | Date: |
10 Mar 2015 | | Abstract: | In this article, we define and study a geometry on the set of partitions of
an even number of objects. One of the definitions involves the partition
algebra, a structure of algebra on the set of such partitions depending on an
integer parameter N. Then we emulate the theory of random matrices in a
combinatorial framework: for any parameter N, we introduce a family of linear
forms on the partition algebras which allows us to define a notion of weak
convergence similar to the convergence in moments in random matrices theory. A
renormalization of the partition algebras allows us to consider the weak
convergence as a simple convergence in a fixed space. This leads us to the
definition of a deformed partition algebra for any integer parameter N and to
the definition of two transforms: the cumulants transform and the exclusive
moments transform. Using an improved triangular inequality for the distance
defined on partitions, we prove that the deformed partition algebras, endowed
with a deformation of the linear forms converge as N go to infinity. This
result allows us to prove combinatorial properties about geodesics and a
convergence theorem for semi-groups of functions on partitions. At the end we
study a sub-algebra of functions on infinite partitions with finite support : a
new addition operation and a notion of R-transform are defined. We introduce
the set of multiplicative functions which becomes a Lie group for the new
addition and multiplication operations. For each of them, the Lie algebra is
studied. The appropriate tools are developed in order to understand the
algebraic fluctuations of the moments and cumulants for converging sequences.
This allows us to extend all the results we got for the zero order of
fluctuations to any order. | | Source: | arXiv, 1503.2792 | | Services: | Forum | Review | PDF | Favorites | |
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