Three-dimensional Gaussian fluctuations of non-commutative random surfaces along time-like paths
Abstract
We construct a continuous-time non-commutative random walk on $U(\mathfrak{gl}_N)$ with dilation maps $U(\mathfrak{gl}_N)\rightarrow L^2(U(N))^{\otimes\infty}$. This is an analog of a continuous-time non-commutative random walk on the group von Neumann algebra $vN(U(N))$ constructed in [15], and is a variant of discrete-time non-commutative random walks on $U(\mathfrak{gl}_N)$ [2,9]. It is also shown that when restricting to the Gelfand-Tsetlin subalgebra of $U(\mathfrak{gl}_N),$ the non-commutative random walk matches a (2+1)-dimensional random surface model introduced in [7]. As an application, it is then proved that the moments converge to an explicit Gaussian field along time-like paths. Combining with [7] which showed convergence to the Gaussian free field along space-like paths, this computes the entire three-dimensional Gaussian field. In particular, it matches a Gaussian field from eigenvalues of random matrices [5].
- Publication:
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arXiv e-prints
- Pub Date:
- January 2014
- DOI:
- 10.48550/arXiv.1401.5834
- arXiv:
- arXiv:1401.5834
- Bibcode:
- 2014arXiv1401.5834K
- Keywords:
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- Mathematics - Representation Theory;
- Mathematical Physics;
- Mathematics - Probability
- E-Print:
- 25 pages