Three-dimensional Gaussian fluctuations of non-commutative random surfaces along time-like paths

详细信息    查看全文

• 作者：Jeffrey Kuan ^{kuan@math.columbia.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Asymptotic representation theory ; Quantum random walk ; Gaussian free field ; Anisotropic Kardar&ndash ; Parisi&ndash ; Zhang ; Random surface growth
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：5 November 2016
• 年：2016
• 卷：303
• 期：Complete
• 页码：716-744
• 全文大小：612 K

文摘

We construct a continuous-time non-commutative Markov operator on U(gl_N)$U({\mathfrak{gl}}_N)$ with corresponding morphisms U(gl_N)→End(L²(U(N)))^⊗∞$U({\mathfrak{gl}}_N)\to \mathrm{End}{(L^2(U(N)))}^{\otimes \infty }$ at any fixed times 0≤t₁<t₂<…$0\le t_1<t_2<\dots$. This is an analog of a continuous-time non-commutative Markov operator on the group von Neumann algebra vN(U(N))$vN(U(N))$ constructed in [16], and is a variant of discrete-time non-commutative random walks on U(gl_N)$U({\mathfrak{gl}}_N)$ and .

It is also shown that when restricting to the Gelfand–Tsetlin subalgebra of U(gl_N)$U({\mathfrak{gl}}_N)$, the non-commutative Markov operator matches a (2+1$2+1$)-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].