Perturbed GUE Minor Process and Warren’s Process with Drifts

Abstract

We consider the minor process of (Hermitian) matrix diffusions with constant diagonal drifts. At any given time, this process is determinantal and we provide an explicit expression for its correlation kernel. This is a measure on the Gelfand–Tsetlin pattern that also appears in a generalization of Warren’s process (Electron. J. Probab. 12:573–590, 2007), in which Brownian motions have level-dependent drifts. Finally, we show that this process arises in a diffusion scaling limit from an interacting particle system in the anisotropic KPZ class in 2+1 dimensions introduced in Borodin and Ferrari (Commun. Math. Phys., 2008). Our results generalize the known results for the zero drift situation.

Appendix: Determinantal Correlations

Since we refer several times to Lemma 3.4 of [12], we report it here.

Lemma A.1

(Lemma 3.4 of [12])

Assume we have a signed measure on {$x_{i}^{n},n=1,\ldots,N$, i=1,…,n} given in the form,

$$ \frac{1}{Z_N}\prod_{n=1}^{N-1} \det\bigl[\phi_n\bigl(x_i^n,x_j^{n+1} \bigr)\bigr]_{1\leq i,j\leq n+1} \det\bigl[\varPsi_{N-i}^{N} \bigl(x_{j}^N\bigr)\bigr]_{1\leq i,j \leq N}, $$

(106)

where $x_{n+1}^{n}$ are some “virtual” variables and Z _N is a normalization constant. If Z _N ≠0, then the correlation functions are determinantal.

To write down the kernel we need to introduce some notations. Define

$$ \phi^{(n_1,n_2)}(x,y)= \begin{cases} (\phi_{n_1} \ast\cdots\ast\phi_{n_2-1})(x,y),& n_1<n_2,\\ 0,& n_1\geq n_2, \end{cases} $$

(107)

where $(a* b)(x,y)=\sum_{z\in\mathbb{Z}}a(x,z) b(z,y)$, and, for 1≤n<N,

$$ \varPsi_{n-j}^{n}(x) := \bigl( \phi^{(n,N)} * \varPsi_{N-j}^{N}\bigr) (y), \quad j=1,\ldots,N. $$

(108)

Set $\phi_{0}(x_{1}^{0},x)=1$. Then the functions

$$ \bigl\{ \bigl(\phi_0*\phi^{(1,n)}\bigr) \bigl(x_1^0,x\bigr), \dots,\bigl(\phi_{n-2}*\phi ^{(n-1,n)}\bigr) \bigl(x_{n-1}^{n-2},x\bigr), \phi_{n-1}\bigl(x_{n}^{n-1},x\bigr)\bigr\} $$

(109)

are linearly independent and generate the n-dimensional space V _n . Define a set of functions $\{\varPhi_{j}^{n}(x), j=0,\ldots,n-1\}$ spanning V _n defined by the orthogonality relations

$$ \sum_x \varPhi_i^n(x) \varPsi_j^n(x) = \delta_{i,j} $$

(110)

for 0≤i,j≤n−1.

Further, if $\phi_{n}(x_{n+1}^{n},x)=c_{n} \varPhi_{0}^{(n+1)}(x)$, for some c _n ≠0, n=1,…,N−1, then the kernel takes the simple form

$$ K(n_1,x_1;n_2,x_2)= -\phi^{(n_1,n_2)}(x_1,x_2)+ \sum _{k=1}^{n_2} \varPsi_{n_1-k}^{n_1}(x_1) \varPhi_{n_2-k}^{n_2}(x_2). $$

(111)