Critical Behavior of Non-intersecting Brownian Motions

Abstract

We study n non-intersecting Brownian motions corresponding to initial configurations which have a vanishing density in the large n limit at an interior point of the support. It is understood that the point of vanishing can propagate up to a critical time, and we investigate the nature of the microscopic space-time correlations near the critical point and critical time. We show that they are described either by the Pearcey process or by the Airy line ensemble, depending on whether a simple integral related to the initial configuration vanishes or not. Since the Airy line ensemble typically arises near edge points of the macroscopic density, its appearance in the interior of the spectrum is surprising. We explain this phenomenon by showing that, even though there is no gap of macroscopic size near the critical point, there is with high probability a gap of mesoscopic size. Moreover, we identify a path which follows the Airy$_2$ process.

Appendix A. Space-Time Correlation Functions

In this appendix, we present some background on the space-time correlation functions of the NIBM process.

We start by recalling that by definition, the transition density of the Markov process $(X(t))_t$ can be obtained as the joint probability density function of the eigenvalues of

$$\begin{aligned} M\mapsto Z_n^{-1}e^{-\frac{n}{2t}{{\,\mathrm{Tr}\,}}(M(0)-M)^2}, \end{aligned}$$

where $Z_n$ is the normalization constant. It is well-known [14, 15, 40] that employing the Harish–Chandra/Itzykson–Zuber formula, the transition density $\mathbf{x }^{(1)}\mapsto p_t(\mathbf{x }^{(0)};\mathbf{x }^{(1)})$ of $(X(t))_t$ (given X(0)) can then be computed as

$$\begin{aligned} p_t(\mathbf{x }^{(0)};\mathbf{x }^{(1)})=\left( \frac{n}{2\pi t}\right) ^{\frac{n}{2}}\frac{\prod _{i<j} (x^{(1)}_j-x_i^{(1)})}{\prod _{i<j} (x^{(0)}_j-x_i^{(0)})}\det \left( e^{-\frac{n}{2t}(x_i^{(0)}-x_j^{(1)})^2}\right) _{1\le i,j\le n}, \end{aligned}$$

where $\mathbf{x }^{(i)}=(x^{(i)}_1,\ldots ,x_n^{(i)})\in W_n:=\{\mathbf{x }\in {\mathbb {R}}^n:x_1\le \ldots \le x_n\},\, i=0,1$ and $\mathbf{x }^{(0)}:=X(0)$. A priori, this density is only defined for distinct initial values $x_j^{(0)}$ but this condition is readily removed by invoking continuity arguments. By the Chapman–Kolmogorov equations, the finite-dimensional distributions of $(X(t))_t$, i.e. the joint distributions of all finite collections of vectors $X(t_1),\ldots ,X(t_k)\in {\mathbb {R}}^{n}$ for any choice $0=t_0<t_1<\ldots <t_k$, $k\in {\mathbb {N}}$, have the densities

$$\begin{aligned}&W_n^k\ni (\mathbf{x }^{(1)},\ldots ,\mathbf{x }^{(k)})\mapsto p_{t_1\dots ,t_k}(\mathbf{x }^{(0)};\mathbf{x }^{(1)},\ldots ,\mathbf{x }^{(k)})\\&\quad :=p_{t_1}(\mathbf{x }^{(0)};\mathbf{x }^{(1)})p_{t_2-t_1}(\mathbf{x }^{(1)};\mathbf{x }^{(2)})\dots p_{t_{k}-t_{k-1}}(\mathbf{x }^{(k-1)};\mathbf{x }^{(k)})\\&\quad =\left( \prod _{l=1}^k\frac{n}{2\pi (t_l-t_{l-1})}\right) ^{\frac{n}{2}}\frac{\prod _{i<j} (x^{(k)}_j-x_i^{(k)})}{\prod _{i<j} (x^{(0)}_j-x_i^{(0)})}\prod _{l=1}^k\det \left( e^{-\frac{n}{2(t_l-t_{l-1})}(x^{(l-1)}_i-x^{(l)}_j)^2}\right) _{1\le i,j\le n}. \end{aligned}$$

An important property of NIBM is the determinantality of its correlation functions. In order to properly introduce these functions, we consider the symmetrized density ${\widehat{p}}_{t_1\dots ,t_k}(\mathbf{x }^{(0)};\mathbf{x }^{(1)},\ldots ,\mathbf{x }^{(k)})$ on $\left( {\mathbb {R}}^{n}\right) ^k$, defined for $\mathbf{x }^{(1)},\ldots ,\mathbf{x }^{(k)}\in {\mathbb {R}}^n$

$$\begin{aligned} {\widehat{p}}_{t_1\dots ,t_k}(\mathbf{x }^{(0)};\mathbf{x }^{(1)},\ldots ,\mathbf{x }^{(k)}):=\frac{1}{(n!)^k} p_{t_1\dots ,t_k}(\mathbf{x }^{(0)};\mathbf{x }^{(1)}_\le ,\ldots ,\mathbf{x }^{(k)}_\le ), \end{aligned}$$

(A.1)

where for $\mathbf{x }^{(i)}\in {\mathbb {R}}^n$, $\mathbf{x }^{(i)}_\le$ denotes the unique permuted vector built from $\mathbf{x }^{(i)}$ that lies in $W_n$. Now, the space-time correlation functions are for any collection of integer numbers $1\le m_1,\ldots ,m_k\le n$ defined as

$$\begin{aligned}&\rho ^{(n)}_{t_1,\ldots ,t_k}(\mathbf{x }^{(1)}_{m_1},\ldots ,\mathbf{x }^{(k)}_{m_k}):=\frac{(n!)^k}{\prod _{l=1}^k(n-m_l)!}\\&\quad \times \int \limits _{{\mathbb {R}}^{kn-\sum _{j=1}^km_k}} {\widehat{p}}_{t_1,\ldots ,t_k}(\mathbf{x }^{(1)},\ldots ,\mathbf{x }^{(k)})dx^{(1)}_{m_1+1}\dots dx^{(1)}_{n}\dots dx^{(k)}_{m_k+1}\dots dx^{(k)}_{n}. \end{aligned}$$

The correlation functions are multiples of the marginal densities of ${{\widehat{p}}}_{t_1\dots ,t_k}(\mathbf{x }^{(0)};\cdot ,\ldots ,\cdot )$ and allow to express the expectation of statistics in terms of integrals. Note that due to the symmetrization, the correlation functions do not directly describe individual paths of the process, i.e. $(x_1,x_2)\mapsto \rho ^{(n)}_{t}(x_1,x_2)$ does not describe the correlations of the two smallest paths but rather the correlations of paths around the point $x_1$ and paths around $x_2$ (at time t). However, statistics of special paths, e.g. those with gaps to one side, admit an effective representation in terms of correlation functions, as for instance used in the proof of Theorem 1.4. Alternatively, one may interpret the correlation functions as joint intensities of the time-dependent point process $\sum _{j=1}^n\delta _{X_j(t)}$.

The structure of the density (A.1) being a product of $k+2$ determinants in the variables $\mathbf{x }^{(0)},\ldots ,\mathbf{x }^{(k)}$ allows for an application of the Eynard–Mehta theorem [12, 31] that yields the determinantality of the correlation functions, i.e. the ability to express the correlation functions as determinants of matrices given by a certain kernel function. This leads to the formulas (1.12)–(1.13).