A phase transition for non-intersecting Brownian motions, and the Painleve II equation

Introduction and statement of results

This paper deals with non-intersecting Brownian motions with two fixed starting positions $a_{1}$, $a_{2}$ and two fixed ending positions $b_{1}$, $b_{2}$. Let us first describe the general framework in which this paper fits.

$\sum_{k=1}^{p}n_{k}=\sum_{l=1}^{q}m_{l}=:n$.

Consider $n$ one-dimensional Brownian motions (actually Brownian bridges) such that $n_{k}$ of them start from the point $a_{k}$ at time $t=0$, for $k=1,\ldots,p$, and $m_{l}$ of them arrive in the point $b_{l}$ at time $t=1$, for $l=1,\ldots,q$, and such that the $n$ particles are conditioned not to collide with each other in the time interval $t\in(0,1)$.

We are interested in the limiting behavior where the number of Brownian particles $n$ tends to infinity in such a way that each of the fractions $n_{k}/n$, $k=1,\ldots,p$ and $m_{l}/n$, $l=1,\ldots,q$ has a limit in $(0,1)$. To obtain non-trivial limiting behavior we assume that the transition probability density of the Brownian motions scales as

where $N$ is a parameter (the inverse of the overall variance) that increases as $n$ increases. In a non-critical regime we will simply take $N=n$.

Under these assumptions, it is expected that the particles will asymptotically fill a bounded region in the time-space plane ($tx$-plane). It is also expected that for every $t\in(0,1)$, the particles at time $t$ are asymptotically distributed according to some well-defined limiting distribution. See Figures 1 and 2 for possible behaviors when $p=q=1$ and $p=q=2$, respectively.

It is natural to ask for an explicit description of this limiting distribution. Such a result is known in the classical case where $p=q=1$, which is connected to Dyson’s Brownian motion . Here it is known that up to a suitable scaling and translation, the Brownian particles at time $t\in(0,1)$ have the same joint distribution as the eigenvalues of the Gaussian unitary ensemble (GUE) of size $n\times n$, and the limiting distribution as $n\to\infty$ is given by Wigner’s semicircle law. More precisely, if all Brownian motions start from $a:=a_{1}$ at time $t=0$ and end up at $b:=b_{1}$ at time $t=1$, and if $N=n/T$, then the non-intersecting Brownian particles at time $t\in(0,1)$ are asymptotically distributed on the interval $[\alpha(t),\beta(t)]$ with endpoints

and with limiting density of the particles given by the semicircle law on that interval

By varying $t\in$, the endpoints (1.2), (1.3) parameterize an ellipse in the time-space plane, see Figure 1.

Explicit descriptions of the limiting distribution of the non-intersecting Brownian motions are also known when $p=1$, $q>1$. Also in this case there is an underlying random matrix model .

The limiting distribution is also known when $p=q=2$, $m_{1}=n_{1}=m_{2}=n_{2}=n/2$, $N=n$, and $(a_{1}-a_{2})(b_{1}-b_{2})<2$. In this case the limiting distribution as $n\to\infty$ of the Brownian particles at time $t\in(0,1)$ is obtained from a certain algebraic curve of degree four . In contrast to the previous cases, however, it is not known if the Brownian particles for finite $n$ can be described as the eigenvalues of a random matrix ensemble. See Figure 2 for an illustration of this case.

2 Separation of the endpoints

The main results of this paper will concern non-intersecting Brownian motions with two starting points $a_{1}>a_{2}$ at time $t=0$, two ending points $b_{1}>b_{2}$ at time $t=1$, and in addition

which are varying with $n$, then we assume that

for certain limiting values $p_{1}^{*},p_{2}^{*}\in(0,1)$. We also assume that $N$ increases with $n$ such that

If the assumptions (1.5)–(1.9) hold, and if the separation between the starting points $a_{1}$ and $a_{2}$, and the ending points $b_{1}$ and $b_{2}$ is large enough, then in analogy with (1.2)–(1.4), one would expect the Brownian particles to be asymptotically distributed on two disjoint ellipses in the $tx$-plane, whose intersections with the vertical line through $t$ are given by the two intervals $[\alpha_{1}^{*},\beta_{1}^{*}]$ and $[\alpha_{2}^{*},\beta_{2}^{*}]$ with

for $j=1,2$, and with limiting densities on these two intervals given by the semicircle laws

for each $t\in(0,1)$. This situation is illustrated in Figure 2. Note that

We derive the precise condition for this two-ellipse scenario to happen. It is clear that a necessary condition is the disjointness of the two ellipses.

(Disjointness of the two ellipses) The two ellipses parameterized by (1.10)–(1.11) are disjoint if and only if

proof. The two ellipses are disjoint if and only if $\alpha_{1}^{*}(t)>\beta_{2}^{*}(t)$ for all $t\in(0,1)$. From (1.10)–(1.11) this leads to the condition

which after putting $u=\sqrt{\frac{t}{1-t}}$ is equivalent to

The left-hand side is a quadratic expression in $u$, whose discriminant is negative if and only if (1.13) holds. The lemma then easily follows since $b_{1}>b_{2}$ and $a_{1}>a_{2}$. $\hfill\square\\$

We will call (1.13) the case of large separation.

(Large, critical and small separation) For each $n$, consider $n$ non-intersecting Brownian motions with two starting points $a_{1}>a_{2}$ at time $t=0$ and two ending points $b_{1}>b_{2}$ at time $t=1$, and assume that the hypotheses (1.5)–(1.9) hold. If (1.13) holds, we say that we are in a situation of large separation of the endpoints. If instead

we are in a situation of small separation, and if

we are in a situation of critical separation between the endpoints. In the latter case, we define the critical time $t_{\textrm{crit}}\in(0,1)$ as

The non-intersecting Brownian motions corresponding to each of these three cases are illustrated in Figure 2–2. In the case of critical separation, the time $t_{\textrm{crit}}$ is precisely the time where the two ellipses with endpoints parameterized by (1.10)–(1.11) are tangent, cf. Figure 2.

We may equivalently view the critical behavior in terms of the temperature parameter $T=n/N$. With $a_{j}$, $b_{j}$, and $p_{j}^{*}$ fixed for $j=1,2$, there is a critical temperature

such that $T<T_{\textrm{crit}}$ (low temperature), $T>T_{\textrm{crit}}$ (high temperature), and $T=T_{\textrm{crit}}$ (critical temperature) correspond to large, small, and critical separation, respectively.

3 Large separation: decoupling of the Brownian motions

The case of small separation of the endpoints was considered in for the special case when $p_{1}=p_{2}=1/2$. In the present paper, we will focus instead on the cases of large and critical separation. As is to be expected, in these regimes the Brownian motions asymptotically decouple into two separate groups, as is evidenced by Figures 2 and 2. This is our first main theorem.

(Decoupling of Brownian motions) Consider $n$ non-intersecting Brownian motions with two starting points $a_{1}>a_{2}$ at time $t=0$ and two ending points $b_{1}>b_{2}$ at time $t=1$. Assume that the hypotheses (1.5)–(1.9) hold, and assume that either

there is large separation of the endpoints (1.13), or

there is critical separation of the endpoints (1.16) and $t\neq t_{\textrm{crit}}$, where $t_{\textrm{crit}}$ is given by (1.17).

Then as $n\to\infty$ the Brownian particles at time $t\in(0,1)$ are asymptotically supported on the two disjoint intervals $[\alpha_{1}^{*}(t),\beta_{1}^{*}(t)]$ and $[\alpha_{2}^{*}(t),\beta_{2}^{*}(t)]$ given by (1.10)–(1.11), with limiting densities given by the semicircle laws (1.12).

In the case of critical separation, we strongly expect that the conclusion of the theorem should also be valid when $t=t_{\textrm{crit}}$. However, we will not consider the critical time in this paper.

Our proof of Theorem 1.3 follows from a steepest descent analysis of the underlying Riemann-Hilbert problem that will be described in Section 1.4. The Riemann-Hilbert problem is of size $4\times 4$ and it was also used in to analyze the case of small separation. In the (apparently) simpler case of large separation one expects that for large $n$, the $4\times 4$ RH problem asymptotically decouples into two smaller RH problems of size $2\times 2$. This is indeed the case, but in order to show this, we need a preliminary transformation where we introduce auxiliary curves in the complex plane and subsequently perform a Gaussian elimination step in the jump matrix of the RH problem, serving to annihilate some undesired entries of this matrix. This Gaussian elimination step is similar to the so-called global opening of the lens discussed in .

It follows from our analysis that the interaction between the two groups of Brownian particles decays exponentially with $n$ in case of large separation, and polynomially (like a power $n^{-1/3}$) in case of critical separation at a non-critical time. In the limit when $n\to\infty$ the particles will then indeed be distributed inside two disjoint ellipses in the time-space plane.

4 Riemann-Hilbert problem

The non-intersecting Brownian motions described in the previous subsections are related to the following Riemann-Hilbert problem (RH problem) which we already alluded to above. The RH problem was introduced in as a generalization of the RH problem for orthogonal polynomials in , see also . In accordance with Section 1.1 we will state the RH problem for general numbers of starting and ending positions $p,q$ of the Brownian motions, although in our applications we will eventually take $p=q=2$.

The RH problem consists in finding a matrix-valued function $Y(z)=$ $Y_{n_{1},\ldots,n_{p};m_{1},\ldots,m_{q}}(z)$ of size $p+q$ by $p+q$ such that

where $I_{k}$ denotes the identity matrix of size $k$; where $W(x)$ denotes the rank-one matrix (outer product of two vectors)

The RH problem has a unique solution that can be described in terms of certain multiple orthogonal polynomials (actually multiple Hermite polynomials); details will be given in Section 1.6.

Let us explain the connection between the non-intersecting Brownian motions and the RH problem in the case $p=q=2$. It is well-known , see also , that the distribution of the non-intersecting Brownian motions at time $t\in(0,1)$ describes a determinantal point process, determined by an associated correlation kernel. According to the correlation kernel $K(x,y)=K_{n_{1},n_{2};m_{1},m_{2}}(x,y)$ can be expressed in terms of the solution to the RH problem as

By general properties of determinantal point processes, Theorem 1.3 then comes down to the statement that under the conditions of Theorem 1.3, the limit of $\frac{1}{n}K_{n_{1},n_{2};n_{1},n_{2}}(x,x)$ as $n\to\infty$ exists and is equal to

for $j=1,2$. This is what we will show in Sections 2 and 3.

Our method will also allow us to obtain the local scaling limits of the correlation kernel that are common in random matrix theory and related areas, namely the sine kernel in the bulk and the Airy kernel at the endpoints of the intervals $[\alpha_{1}^{*},\beta_{1}^{*}]$ and $[\alpha_{2}^{*},\beta_{2}^{*}]$. We will not go into details about this in this paper.

5 Critical separation and the double scaling limit

where $T$ can be interpreted as a temperature variable. For $T=1$ (critical temperature) we have by (1.16), (1.25) and Theorem 1.3 that in the large $n$ limit, the particles fill out two ellipses, which are tangent to each other at the critical time $t_{\textrm{crit}}$, see Figure 2. By varying $T$ around the critical value $T_{\textrm{crit}}=1$ we move from a case of disjoint ellipses (for $T<1$) to a case of small separation (for $T>1$), where the two-ellipses scenario is not valid anymore. Hence we see a phase transition in the case of critical separation, which is clearly seen at the critical time $t_{\textrm{crit}}$. At a non-critical time $t\neq t_{\textrm{crit}}$ the phase transition is less obvious, but there is also a nontrivial transitional effect. Indeed, the distribution of particles at time $t\neq t_{\textrm{crit}}$ in the case of small separation differs from the distribution of two semicircle laws of two disjoint intervals, which it is for large separation. Hence the endpoints of the intervals do not depend analytically on the starting and ending points, which indicates the phase transition. It is a surprising outcome of our analysis that the phase transition for the case where $t\neq t_{\textrm{crit}}$ can be described by the Painlevé II equation.

In the case of critical separation we investigate the behavior of the Brownian particles in a double scaling limit where $n,N\to\infty$, and simultaneously $T\to T_{\textrm{crit}}=1$. More precisely, we consider the endpoints $a_{1}$, $a_{2}$, $b_{1}$, $b_{2}$ fixed so that (1.25) holds for all $n$. The temperature $T=T_{n}=n/N$ is varying with $n$ as follows:

We will show that in the double scaling regime described above, the steepest descent analysis of the Riemann-Hilbert problem leads in a natural way to a model Riemann-Hilbert problem related to the Painlevé II equation. More precisely, we will be led to the construction of a local parametrix that can be mapped onto the model RH problem satisfied by the $\Psi$-functions (Lax pair) associated with the Hastings-McLeod solution of the Painlevé II equation

The Hasting-McLeod solution is the special solution $q(s)$ of (1.27) which is real for real $s$ and satisfies $q(s)\sim\textrm{Ai}(s)$ as $s\to\infty$, where Ai denotes the usual Airy function. The precise form of the model RH problem will be described in Section 3.3.

The Hastings-McLeod solution of the Painlevé II equation also appears in the famous Tracy-Widom distributions for the largest eigenvalues of large random matrices. It also appears in the critical unitarily invariant matrix models, where the parameters in the model are fine-tuned so that the limiting mean eigenvalue density vanishes quadratically at an interior point of its support . In this case it leads to a new family of local scaling limits of the eigenvalue correlation kernel that involve the $\Psi$-functions associated with $q(s)$.

In our situation the Painlevé II equation does not manifest itself in the local scaling limits of the correlation kernel. The construction of the local parametrix is done at a point $x_{0}$ strictly outside of the support and it does not influence the local correlation functions for the positions of any of the particles. The point $x_{0}$ does not seem to have any physical meaning.

We emphasize that our asymptotic analysis will be only valid when $t\neq t_{\textrm{crit}}$. At the critical time $t=t_{\textrm{crit}}$ where the two ellipses are tangent, one is led to a considerably more difficult, multi-critical situation. Here one expects the appearance of a model RH problem related to some as yet unknown fourth order ODE. As already mentioned, we will not attempt to study this case in the present paper.

6 Generalities on multiple orthogonal polynomials

While the appearance of the Hastings-McLeod solution of the RH problem does not affect any of the local scaling limits, it is felt by the recurrence coefficients of the multiple Hermite polynomials. These polynomials appear in the solution of the RH problem given in Section 1.4.

To state the results, let us first recall some generalities on multiple orthogonal polynomials in the sense of . In accordance with Sections 1.1 and 1.4 we will again give the definitions for general values of $p$ and $q$, although in our applications we will eventually take $p=q=2$.

Put $\mathbf{n}:=(n_{1},\ldots,n_{p})$, $|\mathbf{n}|:=\sum_{k=1}^{p}n_{k}$ and similarly for $\mathbf{m}$ and $|\mathbf{m}|$. Assume $|\mathbf{n}|=|\mathbf{m}|+1$. We say that a sequence of polynomials $A_{1}(x),A_{2}(x),\ldots,A_{p}(x)$ is multiple orthogonal with respect to the above data if (i) the polynomials $A_{k}(x)$ have degrees bounded by $n_{k}-1$:

for $j=0,1,\ldots,m_{l}-1$ and $l=1,\ldots,q$.

Note that (1.30) states that $Q(x)$ has $m_{l}$ vanishing moments with respect to the weight $w_{2,l}(x)$, $l=1,\ldots,q$.

A schematic illustration of Definition 1.4 is shown in Figure 3. Let us comment on this figure. The left part of the figure shows the polynomials $A_{k}(x)$ and their corresponding number of free coefficients $n_{k}$, $k=1,\ldots,p$. The middle part of the figure shows how the polynomials $A_{k}(x)$ should be assembled into the function $Q(x)$ defined in (1.29). Finally, the right part of the figure schematically shows the orthogonality relations of $Q(x)$ with respect to the different weights $w_{2,l}(x)$, and it shows next to each weight $w_{2,l}(x)$ also the number of vanishing moments $m_{l}$ of $Q(x)$ with respect to this weight.

We will refer to the polynomials $A_{k}(x)$ in Definition 1.4 as multiple orthogonal polynomials (MOP). Note that these polynomials were called multiple orthogonal polynomials of mixed type in and mixed MOPs in . We will also find it convenient to use the vectorial notation $\mathbf{A}(x)=(A_{1}(x),\ldots,A_{p}(x))$. To stress the dependence on the multi-indices $\mathbf{n}$, $\mathbf{m}$ we will sometimes write $\mathbf{A}(x)=\mathbf{A}_{\mathbf{n},\mathbf{m}}(x)$ and similarly $A_{j}(x)=(A_{j})_{\mathbf{n},\mathbf{m}}(x)$.

The coefficients of the multiple orthogonal polynomials in Definition 1.4 can be found from a homogeneous linear system with $|\mathbf{n}|$ unknowns (polynomial coefficients) and $|\mathbf{m}|$ equations (orthogonality conditions). The restriction $|\mathbf{n}|=|\mathbf{m}|+1$ in Definition 1.4 guarantees that this system has a nontrivial solution. In fact, the solution space to this linear system will be at least one-dimensional. This corresponds to the fact that the MOP are only determined up to some multiplicative factor. If the MOP are unique up to a multiplicative factor then the pair of indices $\mathbf{n},\mathbf{m}$ is called normal .

The fact that the multiple orthogonal polynomials are only determined up to some multiplicative factor allows for different choices of normalization.

(Normalization types of MOP; cf. ) Assume the data in Definition 1.4 and assume that $\mathbf{n},\mathbf{m}$ is a normal pair of indices. Then the MOP in Definition 1.4 are said to satisfy

the normalization of type I with respect to the $l$th index, $l\in\{1,\ldots,q\}$, if the $m_{l}$th moment of $Q(x)$ with respect to $w_{2,l}(x)$ is equal to one, i.e., if

the normalization of type II with respect to the $k$th index, $k\in\{1,\ldots,p\}$, if the leading coefficient of $A_{k}(x)$ is equal to one, i.e., if

The vectors of MOP $\mathbf{A}_{\mathbf{n},\mathbf{m}}(x)$ corresponding to the above normalizations will be denoted as $\mathbf{A}_{\mathbf{n},\mathbf{m}}^{(I,l)}(x)$ and $\mathbf{A}_{\mathbf{n},\mathbf{m}}^{(II,k)}(x)$, respectively.

The above normalizations might not always be possible. The type $(I,l)$ normalization is not possible in those cases where the integral on the left side of (1.31) is equal to zero. Similarly, the type $(II,k)$ normalization is not possible in those cases where the $k$th polynomial $A_{k}(x)$ has degree strictly smaller than $n_{k}-1$.

As mentioned above, the MOP appear in the solution of the RH problem in Section 1.4. Let us describe this is somewhat more detail.

Recall that in Definition 1.4 we needed the condition $|\mathbf{n}|=|\mathbf{m}|+1$ to ensure the existence of the multiple orthogonal polynomials. But let us now assume that $|\mathbf{n}|=|\mathbf{m}|$. In this case, the definition of MOP makes no sense. Indeed, the coefficients of the MOP would then solve a homogeneous linear system with as many equations as unknowns, which has in general only the trivial solution.

Therefore, to apply the definition of MOP in a meaningful way for a pair of multi-indices satisfying $|\mathbf{n}|=|\mathbf{m}|$, we should first adapt the multi-indices. There are essentially $p+q$ ways to proceed:

One can increase one of the components $n_{k}$, i.e., one can work with the pair of multi-indices $\mathbf{n}+\mathbf{e}_{k},\mathbf{m}$, for some $k\in\{1,2,\ldots,p\}$.

One can decrease one of the components $m_{l}$, i.e., one can work with the pair of multi-indices $\mathbf{n},\mathbf{m}-\mathbf{e}_{l}$, for some $l\in\{1,2,\ldots,q\}$.

Here $\mathbf{e}_{k}$ denotes the vector which has all its entries equal to zero, except for the $k$th entry which equals one. The length of $\mathbf{e}_{k}$ should be clear from the context.

Let us discuss the MOP corresponding to each of the $p+q$ pairs of multi-indices above. In the case of a pair of multi-indices $\mathbf{n}+\mathbf{e}_{k},\mathbf{m}$, we are dealing with multiple orthogonal polynomials where the $k$th polynomial $A_{k}$, $k\in\{1,2,\ldots,p\}$ has increased degree; it will be natural to normalize the resulting MOP such that this $k$th polynomial $A_{k}$ is monic, i.e., to work with a normalization of type $(II,k)$. This leads to the vector of MOP

In the case of a pair of multi-indices $\mathbf{n},\mathbf{m}-\mathbf{e}_{l}$, we are dealing with multiple orthogonal polynomials where the $l$th orthogonality condition, $l\in\{1,2,\ldots,q\}$ has a decreased number of vanishing moments; it will be natural to normalize the resulting MOP such that this omitted moment equals one, i.e., to work with a normalization of type $(I,l)$. This leads to the vector of MOP

Of course we are assuming here that all type I and type II normalizations in (1.33) and (1.34) exist. It turns out that the existence of each of these $p+q$ vectors of MOP is equivalent to a single condition , to which we will loosely refer here as the solvability condition. This condition will always be satisfied in the case of Gaussian weight functions (1.18)–(1.19).

Now we can use the above MOP to solve the RH problem in Section 1.4. Indeed, the $p+q$ vectors of MOP in (1.33), (1.34) are row vectors of length $p$, and they can therefore be stacked into the first $p$ columns of a matrix of size $(p+q)\times(p+q)$. Denote with $\widetilde{Y}(z)$ such a matrix. The entries in the remaining $q$ columns of $\widetilde{Y}(z)$ are defined as Cauchy transforms of the functions $Q^{(II,k)}$ and $Q^{(I,l)}$ defined as in (1.29). More precisely, the last $q$ entries in the rows $k=1,\ldots,p$ of $\widetilde{Y}(z)$ are defined by

(Solution to the Riemann-Hilbert problem; cf. ) Let $\mathbf{n},\mathbf{m}$ with $|\mathbf{n}|=|\mathbf{m}|$ be such that the solvability condition holds. Then there exists a unique solution $Y(z)=Y_{\mathbf{n},\mathbf{m}}(z)$ of the RH problem in Section 1.4. This solution is given by

where $\widetilde{Y}(z)$ is the matrix constructed from the MOP as described above, and where $D=\operatorname{diag}(I_{p},-2\pi iI_{q})$.

For example, in the special case where $p=q=2$, the solution $Y(z)$ of the Riemann-Hilbert problem in Section 1.4 is given by the $4\times 4$ matrix

where $D:=\operatorname{diag}(1,1,-2\pi i,-2\pi i)$, and where the entries denoted with $*$ are certain Cauchy transforms as in (1.35) and (1.36). The latter entries will be irrelevant in what follows.

7 Recurrence relations for multiple Hermite polynomials

In analogy with the three-term recurrence relations for classical orthogonal polynomials on the real line, one can show that the multiple orthogonal polynomials in Section 1.6 satisfy certain $p+q+1$ term recurrence relations. We will state these relations in the case of multiple Hermite polynomials, i.e., when the weight functions of the MOP are given by the Gaussians (1.18)–(1.19). For simplicity, we assume throughout that $p=q=2$, as in (1.37).

Define the next term in the asymptotic expansion of $Y(z)$ in (1.22) as

The entries of the matrix $Y_{1}=(Y_{1})_{\mathbf{n},\mathbf{m}}$ in (1.38) will be denoted by $(c_{i,j})_{i,j=1}^{4}$.

(Recurrence relations for multiple Hermite polynomials) Assume $p=q=2$ and let the weight functions be defined by (1.18)–(1.19). Then the multiple Hermite polynomials in (1.37) satisfy the $5$-term recurrence relations

Here we denote with $c_{i,j}$ the entries of the matrix $Y_{1}=(Y_{1})_{\mathbf{n},\mathbf{m}}$ in (1.38).

The proof of Proposition 1.7 will be given in a more general setting in Section 5.1. The explicit form of the first term in the right-hand side of each of (1.39)–(1.42) is only valid under the assumption of Gaussian weight functions (1.18)–(1.19); the explicit form of these terms will be established in Section 5.6.1.

Note that the recurrence relations (1.39)–(1.42) contain several recurrence coefficients of the form $c_{i,j}c_{j,i}$ with $i<j$; it will be convenient to collect them in the $4$ by $4$ matrix

It turns out that there exist certain connections between these recurrence coefficients.

(Relations between recurrence coefficients) Assume $p=q=2$ and let the weight functions be defined by (1.18)–(1.19). Then the $2$ by $2$ submatrix

of (1.43) has row sums equal to $t(1-t)\frac{n_{k}}{N}$, $k=1,2$, and column sums equal to $t(1-t)\frac{m_{l}}{N}$, $l=1,2$. Next, assume that $n_{1}=m_{1}$ and $n_{2}=m_{2}$. Then all the recurrence coefficients $c_{i,j}c_{j,i}$ in (1.43) can be expressed in terms of $c_{1,2}c_{2,1}$ and $c_{1,4}c_{4,1}$ alone, by means of the following relations:

Note that (1.45)–(1.47) follow immediately from the stated row and column sum relations for the matrix (1.44). The latter will be established for general values of $p$ and $q$ in Section 5.4; see also Section 5.5 for a spectral curve interpretation of these relations. On the other hand, the equation (1.48) will be established (in a slightly more general form) in Section 5.6.2.

8 Painlevé II asymptotics for recurrence coefficients

Finally we are in position to formulate the Painlevé II asymptotics of the recurrence coefficients in (1.39)–(1.42) under the double scaling regime in Section 1.5. This is our second main result. We first consider the off-diagonal recurrence coefficients, i.e., the recurrence coefficients of the form $c_{i,j}c_{j,i}$ with $i<j$.

(Asymptotics of off-diagonal recurrence coefficients) Assume the double scaling regime (1.25)–(1.26), and let $t\in(0,1)$ be a non-critical time, i.e., $t\neq t_{\textrm{crit}}$. Define the constants

where $L$ is defined in (1.26). Then we have

as $n\to\infty$, where $q(s)$ denotes the Hastings-McLeod solution to the Painlevé II equation. The asymptotic behavior of the other recurrence coefficients $c_{i,j}c_{j,i}$ with $i<j$ is then determined by (1.45)–(1.48) in Proposition 1.8.

The key point of Theorem 1.9 is that it shows that the Painlevé II equation shows up in the large $n$ behavior of the recurrence coefficients in the case of critical separation at a non-critical time.

(The case of large separation) Using the results in this paper, one can prove a similar result for the case of large separation of the endpoints (1.13). In this case, it can be shown that there exists a constant $c>0$ such that

(The case of small separation) Performing a similar analysis of the results in , one can show that in case of small separation of the endpoints (1.15), and assuming the additional hypothesis $p_{1}=p_{2}=1/2$, the following expansions hold:

We have a similar theorem for the diagonal recurrence coefficients in (1.39)–(1.42), i.e., for the first terms in the right-hand side of each of these equations.

(Asymptotics of diagonal recurrence coefficients) Under the same assumptions as in Theorem 1.9 we have that

9 Phase diagram

The main results of this paper and their relation to the other results known in the literature can be nicely summarized by means of a phase diagram. See Figure 4.

Let us comment on Figure 4. Assume that the endpoints $a_{j}$, $b_{j}$, $j=1,2$ are fixed and satisfy the critical separation (1.25). The horizontal axis in the figure denotes the time $t\in(0,1)$ and the vertical axis denotes the temperature $T>0$. The diagram is divided into different regions according to the behavior of the limiting distribution for $n\to\infty$ of the non-intersecting Brownian motions at time $t$ and temperature $T$. The region where $T<1$ corresponds to the case of large separation; according to Theorem 1.3 the limiting distribution is given here by two semicircles on the two disjoint intervals $[\alpha_{1}^{*}(t),\beta_{1}^{*}(t)]$ and $[\alpha_{2}^{*}(t),\beta_{2}^{*}(t)]$. At temperature $T=1$, these two intervals meet each other at a certain critical time $t_{\textrm{crit}}$. When $T$ further increases, the intersection region between the two groups of Brownian motions starts to grow; this is indicated by the boldface curve in the middle of the picture. In the region below this curve but above $T=1$, the limiting distribution at time $t$ is still supported on two disjoint intervals, but now with distribution described in terms of a certain algebraic curve of degree 4 , rather than semicircle laws. In the region above the curve, the limiting distribution is on one interval .

In case where $p_{1}^{*}=p_{2}^{*}=1/2$, one can find an explicit description of the boundary curve in Figure 4 from the results in ; it turns out that this curve is given by the equation

The curve plotted in Figure 4 corresponds to the choice of endpoints $a_{1}=-a_{2}=b_{1}=-b_{2}=1/\sqrt{2}$ and hence $t_{\textrm{crit}}=1/2$.

Figure 4 also displays the phase transitions between the different regions. On the horizontal line $T=1$ the phase transition is described in terms of the Painlevé II equation as shown in this paper. On the curve (1.61) one expects a description in terms of the Pearcey kernels; although for the case of two starting and two ending points this has not been strictly proven. For the case of non-intersecting Brownian motion with one starting and two ending points, the Pearcey kernels were obtained in (in the equivalent setting of Gaussian random matrices with external source), see also . Finally, at the place where the two curves in Figure 4 meet, one expects a phase transition in terms of an as yet unknown family of kernels. This is indicated by the question mark in the figure.

10 Outline of the paper

The remainder of this paper is organized as follows. In Sections 2 and 3 we apply the Deift-Zhou steepest descent analysis to the Riemann-Hilbert problem for multiple Hermite polynomials. We perform this analysis for the case of large separation in Section 2 and for the case of critical separation at a non-critical time in Section 3, leading to the proof of Theorem 1.3. In the critical case we are also led to a local parametrix for the RH problem in terms of the Painlevé II equation. Next, we investigate the large $n$ asymptotics of the recurrence coefficients of the multiple Hermite polynomials under the double scaling regime. This is done in Section 4, leading to the proof of Theorems 1.9 and 1.12. Finally, the Propositions 1.7 and 1.8 are established in Section 5.

Steepest descent analysis in the case of large separation

In this section we analyse the non-intersecting Brownian motions in case of large separation between the endpoints (1.13). Using the Deift/Zhou steepest descent analysis of the Riemann-Hilbert problem, we show that the interaction between the two groups of Brownian particles decays exponentially with $n$, thereby establishing Theorem 1.3.

Our starting point is the RH problem (1.20)–(1.22) with $p=q=2$ and in addition $n_{1}=m_{1}$ and $n_{2}=m_{2}$. As in (1.6) we write $p_{j}=n_{j}/n$. Without loss of generality we take $N=n$ (i.e., $T=1$). For $T=1$ the case of large separation corresponds to $(a_{1}-a_{2})(b_{1}-b_{2})>(\sqrt{p_{1}^{*}}+\sqrt{p_{2}^{*}})^{2}$. Since $p_{j}\to p_{j}^{*}$ as $n\to\infty$, we already assume that $n$ is so large that

Thus $Y$ satisfies the following RH problem.

The entries of the rank-one block $W=W(x)$ in (2.1) can be written explicitly as

It will be convenient to write the diagonal entries of (2.1) as

Our goal is to show that in the large $n$ limit the $4\times 4$ matrix valued RH problem for $Y(z)$ essentially decouples into two smaller $2\times 2$ problems with weight functions (2.3) and (2.4). To show that this decoupling indeed occurs, we need to show that in some sense, the off-diagonal entries in (2.1) can be neglected with respect to the diagonal entries of (2.1). More precisely, one expects that

around the interval $(\alpha_{1}^{*},\beta_{1}^{*})$, the $(1,1)$ entry of (2.1) is dominant (as $n\to\infty$) with respect to the other three entries.

around the interval $(\alpha_{2}^{*},\beta_{2}^{*})$, the $(2,2)$ entry of (2.1) is dominant with respect to the other three entries.

Here $\alpha_{j}^{*}$, $\beta_{j}^{*}$ for $j=1,2$ are as in (1.10)–(1.11).

Remarkably, these expectations are not confirmed by a straightforward steepest descent analysis in which the first transformation of the RH problem is based on the two semicircle densities (1.12) and the corresponding $g$-functions. This approach turns out to be successful only for $t$ near the critical time $t_{\textrm{crit}}$ defined in (1.17). When $t$ is sufficiently close to however, one runs into difficulties since then

the $(1,2)$ entry of (2.1) blows up (i.e., becomes exponentially large when $n\to\infty$) somewhere in the interval $(\alpha_{1}^{*},\beta_{1}^{*})$, and

the $(2,1)$ entry of (2.1) blows up somewhere in the interval $(\alpha_{2}^{*},\beta_{2}^{*})$.

Similar problems occur when $t$ is close to $1$, but then with the roles of the $(1,2)$ and $(2,1)$ entries of $W$ reversed.

In order to prevent the blow-up of undesired entries, we make a first preliminary transformation that is described in the next subsection. The transformation is different for the two cases $0<t\leq t_{\textrm{crit}}$ and $t_{crit}\leq t<1$. For definiteness we assume from now on

The case where $t_{\textrm{crit}}\leq t<1$ is similar and the corresponding modifications will be briefly commented on later.

2 First transformation: Gaussian elimination in the jump matrix

The first transformation of the RH problem is a Gaussian elimination step for the jump matrix, serving to annihilate some of the undesired entries in (2.1). This elimination step will be at the price of introducing new jump matrices on certain contours $\Gamma_{1}$, $\Gamma_{2}$ in the complex plane. This transformation is similar to the so-called global opening of the lens discussed in .

for $j=1,2$. Since $p_{j}$ is varying with $n$, the quantities (2.8)–(2.9) are also varying with $n$. For $n\to\infty$ they tend to $\alpha_{j}^{*}$ and $\beta_{j}^{*}$ given by (1.10)–(1.11) with $T=1$. Define the corresponding semicircle laws

which are also (slightly) varying with $n$.

We take a reference point $x_{0}\in(\beta_{2},\alpha_{1})$ and we choose unbounded contours $\Gamma_{1}$, $\Gamma_{2}$ in the complex plane, crossing the real axis in points $x_{1}$, $x_{2}$ so that

Thus $X$ satisfies the following RH problem.

For the asymptotic condition (2.14) we note that

Note also that the jump matrices on $\Gamma_{1}$ and $\Gamma_{2}$ tend to the identity matrix as $z\to\infty$.

What we have gained is that in the jump matrices on the intervals $(-\infty,x_{2})$ and $(x_{1},\infty)$, the first and second columns of the top right $2\times 2$ block (2.1) of the jump matrix have been eliminated, respectively.

The description of the Gaussian elimination step above has been done under the assumption (2.7), i.e., $0<t\leq t_{\textrm{crit}}$. The case where $t_{\textrm{crit}}<t<1$ is different since it requires other entries of the jump matrix to be eliminated in the different regions of the complex plane. In order to do so one would define $X$ differently as (compare with (2.11), (2.12))

The further steps in the steepest descent analysis will then be similar to the case where $0<t\leq t_{\textrm{crit}}$ and we will not discuss this any further.

Alternatively, the results for $t_{\textrm{crit}}<t<1$ could be reduced to those for $0<t\leq t_{\textrm{crit}}$ by virtue of the involution symmetry to be discussed in Section 5.3.3; a further note about this will be given in Remark 4.5.

3 Second transformation: g𝑔g-functions

The second transformation is the normalization of the RH problem at infinity. To this end we use the so-called $g$-functions.

As said before, in the limit $n\to\infty$ we expect the Brownian particles to be distributed on two separate intervals $[\alpha_{1}^{*},\beta_{1}^{*}]$ and $[\alpha_{2}^{*},\beta_{2}^{*}]$ (recall $t$ is fixed), with limiting densities given by the two Wigner semicircle laws (1.12).

For finite $n$, we have defined $\alpha_{j}$ and $\beta_{j}$ by (2.8)–(2.9) and the densities (2.10) that are varying with $n$. We use the $g$-functions corresponding to these varying semicircle densities. We define

It follows from the above observations and from the definitions (2.5), (2.6), (1.10)–(1.11) that

Moreover, from the signs of the square roots in (2.18) and (2.19) we have

The above equations (2.22)–(2.25) are the Euler-Lagrange variational conditions for the equilibrium measure under an external field. Indeed, the Wigner semicircle laws (2.10) (after normalization by a factor $1/p_{j}$) are the equilibrium measures in the presence of the quadratic external fields $V_{1}$, $V_{2}$, respectively, see . The fact that $V_{1}$ in (2.5) has a factor $p_{1}$ in its denominator can be interpreted by noting that for $p_{1}\to 0$, the external field $V_{1}$ gets stronger and stronger and hence the (potential theoretic) electrostatic particles are pushed together onto a narrower and narrower interval, which is confirmed by (2.8)–(2.9).

Now we use the $g$-functions to normalize the RH problem at infinity. We define a new $4\times 4$ matrix-valued function $T=T(z)$ by

The RH problem for $T=T(z)$ is normalized at infinity in the sense that

as $z\to\infty$. This follows from the fact that $g_{j}(z)=p_{j}\log z+O(1/z)$ as $z\to\infty$ for $j=1,2$, so that

as $z\to\infty$. Hence the factor $G^{n}$ in (2.26) cancels the powers of $z$ appearing in (2.14). Also note that the similarity relation with the matrix $L$ in (2.26) does not change the asymptotics (2.29).

Using (2.34) and after a little calculation, we find that we can write all jump matrices in the RH problem for $T(z)$ as shown in Figure 6. On $(x_{2},x_{1})$ the jump matrix is

Thus $T$ satisfies the following RH problem.

4 Asymptotic behavior of the jump matrices

In this subsection, we investigate the asymptotic behavior in the large $n$ limit of each of the jump matrices in the RH problem for $T=T(z)$ in Figure 6. Our goal is to show that all the jump matrices are exponentially close (as $n\to\infty$) to the identity matrix, except for the jump matrices on the intervals $(\alpha_{1},\beta_{1})$ and $(\alpha_{2},\beta_{2})$ which have an oscillatory behavior. To be able do so we still have the freedom to take the reference point $x_{0}$ and the contours $\Gamma_{1}$ and $\Gamma_{2}$ in an appropriate way. We also have the constant $\kappa$ in (2.28) at our disposal.

In what follows we will make extensive use of the $\xi$-functions which are essentially the derivatives of the $\lambda$-functions. We define

Inserting (2.18)–(2.19) and (2.8)–(2.9) into these expressions, we obtain after some simplification,

proof. All statements immediately follow from (2.43), except maybe the statement of part (c). Part (c) follows from the fact that

which is indeed non-negative because of our assumption (2.7). $\hfill\square\\$

The main property of the $\xi$-functions is contained in the following lemma. It will only be valid under the large separation assumption, as we will see in the proof.

There is a value $x_{0}\in(\beta_{2},\alpha_{1})$ such that

proof. We prove the lemma in two steps. In the first step we show that there exists $x_{0}\in(\beta_{2},\alpha_{1})$ such that

This we can do by giving $x_{0}$ the explicit value

which means that $x_{0}$ lies between the midpoints of the two intervals, and so in particular $\alpha_{2}<x_{0}<\beta_{1}$.

Because we are assuming that we are in a situation of large separation we have $(\sqrt{p_{1}}+\sqrt{p_{2}})^{2}<(a_{1}-a_{2})(b_{1}-b_{2})$ so that we obtain from (2.46)

It then follows that both factors on the left-hand sides of (2.47) and (2.48) are positive, so that

Furthermore, we obtain from (2.47) and (2.48)

it follows from (2.49) that $\xi_{3}(x_{0})>\xi_{4}(x_{0})$, as claimed as well.

If $\xi_{3}(\beta_{2})<\Xi_{2}$, then $\xi_{3}(\beta_{2})<\xi_{4}(\beta_{2})$, and since by what we already proved, the inequality $\xi_{3}(x)>\xi_{4}(x)$ holds for some $x\in(\beta_{2},\alpha_{1})$, there also exists $x^{*}\in(\beta_{2},\alpha_{1})$ where equality $\xi_{3}(x^{*})=\xi_{4}(x^{*})$ holds. Then by the monotonicity properties of the $\xi$-functions (stated in parts (a) and (b) of Lemma 2.2)

Then by slightly increasing $x^{*}$ we find $x_{0}$ such that the inequalities (2.44) hold (with strict inequalities).

If $\xi_{3}(\beta_{2})=\Xi_{2}$, then as above we have

Then for $x_{0}$ slightly larger than $\beta_{2}$, we have (2.44) with strict inequalities, since $\xi_{2}^{\prime}(x_{0})=+\infty$ and $\xi_{4}^{\prime}(x_{0})=-\infty$, see formulas (2.43).

If $\xi_{3}(\beta_{2})>\Xi_{2}$, then $\xi_{3}(\beta_{2})>\xi_{2}(\beta_{2})$. Since $\xi_{2}$ is increasing we find $\xi_{2}(\alpha_{1})>\xi_{2}(\beta_{2})=\Xi_{2}\geq\Xi_{1}=\xi_{3}(\alpha_{1})$, see Lemma 2.2. Thus there exists $x_{0}\in(\beta_{2},\alpha_{1})$ with $\xi_{2}(x_{0})=\xi_{3}(x_{0})$. We have $\xi_{4}(x_{0})<\xi_{2}(x_{0})$, $\xi_{1}(x_{0})<\xi_{3}(x_{0})$, and

again by Lemma 2.2, so that $\xi_{1}(x_{0})\leq\xi_{4}(x_{0})$ and the inequalities (2.44) hold. $\hfill\square\\$

It follows from the proof that in most cases we may assume that the inequalities (2.44) are strict. Only if $t=t_{\textrm{crit}}$ then $\Xi_{1}=\Xi_{2}$ and then we can only obtain

After the preliminaries on the $\xi$-functions we consider the jump matrices on $\Gamma_{1}$ and $\Gamma_{2}$. We want that the jump matrices on these curves, shown in Figure 6, are exponentially close to the identity matrix as $n\to\infty$. Thus we want to choose $\Gamma_{1}$ and $\Gamma_{2}$ so that

for certain constants $c_{1}$, $c_{2}$ that do not depend on $n$.

We take the point $x_{0}$ satisfying the inequalities (2.44) of Lemma 2.3. We have by (2.34)

The constant $\kappa$ is still at our disposal. We choose it here so that (2.54) vanishes for $z=x_{0}$. Thus

see (2.37), we can then find $x_{1}$ and $x_{2}$ sufficiently close to $x_{0}$ so that $\beta_{2}<x_{2}<x_{0}<x_{1}<\alpha_{1}$ and

We will choose $\Gamma_{1}$ and $\Gamma_{2}$ so that they cross the real axis in $x_{1}$ and $x_{2}$, respectively. It remains to show that they can be extended to infinity, while remaining in the open sets

The fact that we can indeed choose $\Gamma_{1}$ and $\Gamma_{2}$ this way, follows from the following lemma.

For $j=1,2$, let $\Omega_{j}^{o}$ denote the connected component of $\Omega_{j}$ that contains $x_{j}$. Then $\Omega_{j}^{o}$ is unbounded. In addition, for each $\varepsilon>0$ there exists $R>0$ so that

Suppose that $\overline{\Omega_{1}^{o}}$ (the closure of $\Omega_{1}^{o}$) has nonempty intersection with the interval $[\alpha_{2},\beta_{2}]$. Since $\Omega_{1}^{o}$ is connected, and symmetric with respect to the real axis, it will then surround the point $x_{2}$, so that $\Omega_{2}^{o}$ must be bounded and

Then if $\Omega_{1}^{o}$ is bounded we obtain a contradiction with the maximum principle in the same way. Thus $\Omega_{1}^{o}$ is unbounded, and likewise $\Omega_{2}^{o}$ is unbounded as well.

as $z\to\infty$, since $g_{j}(z)=p_{j}\log z+O(1/z)$ as $z\to\infty$. In addition, we have

We conclude from Lemma 2.4 that the curves $\Gamma_{1}$, $\Gamma_{2}$ can be extended to infinity so that

for some constants $c_{1},c_{2}>0$. Since as $n\to\infty$ we are in a non-critical situation, the contours $\Gamma_{j}$ and the constants $c_{j}$ can be taken independently of $n$ for $n$ large enough. Then the off-diagonal entries in the jump matrices on $\Gamma_{1}$ and $\Gamma_{2}$ are uniformly exponentially small as $n\to\infty$, as required.

We next investigate the jump matrices in the RH problem for $T$ on the various parts of the real line. Recall that the jump matrices are given in Figure 6 and so we are interested in the behavior of $\lambda_{k,+}-\lambda_{j,-}$ on the real line for various combinations of $j$ and $k$.

proof. (a) From the definitions (2.34) and (2.5)–(2.6) it follows that for $j=1,2$,

so that part (a) is a restatement of the Euler-Lagrange conditions (2.22)–(2.25).

(b) Recall that the constant $\kappa$ was chosen so that $\lambda_{3}(x_{0})=\lambda_{4}(x_{0})$. Then in view of part (a) we know that

Taking $x_{1}$ and $x_{2}$ sufficiently close to $x_{0}$ (which we can do without loss of generality), we then have the inequalities (2.64)–(2.65) on the interval $(x_{2},x_{1})$.

where the last inequality holds because of Lemma 2.3. Since

The inequality (2.65) for $x>x_{0}$ follows in the same way. $\hfill\square\\$

It follows from Lemma 2.5 that, up to exponentially small corrections, the jump matrices on the real line in the RH problem for $T$ take the following form

and $\begin{pmatrix}I_{2}&0\\ 0&I_{2}\end{pmatrix}$ on $(x_{2},x_{1})$.

The matrices (2.66) and (2.67) are in a standard form. By standard arguments we have that the non-trivial diagonal entries of (2.66) and (2.67) are rapidly oscillating on the intervals $[\alpha_{2},\beta_{2}]$ and $[\alpha_{1},\beta_{1}]$, respectively, and they are equal to $1$ outside these intervals.

From part (a) of Lemma 2.5 we can further deduce that (2.66) reduces to

Outside these intervals, the jump matrices (2.66) and (2.67) are exponentially close to the identity.

4.4 Summary

Summarizing, we have shown that we can take $\Gamma_{1}$ and $\Gamma_{2}$ so that all jumps in the RH problem for $T$ tend to the identity matrix as $n\to\infty$, except for the jump matrices on the two intervals $[\alpha_{j},\beta_{j}]$, $j=1,2$, which are given by (2.68)–(2.69) plus an exponentially small term.

Observe that each of the asymptotic jump matrices above has the sparsity pattern

It follows that the $4\times 4$ RH problem for $T=T(z)$ asymptotically decouples into two RH problems of size $2\times 2$, one involving rows and columns $1$ and $3$ and the other involving rows and columns $2$ and $4$. The coupling between the two RH problems is exponentially small as $n\to\infty$. Thus we are dealing now essentially with two RH problems of size $2\times 2$. The remaining steps in the Deift-Zhou steepest descent analysis can then be done in a standard way on these decoupled $2\times 2$ problems. This will be described in the next subsection.

5 Remaining steps of the steepest descent analysis

The next step in the steepest descent analysis is to open lenses around the intervals $(\alpha_{j},\beta_{j})$, $j=1,2$. This operation serves to transform the oscillating jumps on these intervals into exponentially decaying or constant ones. This can be done in the usual way on each of the decoupled $2\times 2$ problems. We obtain in this way a new matrix function $S=S(z)$ obtained from $T(z)$ by multiplication on the right with a suitable transformation matrix, the precise form of which depends on the different regions of the complex plane.

Let us describe this in detail. First consider the interval $[\alpha_{1},\beta_{1}]$. We have here the matrix factorization

We can then open a lens around $[\alpha_{1},\beta_{1}]$ and define

Similarly we open a lens around $[\alpha_{2},\beta_{2}]$ and define

The matrix function $S=S(z)$ satisfies the following RH problem

$S_{+}(x)=S_{-}(x)\begin{pmatrix}0&0&1&0\\ -e^{n(\lambda_{1,+}-\lambda_{2})}&1&e^{n(\lambda_{3,+}-\lambda_{2})}&0\\ -1&0&0&0\\ 0&0&0&1\end{pmatrix}$ on $(\alpha_{1},\beta_{1})$

$S_{+}(x)=S_{-}(x)\begin{pmatrix}1&-e^{n(\lambda_{2,+}-\lambda_{1})}&0&e^{n(\lambda_{4,+}-\lambda_{1})}\\ 0&0&0&1\\ 0&0&1&0\\ 0&-1&0&0\end{pmatrix}$ on $(\alpha_{2},\beta_{2})$

$S_{+}(x)=S_{-}(x)\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ e^{n(\lambda_{1}-\lambda_{3})}&0&1&0\\ 0&0&0&1\end{pmatrix}$ on the lips of the lens around $[\alpha_{1},\beta_{1}]$,

$S_{+}(x)=S_{-}(x)\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&e^{n(\lambda_{2}-\lambda_{4})}&0&1\end{pmatrix}$ on the lips of the lens around $[\alpha_{2},\beta_{2}]$.

On the remaining contours $\Gamma_{1},\Gamma_{2},(-\infty,\alpha_{2}),(\beta_{2},\alpha_{1})$ and $(\beta_{1},\infty)$, the jumps for $S$ are exactly the same as those for $T$.

As already mentioned before, the entries in positions $(1,2)$, $(1,4)$, $(2,1)$ and $(2,3)$ in the jump matrices for $S$ are all exponentially small when $n\to\infty$ (use parts (a) and (b) of Lemma 2.5).

5.2 Model RH problem: Parametrix away from the branch points

We consider now the model RH problem obtained from the RH problem for $S=S(z)$ by ignoring all exponentially small entries in the jump matrices. The model RH will be defined in the region

The solution $P^{(\infty)}(z)$ to this model RH problem can be constructed in the usual way for each of the two $2\times 2$ problems into which the $4\times 4$ problem decouples. This leads to the parametrix

and where we choose the principal branches of the $1/4$ powers.

5.3 Local parametrices around the branch points

Consider disks around the branch points $\alpha_{j},\beta_{j}$, $j=1,2$ with sufficiently small radius. Inside these disks one can construct local parametrices $P^{(\textrm{Airy})}(z)$ to the RH problem for $S(z)$ in terms of Airy functions. Once again, these parametrices can be constructed in the usual way on each of the two $2\times 2$ RH problems. We omit further details.

5.4 Fourth transformation and completion of the steepest descent analysis

Define a final matrix-valued function $R(z)$ by

From the construction of the parametrices it then follows that $R$ satisfies a RH problem

$R$ has jumps $R_{+}=R_{-}J_{R}$ on $\Sigma_{R}$, that satisfy

as $n\to\infty$, uniformly for $z$ in the complex plane. This completes the RH steepest descent analysis.

6 Proof of Theorem 1.3 in the case of large separation

We will now establish Theorem 1.3 in the case of large separation. The idea is that from the asymptotic decoupling of the $4\times 4$ RH problem for large $n$, there follows a similar decoupling for the kernel in (1.23) and hence for the associated non-intersecting Brownian particles.

First assume that $x,y\in(\alpha_{1}^{*},\beta_{1}^{*})$ and consider the correlation kernel (1.23) $K_{n_{1},n_{2};n_{1},n_{2}}$, which for short we denote by $K_{n}$. By virtue of (2.11) we obtain

Now it follows from (2.77) by standard arguments (e.g. [7, Section 9]) that

uniformly in $n$. Inserting this into (2.78) yields

uniformly in $n$. Now in the limit when $y\to x$, the last two terms in (2.79) become exponentially small by virtue of Lemma 2.5(b). Then by letting $y\to x$ and using l’Hôpital’s rule and find

as $n\to\infty$, where the last step follows from (2.43). We conclude that

For $x\in(\alpha_{2}^{*},\beta_{2}^{*})$ we obtain in a similar way that $\frac{1}{n}K_{n}(x,x)$ tends to the semi-circle density on the interval $[\alpha_{2}^{*},\beta_{2}^{*}]$. Thus by (1.23) we have completed the proof of Theorem 1.3 in the case of large separation (recall that $T=1$). $\hfill\square\\$

Steepest descent analysis in the case of critical separation

In this section we study the non-intersecting Brownian motions in case of critical separation between the endpoints. We will work under the assumption of the double scaling regime (1.25)–(1.26). This means that we take the endpoints $a_{j},b_{j}$, $j=1,2$ fixed such that

and that we consider the temperature $T$ to be varying with $n$ as

Before proceeding further, let us recall the main objects needed for the steepest descent analysis. The points $\alpha_{j},\beta_{j}$, $j=1,2$ are as defined in (2.8)–(2.9), but now with $T$ in (3.2) not identically equal to 1:

for $j=1,2$. The limiting values for $n\to\infty$ of these points are denoted by (cf. (1.10)–(1.11))

$j=1,2$. Recall also the $\lambda$-functions and $\xi$-functions which are given by (2.34) and (2.43). We will denote the limiting values for $n\to\infty$ of these functions by $\lambda_{k}^{*}(z)$, $\xi_{k}^{*}(z)$, $k=1,\ldots,4$. For example, the functions $\xi_{k}^{*}(z)$ are given by

Throughout this section we will assume again that the hypothesis (2.7) holds, but now with strict inequality, i.e.,

The time $t_{\textrm{crit}}$ is now the time where the two ellipses in Figure 2 are tangent to each other. The case where $t_{\textrm{crit}}<t<1$ can be handled in a similar way; cf. Remark 2.1. The behavior at the tangent time $t_{\textrm{crit}}$ itself leads to a multi-critical situation which is outside the scope of this paper.

The main technical tool that made the analysis in Section 2.4 work was the existence of a point $x_{0}\in(\beta_{2},\alpha_{1})$ such that the string of inequalities (2.44) holds, cf. Lemma 2.3. It will turn out that in the present context, this can be achieved by the point $x_{0}^{*}$ given by the explicit formula