\documentclass{slides}\textheight=10in\topmargin=-.8in
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\def\t{\tau} \def\ch{\raisebox{.3ex}{$\chi$}} \def\inv{^{-1}}
\renewcommand{\Pr}{{\rm Pr}\,} \def\iy{\infty} \def\x{\xi}
\def\A{{\rm Ai}} \def\pl{\partial} \def\tl{\tilde} \def\ov{\over}
\def\ph{\varphi} \def\ps{\psi}
% May 1

\begin{center}{\large Differential Equations for Dyson Processes}\end{center}

\begin{center}Joint work with Craig Tracy\end{center}

Start with $n\times n$ GUE matrix, let entries independently undergo 
Ornstein-Uhlenbeck diffusion. (Dyson Brownian motion.) 
Eigenvalues describe $n$ curves. {\it Hermite Process}. 

Let $n\to\iy$, scale near the top. Infinitely many curves, {\it Airy process}.
Top curve $A(\tau)$. (Pr\"ahoffer-Spohn, Johansson.) 

Scale in the bulk, {\it sine process}. Evolution of singular values of complex
matrices leads to {\it Laguerre process}; scaling this at bottom edge gives {\it
Bessel process}.

Differential equations that determine the probabilities


\[\Pr(A(\t_1)\le \x_1,\ldots,A(\t_m)\le \x_m).\]

Tracy-Widom: $\x_k=\eta_k+\x,\ \ \eta_k$ fixed. System of ODEs in $\x$.
($\t_k$ parameters.)\\ 
Adler-van Moerbeke: $m=2,\ \t_2-\t_1=t$. PDE in $t,\,\x_1,\ \x_2$. (Scaled
PDE for Hermite process.)


Present result. $X_k$ finite unions of intervals. Interested in
probability that for $k=1,\ldots,m$ no curve passes through $X_k$ at 
time $\t_k$. For Airy process
when $X_k=\ch_{(\x_k,\,\iy)}$ this is
\[\Pr(A(\t_1)\le \x_1,\ldots,A(\t_m)\le \x_m).\]

Found systems of PDEs with endpoints of
the $X_k$ as independent variables whose solutions determine  
this probability. Did this for all but Laguerre process. 


Airy process simplest. Probability given as Fredholm determinant of 
{\it extended Airy kernel}, an $m\times m$ matrix kernel. Entries 
$L_{ij}(x,y)$ given by
\[ \begin{array}{ll} {\displaystyle{\int_0^\iy}} e^{-z\,(\t_i-\t_j)}\,
\A(x+z)\,\A(y+z)\,dz&{\rm if}\ i\ge j,\\&\\ {\displaystyle -\int_{-\iy}^0}e^{-z\,(\t_i-\t_j)}
\,\A(x+z)\,\A(y+z)\,dz&{\rm if}\ i<j.\end{array}\]
\[K_{ij}(x,y)=L_{ij}(x,y)\,\ch_{X_j}(y).\]
Probability equals $\det\,(I-K)$. 

Case $X_k=(\x_k,\,\iy)$. Set $R=K\,(I-K)\inv$. 
\[\pl_{\x_k}\log\,\det\,(I-K)=R_{kk}(\x_k,\,\x_k).\]
\newpage
Unknowns will be five matrix functions of the $\x_k$. First is
\[r_{ij}=R_{ij}(\x_i,\,\x_j).\]
To define others, let 
\[A={\rm diag}\,(\A),\ \ \ \ch={\rm diag}\,(\ch_{(\x_k,\iy)}),\]
\[Q=(I-K)\inv A,\ \ \ \tl Q=A\ch(I-K)\inv.\]
Other unknowns are
\[q_{ij}=Q_{ij}(\x_i),\ \ \tl q_{ij}=\tl Q_{ij}(\x_j),\]
\[q'_{ij}=Q'_{ij}(\x_i),\ \ \ \tl q'_{ij}=\tl Q_{ij}'(\x_j).\]
Define $r_x$ and $r_y$ by
\[(r_x)_{ij}=(\pl_x R)_{ij}(\x_i,\,\x_j),\ \ \ (r_y)_{ij}=
(\pl_y R)_{ij}(\x_i,\,\x_j).\]
These are not unknowns.

Set $\x={\rm diag}\,(\x_k)$. Equations are
\begin{eqnarray*}
dr&=&-r\,d\x\,r+d\x\,r_x+r_y\,d\x,\\
dq&=&d\x\,q'-r\,d\x\,q,\\
d\tl q&=&\tl q'\,d\x-\tl q\,d\x\,r,\\
dq'&=&d\x\,\x\,q-(r_x\,d\x+d\x\,r_y)\,q+d\x\,r\,q',\\
d\tl q'&=&\tl q\,\x\,d\x-\tl q\,(d\x\,r_y+r_x\,d\x)+\tl q'\,r\,d\x.
\end{eqnarray*}

Have to show diagonal entries of $r_x+r_y$ and off-diagonal entries of
$r_x$ and $r_y$ are expressible in terms of the unknowns. 
Here is where the $\t_k$ enter. Let $\t={\rm diag}\,(\t)$.
Commutators
\[[D,\,L]=-A\otimes A+[\t,L],\]
\[[D^2-M,\,L]=0.\]
>From these can derive
\[r_x+r_y=-q\,\tl q+r^2+[\t,\,r],\]
\[[\t,\,r_x-r_y]=q'\,\tl q-q\,\tl q'+[r,\,r_x+r_y]+[\x,\,r].\]



Case $m=1$. Then $\tl q=q,\ \tl q'=q',\ r=R(\x,\,\x)$ and equations give
\[{d^2q\ov d\x^2}=\x\,q+2\,(r^2-r_x-r_y)q.\]
Thus
\[{d^2q\ov d\x^2}=\x\,q+2\,q^3.\]
Painlev\'e II. Also,
\[{d^2\ov d\x^2}\log\,\det\,(I-K)={dr\ov d\x}=-r^2+r_x+r_y=-q^2.\]

Hermite process. Extended Hermite kernel (Johansson, Eynard-Mehta) has entries 
$L_{ij}(x,y)$ given by
\[\begin{array}{ll}\sum\limits_{k=0}^{n-1}e^{(k-n)\,(\t_i-\t_j)}\,
\ph_k(x)\,\ph_k(y)&{\rm if}\ i\ge j,
\\&\\-\sum\limits_{k=n}^\iy e^{(k-n)\,(\t_i-\t_j)}\,\ph_k(x)\,\ph_k(y)&{\rm if}\ i<j.
\end{array}\]

Set
\[\ph=(2n)^{1/4}\,\ph_n,\ \ \ \ps=(2n)^{1/4}\,\ph_{n-1},\]
and define 
\[Q=(I-K)\inv\,\ph,\ \ \ P=(I-K)\inv\,\ps,\]
\[\tl Q=\ph\ch\,(I-K)\inv, \ \ \ \tl P=\ps\ch\,(I-K)\inv.\]
Unknowns $r_{ij}=R_{ij}(\x_i,\,\x_j)$ and 
$q,\ \tl q,\ p,\ \tl p$ given by
\[q_{ij}=Q_{ij}(\x_i),\ \ \ \tl q_{ij}=\tl Q_{ij}(\x_j),\]
\[p_{ij}=P_{ij}(\x_i),\ \ \ \tl p_{ij}=\tl P_{ij}(\x_j),\]
\[q'_{ij}=Q'_{ij}(\x_i),\ \ \ \tl q'_{ij}=\tl Q_{ij}(\x_j),\]
\[p'_{ij}=P'_{ij}(\x_i),\ \ \ 
\tl p'_{ij}=\tl P'_{ij}(\x_j).\]
Equations
\begin{eqnarray*}
dr&=&-r\,d\x\,r+d\x\,r_x+r_y\,d\x,\label{hpde1}\\
dq&=&d\x\,q'-r\,d\x\,q,\label{hpde2}\\
d\tl q&=&\tl q'\,d\x-\tl q\,d\x\,r,\label{hpde3}\\
dq'&=&d\x\,(\x^2-2n-1)\,q-(r_x\,d\x+d\x\,r_y)\,q+d\x\,r\,q',\label{hpde4}\\
d\tl q'&=&\tl q\,(\x^2-2n-1)\,d\x-\tl q\,(d\x\,r_y+r_x\,d\x)+\tl q'\,r\,d\x,\label{hpde5}\\
dp&=&d\x\,p'-r\,d\x\,p,\label{hpde6}\\
d\tl p&=&\tl p'\,d\x-\tl p\,d\x\,r,\label{hpde7}\\
dp'&=&d\x\,(\x^2-2n+1)\,p-(r_x\,d\x+d\x\,r_y)\,p+d\x\,r\,p',\label{hpde8}\\
d\tl p'&=&\tl p\,(\x^2-2n+1)\,d\x-\tl p\,(d\x\,r_y+r_x\,d\x)+\tl p'\,r\,d\x.\label{hpde9}
\end{eqnarray*}

Commutators with $D^2-M^2$ and $e^{\pm\t}(D\mp M)$.

Case $m=1$. Can eliminate $q$ and $p$, arrive at
\[{d^3r\ov d\x^3}=4(\x^2-2n){dr\ov d\x}-4\x r-6\left({dr\ov d\x}\right)^2.\]
Integrates to Painlev\'e IV.









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