# Painlevé IV, $\sigma-$Form and the Deformed Hermite Unitary Ensembles

Mengkun<EMAIL_ADDRESS>School of Mathematics and Statistics, Qilu
University of Technology (Shandong Academy of Sciences), Jinan 250353, China
Department of Mathematics, Faculty of Science and Technology, University of
Macau, Avenida da Universidade, Taipa, Macau, China  Dan Wang†††Corresponding
author<EMAIL_ADDRESS>Department of Mathematics, Faculty of Science and
Technology, University of Macau, Avenida da Universidade, Taipa, Macau, China
Yang<EMAIL_ADDRESS>Department of Mathematics, Faculty of
Science and Technology, University of Macau, Avenida da Universidade, Taipa,
Macau, China

###### Abstract

We study the Hankel determinant generated by a deformed Hermite weight with
one jump $w(z,t,\gamma)=e^{-z^{2}+tz}|z-t|^{\gamma}(A+B\theta(z-t))$, where
$A\geq 0$, $A+B\geq 0$, $t\in\textbf{R}$, $\gamma>-1$ and $z\in\textbf{R}$. By
using the ladder operators for the corresponding monic orthogonal polynomials,
and their relative compatibility conditions, we obtain a series of difference
and differential equations to describe the relations among $\alpha_{n}$,
$\beta_{n}$, $R_{n}(t)$ and $r_{n}(t)$. Especially, we find that the auxiliary
quantities $R_{n}(t)$ and $r_{n}(t)$ satisfy the coupled Riccati equations,
and $R_{n}(t)$ satisfies a particular Painlevé IV equation. Based on above
results, we show that $\sigma_{n}(t)$ and $\hat{\sigma}_{n}(t)$, two
quantities related to the Hankel determinant and $R_{n}(t)$, satisfy the
continuous and discrete $\sigma-$form equations, respectively. In the end, we
also discuss the large $n$ asymptotic behavior of $R_{n}(t)$, which produce
the expansion of the logarithmic of the Hankel determinant and the asymptotic
of the second order differential equation of the monic orthogonal polynomials.

Keywords: Ladder operators, Hermite unitary ensembles, $\sigma-$form, Painlevé
IV.
MSC: 15B52, 42C05, 33E17

##  1 Introduction

The joint probability density function of the eigenvalues
$\\{z_{i}\\}_{i=1}^{n}$,

$p(z_{1},z_{2},\cdots,z_{n})=\frac{1}{n!D_{n}(w_{0})}\prod\limits_{1\leq
i<k\leq n}(z_{k}-z_{i})^{2}\prod\limits_{j=1}^{n}w_{0}(z_{j}),$

is a well known fact in the theory of random matrix ensembles [6, 7, 8, 13],
where $w_{0}(z)$ is a weight function on the interval $(a,b)$ and has the
finite moments, i. e.

$\mu_{k}:=\int_{a}^{b}z^{k}w_{0}(z)dz,~{}~{}~{}~{}k\in\\{0,1,2,\cdots\\}.$

Here $D_{n}(w_{0})$ is the normalization constant

$\displaystyle D_{n}(w_{0})=\frac{1}{n!}\int_{[a,b]^{n}}\prod\limits_{1\leq
i<k\leq n}(z_{k}-z_{i})^{2}\prod\limits_{j=1}^{n}w_{0}(z_{j})dz_{j},$ (1.1)

so that

$\int_{[a,b]^{n}}p(z_{1},z_{2},\cdots,z_{n})dz_{1}dz_{2}\cdots dz_{n}=1.$

In this paper, we consider

$\displaystyle w(z,t,\gamma)=e^{-z^{2}+tz}|z-t|^{\gamma}(A+B\theta(z-t)),$
(1.2)

where $A\geq 0$, $A+B\geq 0$, $t\in\textbf{R}$, $\gamma>-1$ and
$z\in\textbf{R}$, which

$w_{0}(z)=e^{-z^{2}+tz},~{}~{}t\in\textbf{R},~{}~{}z\in\textbf{R},$

corresponds to the deformed Hermite (or Gaussian) unitary ensemble. Here
$\theta(x)$ is the Heaviside function, i.e.

$\theta(x)=\left\\{\begin{aligned} 1&,~{}~{}x>0\\\ 0&,~{}~{}x\leq
0\end{aligned}\right..$

According to the general theory of orthogonal polynomials of one variable, the
normalization constant (1.1) has the two more alternative representation

$\displaystyle
D_{n}(w)=\det(\mu_{i+j}(t))_{i,j=0}^{n-1}=\det\left(\int_{\textbf{R}}z^{i+j}w(z,t)dz\right)_{i,j=0}^{n-1}=\prod\limits_{k=0}^{n-1}h_{k}(t),$

where the determinant of the moment matrix $\mu_{i+j}(t)$ is the Hankel
determinant, and $\\{h_{k}(t)\\}_{k=0}^{n}$ is the square of the $L^{2}$ norm
of the sequence of polynomials $\\{P_{k}(z)\\}_{k=0}^{n}$.

Min and Chen [14] have studied the Painlevé transcendents and the Hankel
determinants generated by a discontinuous Gaussian weight

$w(x,t_{1},t_{2})=e^{-x^{2}}(A+B_{1}\theta(x-t_{1})+B_{2}\theta(x-t_{2})),$
(1.3)

where $A_{1}$, $B_{1}$ and $B_{2}$ are constants, $A\geq 0$, $A+B_{1}\geq 0$,
$A+B_{1}+B_{2}\geq 0$, $t_{1}<t_{2}$, $x\in\textbf{R}$. They considered the
Gaussian weight with a single jump ($B_{1}$ or $B_{2}$ $=0$) or two jumps
($B_{1}\neq 0$ and $B_{2}\neq 0$), and proved their auxiliary quantities
satisfy the second order difference and differential equations, respectively,
via the ladder operators and supplementary conditions.

Let $P_{n}(z,t,\gamma)$ be the monic polynomials of degree $n$ orthogonal with
respect to (1.2),

$\int_{\textbf{R}}P_{j}(z,t,\gamma)P_{i}(z,t,\gamma)w(z,t,\gamma)dz=h_{j}(t,\gamma)\delta_{ji},$
(1.4)

where $j,i\in\\{0,1,2,\cdots\\}$, and $\delta_{ji}$ denotes the Kronecker
delta. It follows from the orthogonality relations that

$zP_{n}(z,t,\gamma)=P_{n+1}(z,t,\gamma)+\alpha_{n}P_{n}(z,t,\gamma)+\beta_{n}P_{n-1}(z,t,\gamma),~{}~{}n\geq
0,$ (1.5)

together with the initial conditions

$P_{0}(z,t,\gamma):=1,~{}~{}~{}\beta_{0}P_{-1}(z,t,\gamma):=0.$

The monic polynomials $P_{n}(z,t)$ have the monomial expansion

$\displaystyle
P_{n}(z,t,\gamma)=z^{n}+\textbf{p}(n,t,\gamma)z^{n-1}+\cdots+P_{n}(0,t,\gamma),$
(1.6)

and we will see that the coefficient of $z^{n-1}$, $\textbf{p}(n,t,\gamma)$,
play a significant role in the following discussions. Note that due to the $t$
dependence of the weight, the coefficients of the polynomials and the
recurrence coefficients $\alpha_{n}$ and $\beta_{n}$ also depend on $t$, the
position of the jump.

From (1.5) and (1.6), for $n\in\\{0,1,2,\cdots\\}$,

$\alpha_{n}=\textbf{p}(n,t,\gamma)-\textbf{p}(n+1,t,\gamma),$ (1.7)

and

$\beta_{n}=\frac{h_{n}(t,\gamma)}{h_{n-1}(t,\gamma)}.$ (1.8)

A telescopic sum of (1.7), it yields

$\sum\limits_{j=0}^{n-1}\alpha_{j}=-\textbf{p}(n,t,\gamma).$ (1.9)

The paper is organized up as follow. In section 2, we derive the ladder
operators and the compatibility conditions ($S_{1}$), ($S_{2}$) and
($S_{2}^{\prime}$) with respect to the weight (1.2). In section 3, we obtain
some important identities on the auxiliary quantities $R_{n}(t)$ and
$r_{n}(t)$ via applying the ladder operators. In section 4, we are interested
in the parameter of the weight (1.2), and we show that $r_{n}(t)$ and
$R_{n}(t)$ satisfy the coupled Riccati equations. Section 5 is devoted to
discuss two quantities $\sigma_{n}(t)$ and $\hat{\sigma}_{n}(t)$, which are
allied to the Hankel determinant and $R_{n}(t)$. We prove that they satisfy
the continuous or discrete $\sigma-$form equations, respectively. The final
section, we make use of the equations in section $2-5$ to show that $R_{n}(t)$
satisfy a particular Painlevé IV equation. Based on this equation, we obtain
the large $n$ asymptotic of the auxiliary quantity $R_{n}(t)$. Moreover, the
logarithmic of the Hankel determinant $D_{n}(t)$ and the asymptotic of the
second order differential equation of $P_{n}(z,t,\gamma)$ are also obtained.

##  2 Ladder operators and compatibility conditions

A detailed proof of the lowering ladder operator, which is included in the
ladder operators [1], is shown in the following theorem. For convenience, we
denote

$\displaystyle w(z,t,\gamma)$
$\displaystyle=e^{-z^{2}+tz}|z-t|^{\gamma}(A+B\theta(z-t)),$ $\displaystyle
w_{0}(z,t)$ $\displaystyle=e^{-z^{2}+tz},$ $\displaystyle w_{J}(z,t)$
$\displaystyle=A+B\theta(z-t),$

where $A\geq 0$, $A+B\geq 0$, $t\in\textbf{R}$, $\gamma>-1$ and
$z\in\textbf{R}$. Meanwhile, we do not always write down the $t$ or $\gamma$
dependence in $P_{n}(z,t,\gamma)$, $w(z,t,\gamma)$, $w_{0}(z,t)$,
$w_{J}(z,t)$, $h_{n}(t,\gamma)$, $\alpha_{n}(t,\gamma)$ and
$\beta_{n}(t,\gamma)$ unless it is needed.

######  Theorem 2.1.

The monic orthogonal polynomials with respect to the weight (1.2) satisfy the
following differentiation formula:

$P_{n}^{\prime}(z)=\beta_{n}A_{n}(z)P_{n-1}(z)-B_{n}(z)P_{n}(z),$ (2.1)

where

$\displaystyle
A_{n}(z)=\frac{1}{h_{n}}\int_{\textbf{R}}P_{n}^{2}(y)\frac{v_{0}^{\prime}(z)-v_{0}^{\prime}(y)}{z-y}w(y)dy+a_{n}(t),$
(2.2)
$B_{n}(z)=\frac{1}{h_{n-1}}\int_{\textbf{R}}P_{n}(y)P_{n-1}(y)\frac{v_{0}^{\prime}(z)-v_{0}^{\prime}(y)}{z-y}w(y)dy+b_{n}(t),$
(2.3)
$a_{n}(t)=\frac{\gamma}{h_{n}}\int_{\textbf{R}}\frac{P_{n}^{2}(y)}{(z-y)(y-t)}w(y)dy$
(2.4)

and

$b_{n}(t)=\frac{\gamma}{h_{n-1}}\int_{\textbf{R}}\frac{P_{n}(y)P_{n-1}(y)}{(z-y)(y-t)}w(y)dy.$
(2.5)

Here

$v_{0}(z)=-\ln w_{0}(z).$

###### Proof.

Firstly, we note [4]

$\frac{\partial(|z-t|^{\gamma})}{\partial
z}=\bigg{(}(z-t)^{\gamma}-(t-z)^{\gamma}\bigg{)}\delta(z-t)+\frac{\gamma|z-t|^{\gamma}}{z-t},$
(2.6)

which is obtained by writing

$|z-t|^{\gamma}=(z-t)^{\gamma}\theta(z-t)+(t-z)^{\gamma}\theta(t-z).$

From

$P_{n}^{\prime}(z)=\sum\limits_{k=0}^{n-1}C_{n,k}P_{k}(z),$

where $C_{n,k}$ is determined from the orthogonality relations

$\displaystyle C_{n,k}$
$\displaystyle=\frac{1}{h_{k}}\int_{\textbf{R}}P_{n}^{\prime}(y)P_{k}(y)\omega(y)dy.$

Using the integration by parts and noting the weight (1.2) vanishes at the
endpoints of the orthogonality interval, we find

$\displaystyle C_{n,k}$
$\displaystyle=\frac{1}{h_{k}}\int_{\textbf{R}}P_{n}^{\prime}(y)P_{k}(y)w(y)dy=-\frac{1}{h_{k}}\int_{\textbf{R}}P_{n}(y)P_{k}(y)w^{\prime}(y)dy$
$\displaystyle=-\frac{1}{h_{k}}\int_{\textbf{R}}P_{n}(y)P_{k}(y)(v_{0}^{\prime}(z)-v_{0}^{\prime}(y))w(y)dy-\frac{\gamma}{h_{k}}\int_{\textbf{R}}P_{n}(y)P_{k}(y)\frac{w(y)}{y-t}dy;$

straightforward,

$\displaystyle P_{n}^{\prime}(z)$ $\displaystyle=$
$\displaystyle\sum\limits_{k=0}^{n-1}\bigg{(}-\frac{1}{h_{k}}\int_{\textbf{R}}P_{n}(y)P_{k}(y)(v_{0}^{\prime}(z)-v_{0}^{\prime}(y))w(y)dy-\frac{\gamma}{h_{k}}\int_{\textbf{R}}P_{n}(y)P_{k}(y)\frac{w(y)}{y-t}dy\bigg{)}P_{k}(z)$
$\displaystyle=$
$\displaystyle-\int_{\textbf{R}}P_{n}(y)\sum\limits_{k=0}^{n-1}\frac{P_{k}(y)P_{k}(z)}{h_{k}}(v_{0}^{\prime}(z)-v_{0}^{\prime}(y))w(y)dy-\gamma\int_{\textbf{R}}P_{n}(y)\sum\limits_{k=0}^{n-1}\frac{P_{k}(y)P_{k}(z)}{h_{k}}\frac{w(y)}{y-t}dy$
$\displaystyle=$
$\displaystyle-\frac{1}{h_{n-1}}\int_{\textbf{R}}\frac{v_{0}^{\prime}(z)-v_{0}^{\prime}(y)}{z-y}P_{n}(y)P_{n-1}(y)w(y)dyP_{n}(z)$
$\displaystyle+\frac{1}{h_{n-1}}\int_{\textbf{R}}\frac{v_{0}^{\prime}(z)-v_{0}^{\prime}(y)}{z-y}P_{n}^{2}(y)w(y)dyP_{n-1}(z)$
$\displaystyle-\frac{\gamma}{h_{n-1}}\int_{\textbf{R}}P_{n}(y)P_{n-1}(y)\frac{w(y)}{(z-y)(y-t)}dyP_{n}(z)$
$\displaystyle+\frac{\gamma}{h_{n-1}}\int_{\textbf{R}}P_{n}^{2}(y)\frac{w(y)}{(z-y)(y-t)}dyP_{n-1}(z),$
(2.7)

where we have used the Christoffel-Darboux formula[17]. Applying (1.8), we
arrive the desired result. ∎

######  Corollary 2.2.

Two identities

$\gamma\int_{\textbf{R}}\frac{P_{n}^{2}(y)w(y)}{y-t}dy=\int_{\textbf{R}}P_{n}^{2}(y)v_{0}^{\prime}(y)w(y)dy$
(2.8)

and

$\frac{\gamma}{h_{n-1}}\int_{\textbf{R}}P_{n}(y)P_{n-1}(y)\frac{w(y)}{y-t}dy=\frac{1}{h_{n-1}}\int_{\textbf{R}}P_{n}(y)P_{n-1}(y)v_{0}^{\prime}(y)w(y)dy-n$
(2.9)

###### Proof.

From the proof of theorem 2.1, we see that

$\frac{1}{h_{n}}\int_{\textbf{R}}P_{n}^{\prime}(y)P_{n}(y)w(y)dy=0.$

Applying integration by parts into the left side of above equation,

$\displaystyle\frac{1}{h_{n}}\int_{\textbf{R}}P_{n}^{\prime}(y)P_{n}(y)w(y)dy=\frac{1}{h_{n}}\int_{\textbf{R}}P_{n}^{2}(y)v_{0}^{\prime}(y)w(y)dy-\frac{\gamma}{h_{n}}\int_{\textbf{R}}P_{n}^{2}(y)\frac{w(y)}{y-t}dy,$

which gives (2.8).

The same steps to do below equation,

$\frac{1}{h_{n-1}}\int_{\textbf{R}}P_{n}^{\prime}(y)P_{n-1}(y)w(y)dy=n,$

we obtain (2.9). ∎

######  Remark 1.

The equation (2.1) is called a lowering operator, see[2, 24, 3, 9, 22, 23,
14]. With the help of (2.1) and (2.10), $P_{n}(z)$ also satisfies the raising
operator equation

$P_{n-1}^{\prime}(z)=-A_{n-1}(z)P_{n}(z)+(B_{n}(z)+v_{0}^{\prime}(z))P_{n-1}(z).$

Note that $A_{n}(z)$ and $B_{n}(z)$ are not independent, a direct calculation
produces two fundamental supplementary conditions[2] valid for all $z$.

######  Theorem 2.3.

The functions $A_{n}(z)$ and $B_{n}(z)$ satisfy the conditions:

$B_{n}(z)+B_{n+1}(z)=(z-\alpha_{n})A_{n}(z)-v_{0}^{\prime}(z),$ (2.10)
$1+(z-\alpha_{n})(B_{n+1}(z)-B_{n}(z))=\beta_{n+1}A_{n+1}(z)-\beta_{n}A_{n-1}(z).$
(2.11)

###### Proof.

Using (1.8), the recurrence relation
$(z-\alpha_{n})P_{n}(z,t,\gamma)=P_{n+1}(z,t,\gamma)+\beta_{n}P_{n-1}(z,t,\gamma)$
has a restatement, i. e.

$\frac{(z-\alpha_{n})P_{n}(z,t,\gamma)}{h_{n}}=\frac{P_{n+1}(z,t,\gamma)}{h_{n}}+\frac{P_{n-1}(z,t,\gamma)}{h_{n-1}}.$
(2.12)

From the definitions of $A_{n}(z)$ and $B_{n}(z)$, (2.2) and (2.3), and
substituting (2.12) into below computation, we have

$\displaystyle B_{n}(z)+B_{n+1}(z)$ $\displaystyle=$
$\displaystyle\int_{\textbf{R}}P_{n}(y)\bigg{(}\frac{P_{n+1}(y)}{h_{n}}+\frac{P_{n-1}(y)}{h_{n-1}}\bigg{)}\frac{v_{0}^{\prime}(z)-v_{0}^{\prime}(y)}{z-y}w(y)dy$
$\displaystyle+\gamma\int_{\textbf{R}}P_{n}(y)\bigg{(}\frac{P_{n+1}(y)}{h_{n}}+\frac{P_{n-1}(y)}{h_{n-1}}\bigg{)}\frac{w(y)}{(z-y)(y-t)}dy$
$\displaystyle=\frac{1}{h_{n}}\int_{\textbf{R}}(y-\alpha_{n})P_{n}^{2}(y)\frac{v_{0}^{\prime}(z)-v_{0}^{\prime}(y)}{z-y}w(y)dy+\frac{\gamma}{h_{n}}\int_{\textbf{R}}\frac{(y-\alpha_{n})P_{n}^{2}(y)w(y)}{(z-y)(y-t)}dy$
$\displaystyle=-\frac{1}{h_{n}}\int_{\textbf{R}}P_{n}^{2}(y)(v_{0}^{\prime}(z)-v_{0}^{\prime}(y))w(y)dy-\frac{\gamma}{h_{n}}\int_{\textbf{R}}\frac{P_{n}^{2}(y)w(y)}{y-t}dy$
$\displaystyle+\frac{1}{h_{n}}\int_{\textbf{R}}(z-\alpha_{n})P_{n}^{2}(y)\frac{v_{0}^{\prime}(z)-v_{0}^{\prime}(y)}{z-y}w(y)dy+\frac{\gamma}{h_{n}}\int_{\textbf{R}}\frac{(z-\alpha_{n})P_{n}^{2}(y)w(y)}{(z-y)(y-t)}dy$
$\displaystyle=(z-\alpha_{n})A_{n}(z)-v_{0}^{\prime}(z)+\frac{1}{h_{n}}\int_{\textbf{R}}P_{n}^{2}(y)v_{0}^{\prime}(y)w(y)dy-\frac{\gamma}{h_{n}}\int_{\textbf{R}}\frac{P_{n}^{2}(y)w(y)}{y-t}dy,$
(2.13)

where

$y-\alpha_{n}=(y-z)+(z-\alpha_{n})$

and (2.8) are used.The proof of (2.10) is completed.
Similarly steps, by using the definition of $A_{n}(z)$ and (1.8), we have

$\displaystyle\beta_{n+1}A_{n+1}(z)-\beta_{n}A_{n-1}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{h_{n}}\int_{\textbf{R}}\frac{v_{0}^{\prime}(z)-v_{0}^{\prime}(y)}{z-y}P_{n+1}^{2}(y)w(y)dy+\frac{\gamma}{h_{n}}\int_{\textbf{R}}\frac{P_{n+1}^{2}(y)}{(z-y)(y-t)}w(y)dy$
$\displaystyle-\frac{\beta_{n}}{h_{n-1}}\int_{\textbf{R}}\frac{v_{0}^{\prime}(z)-v_{0}^{\prime}(y)}{z-y}P_{n-1}^{2}(y)w(y)dy-\frac{\gamma\beta_{n}}{h_{n-1}}\int_{\textbf{R}}\frac{P_{n-1}^{2}(y)w(y)}{(z-y)(y-t)}dy$
$\displaystyle=$
$\displaystyle\int_{\textbf{R}}\frac{v_{0}^{\prime}(z)-v_{0}^{\prime}(y)}{z-y}\bigg{(}\frac{P_{n+1}^{2}(y)}{h_{n}}-\frac{\beta_{n}P_{n-1}^{2}(y)}{h_{n-1}}\bigg{)}w(y)dy$
$\displaystyle+\gamma\int_{\textbf{R}}\bigg{(}\frac{P_{n+1}^{2}(y)}{h_{n}}-\frac{\beta_{n}P_{n-1}^{2}(y)}{h_{n-1}}\bigg{)}\frac{w(y)}{(z-y)(y-t)}dy.$
(2.14)

On the other hand, Applying (2.9) and (2.12), we consider

$\displaystyle(z-\alpha_{n})(B_{n+1}(z)-B_{n}(z))$ $\displaystyle=$
$\displaystyle(z-\alpha_{n})\bigg{(}\int_{\textbf{R}}\frac{v_{0}^{\prime}(z)-v_{0}^{\prime}(y)}{z-y}\bigg{(}\frac{P_{n+1}(y)}{h_{n}}-\frac{P_{n-1}(y)}{h_{n-1}}\bigg{)}P_{n}(y)w(y)dy$
$\displaystyle+\gamma\int_{0}^{\infty}\frac{P_{n}(y)}{(z-y)(y-t)}\bigg{(}\frac{P_{n+1}(y)}{h_{n}}-\frac{P_{n-1}(y)}{h_{n-1}}\bigg{)}w(y)dy$
$\displaystyle=$
$\displaystyle\int_{\textbf{R}}(v_{0}^{\prime}(z)-v_{0}^{\prime}(y))\bigg{(}\frac{P_{n+1}(y)}{h_{n}}-\frac{P_{n-1}(y)}{h_{n-1}}\bigg{)}P_{n}(y)w(y)dy$
$\displaystyle+\int_{\textbf{R}}\frac{v_{0}^{\prime}(z)-v_{0}^{\prime}(y)}{z-y}\bigg{(}\frac{P_{n+1}^{2}(y)}{h_{n}^{2}}-\frac{P_{n-1}^{2}(y)}{h_{n-1}^{2}}\bigg{)}h_{n}w(y)dy$
$\displaystyle+\gamma\int_{\textbf{R}}\frac{P_{n}(y)}{y-t}\bigg{(}\frac{P_{n+1}(y)}{h_{n}}-\frac{P_{n-1}(y)}{h_{n-1}}\bigg{)}w(y)dy$
$\displaystyle+\gamma\int_{\textbf{R}}\frac{w(y)}{(z-y)(y-t)}\bigg{(}\frac{P_{n+1}^{2}(y)}{h_{n}^{2}}-\frac{P_{n-1}^{2}(y)}{h_{n-1}^{2}}\bigg{)}h_{n}dy$
$\displaystyle=$
$\displaystyle-\bigg{(}\frac{1}{h_{n}}\int_{\textbf{R}}P_{n+1}(y)P_{n}(y)v_{0}^{\prime}(y)w(y)dy-\frac{\gamma}{h_{n}}\int_{\textbf{R}}\frac{P_{n}(y)P_{n+1}(y)w(y)}{y-t}dy\bigg{)}$
$\displaystyle+\bigg{(}\frac{1}{h_{n-1}}\int_{\textbf{R}}P_{n}(y)P_{n-1}(y)v_{0}^{\prime}(y)w(y)dy-\frac{\gamma}{h_{n-1}}\int_{\textbf{R}}\frac{P_{n-1}(y)P_{n}(y)w(y)}{y-t}dy\bigg{)}$
$\displaystyle+\int_{\textbf{R}}\frac{v_{0}^{\prime}(z)-v_{0}^{\prime}(y)}{z-y}\bigg{(}\frac{P_{n+1}^{2}(y)}{h_{n}}-\frac{\beta_{n}P_{n-1}^{2}(y)}{h_{n-1}}\bigg{)}w(y)dy$
$\displaystyle+\gamma\int_{\textbf{R}}\frac{w(y)}{(z-y)(y-t)}\bigg{(}\frac{P_{n+1}^{2}(y)}{h_{n}}-\frac{\beta_{n}P_{n-1}^{2}(y)}{h_{n-1}}\bigg{)}dy$
$\displaystyle=$
$\displaystyle\int_{\textbf{R}}\frac{v_{0}^{\prime}(z)-v_{0}^{\prime}(y)}{z-y}\bigg{(}\frac{P_{n+1}^{2}(y)}{h_{n}}-\frac{\beta_{n}P_{n-1}^{2}(y)}{h_{n-1}}\bigg{)}w(y)dy$
$\displaystyle+\gamma\int_{\textbf{R}}\frac{w(y)}{(z-y)(y-t)}\bigg{(}\frac{P_{n+1}^{2}(y)}{h_{n}}-\frac{\beta_{n}P_{n-1}^{2}(y)}{h_{n-1}}\bigg{)}dy-1.$
(2.15)

(2) minus (2) follows

$\beta_{n+1}A_{n+1}(z)-\beta_{n}A_{n-1}(z)-(z-\alpha_{n})(B_{n+1}(z)-B_{n}(z))=1,$

and we arrive (2.11). ∎

######  Remark 2.

(2.10) and (2.11) are the well-known supplementary conditions ($S_{1}$) and
($S_{2}$), respectively. ($S_{1}$) and ($S_{2}$) have been applied to random
matrix theory in [20].

We produces an identity involving $\sum\limits_{k=0}^{n-1}A_{k}(z)$ by the
combination of ($S_{1}$) and ($S_{2}$).

######  Theorem 2.4.

$A_{n}(z)$, $B_{n}(z)$ and $\sum\limits_{k=0}^{n-1}A_{k}(z)$ satisfy the
identity

$B_{n}^{2}(z)+v_{0}^{\prime}(z)B_{n}(z)+\sum\limits_{k=0}^{n-1}A_{k}(z)=\beta_{n}A_{n}(z)A_{n-1}(z).$
($S_{2}^{\prime}$)

###### Proof.

Using ($S_{1}$) with multiplying ($S_{2}$) by $A_{n}(z)$ on both sides, we
have

$\displaystyle
A_{n}(z)+B_{n+1}^{2}(z)-B_{n}^{2}(z)+v_{0}^{\prime}(z)(B_{n+1}(z)-B_{n}(z))$
$\displaystyle=$
$\displaystyle\beta_{n+1}A_{n+1}(z)A_{n}(z)-\beta_{n}A_{n}(z)A_{n-1}(z).$

Taking a telescopic sum together with the appropriate ”initial conditions”,
$B_{0}(z)=0$ and $\beta_{0}A_{-1}(z)=0$, we obtain the desired result. ∎

######  Remark 3.

The compatibility conditions ($S_{1}$), ($S_{2}$) and ($S_{2}^{\prime}$) are
valid for $z\in\textbf{C}\cup\\{\infty\\}$.

Using Theorem 2.1 and Theorem 2.3, we obtain the second order linear ordinary
differential equation satisfied by $P_{n}(z)$.

######  Theorem 2.5.

The monic orthogonal polynomials $P_{n}(z)$ satisfy the second order
differential equation,

$P_{n}^{\prime\prime}(z)+Q_{n}(z)P_{n}^{\prime}(z)+T_{n}(z)P_{n}(z)=0$ (2.16)

where

$Q_{n}(z)=-v_{0}^{\prime}(z)-\frac{A_{n}^{\prime}(z)}{A_{n}(z)},$ (2.17)
$T_{n}(z)=B_{n}^{\prime}(z)-\frac{B_{n}(z)A_{n}^{\prime}(z)}{A_{n}(z)}+\beta_{n}A_{n}(z)A_{n-1}(z)-B_{n}^{2}(z)-B_{n}(z)v_{0}^{\prime}(z).$
(2.18)

###### Proof.

See [24, 5, 21, 14]. ∎

##  3 Deformed Hermite weight

Recalling Theorem 2.1,

$\displaystyle
A_{n}(z)=\frac{1}{h_{n}}\int_{\textbf{R}}P_{n}^{2}(y)\frac{v_{0}^{\prime}(z)-v_{0}^{\prime}(y)}{z-y}w(y)dy+a_{n}(t),$
$B_{n}(z)=\frac{1}{h_{n-1}}\int_{\textbf{R}}P_{n}(y)P_{n-1}(y)\frac{v_{0}^{\prime}(z)-v_{0}^{\prime}(y)}{z-y}w(y)dy+b_{n}(t),$
$a_{n}(t)=\frac{\gamma}{h_{n}}\int_{\textbf{R}}\frac{P_{n}^{2}(y)}{(z-y)(y-t)}w(y)dy$

and

$b_{n}(t)=\frac{\gamma}{h_{n-1}}\int_{\textbf{R}}\frac{P_{n}(y)P_{n-1}(y)}{(z-y)(y-t)}w(y)dy,$

we have the following expression of $A_{n}(z)$ and $B_{n}(z)$.

######  Theorem 3.1.

As $n\rightarrow\infty$, it yields

$A_{n}(z)=2+\frac{R_{n}(t)}{z}+\frac{\gamma+tR_{n}(t)}{z^{2}}+\frac{\gamma\alpha_{n}+\gamma
t+t^{2}R_{n}(t)}{z^{3}}+\mathcal{O}(z^{-4})$ (3.1)

and

$B_{n}(z)=\frac{r_{n}(t)}{z}+\frac{tr_{n}(t)}{z^{2}}+\frac{\gamma\beta_{n}+t^{2}r_{n}(t)}{z^{3}}+\mathcal{O}(z^{-4}),$
(3.2)

where

$R_{n}(t)=\frac{\gamma}{h_{n}}\int_{\mathbf{R}}\frac{P_{n}^{2}(y)w(y)}{y-t}dy$
(3.3)

and

$r_{n}(t)=\frac{\gamma}{h_{n-1}}\int_{\mathbf{R}}\frac{P_{n}(y)P_{n-1}(y)w(y)}{y-t}dy.$
(3.4)

###### Proof.

As $n\rightarrow\infty$,

$\displaystyle\frac{1}{z-y}$
$\displaystyle=\frac{1}{z}\cdot\frac{1}{1-\frac{y}{z}}=\frac{1}{z}\bigg{(}1+\frac{y}{z}+\frac{y^{2}}{z^{2}}+\mathcal{O}(z^{-3})\bigg{)}$
$\displaystyle=\frac{1}{z}+\frac{y}{z^{2}}+\frac{y^{2}}{z^{3}}+\mathcal{O}(z^{-4}),$

we can rewrite $a_{n}(t)$ and $b_{n}(t)$, respectively,

$\displaystyle a_{n}(t)=$
$\displaystyle\frac{\gamma}{zh_{n}}\int_{\mathbf{R}}\frac{P_{n}^{2}(y)w(y)}{y-t}dy+\frac{\gamma}{z^{2}h_{n}}\int_{\mathbf{R}}\frac{yP_{n}^{2}(y)w(y)}{y-t}dy+\frac{\gamma}{z^{3}h_{n}}\int_{\mathbf{R}}\frac{y^{2}P_{n}^{2}(y)w(y)}{y-t}dy$
$\displaystyle+\mathcal{O}(z^{-4}),$ (3.5) $\displaystyle b_{n}(t)=$
$\displaystyle\frac{\gamma}{zh_{n-1}}\int_{\mathbf{R}}\frac{P_{n}(y)P_{n-1}(y)w(y)}{y-t}dy+\frac{\gamma}{z^{2}h_{n-1}}\int_{\mathbf{R}}\frac{yP_{n}(y)P_{n-1}(y)w(y)}{y-t}dy$
$\displaystyle+\frac{\gamma}{z^{3}h_{n-1}}\int_{\mathbf{R}}\frac{y^{2}P_{n}(y)P_{n-1}(y)w(y)}{y-t}dy+\mathcal{O}(z^{-4}).$
(3.6)

Note

$\displaystyle\frac{\gamma}{h_{n}}\int_{\mathbf{R}}\frac{yP_{n}^{2}(y)w(y)}{y-t}dy=\frac{\gamma}{h_{n}}\int_{\mathbf{R}}\frac{((y-t)+t)P_{n}^{2}(y)w(y)}{y-t}dy$
$\displaystyle=$
$\displaystyle\frac{\gamma}{h_{n}}\bigg{(}\int_{\mathbf{R}}P_{n}^{2}(y)w(y)dy+t\int_{\mathbf{R}}\frac{P_{n}^{2}(y)w(y)}{y-t}dy\bigg{)}$
$\displaystyle=$ $\displaystyle\gamma+\frac{\gamma
t}{h_{n}}\int_{\mathbf{R}}\frac{P_{n}^{2}(y)w(y)}{y-t}dy,$ (3.7)
$\displaystyle\frac{\gamma}{h_{n}}\int_{\mathbf{R}}\frac{y^{2}P_{n}^{2}(y)w(y)}{y-t}dy=\frac{\gamma}{h_{n}}\int_{\mathbf{R}}\frac{((y^{2}-t^{2})+t^{2})P_{n}^{2}(y)w(y)}{y-t}dy$
$\displaystyle=$
$\displaystyle\frac{\gamma}{h_{n}}\bigg{(}\int_{\mathbf{R}}(y+t)P_{n}^{2}(y)w(y)dy+t^{2}\int_{\mathbf{R}}\frac{P_{n}^{2}(y)w(y)}{y-t}dy\bigg{)}$
$\displaystyle=$
$\displaystyle\frac{\gamma}{h_{n}}\bigg{(}\alpha_{n}h_{n}+th_{n}+t^{2}\int_{\mathbf{R}}\frac{P_{n}^{2}(y)w(y)}{y-t}dy\bigg{)}$
$\displaystyle=$ $\displaystyle\gamma\alpha_{n}+\gamma t+\frac{\gamma
t^{2}}{h_{n}}\int_{\textbf{R}}\frac{P_{n}^{2}(y)w(y)}{y-t}dy,$ (3.8)
$\displaystyle\frac{\gamma}{h_{n-1}}\int_{\mathbf{R}}\frac{yP_{n}(y)P_{n-1}(y)w(y)}{y-t}dy=\frac{\gamma}{h_{n-1}}\int_{\mathbf{R}}\frac{((y-t)+t)P_{n}(y)P_{n-1}(y)w(y)}{y-t}dy$
$\displaystyle=$
$\displaystyle\frac{\gamma}{h_{n-1}}\bigg{(}\int_{\mathbf{R}}P_{n}(y)P_{n-1}(y)w(y)dy+t\int_{\mathbf{R}}\frac{P_{n}(y)P_{n-1}(y)w(y)}{y-t}dy\bigg{)}$
$\displaystyle=$ $\displaystyle\frac{\gamma
t}{h_{n-1}}\int_{\mathbf{R}}\frac{P_{n}(y)P_{n-1}(y)w(y)}{y-t}dy,$ (3.9)

and

$\displaystyle\frac{\gamma}{h_{n-1}}\int_{\mathbf{R}}\frac{y^{2}P_{n}(y)P_{n-1}(y)w(y)}{y-t}dy=\frac{\gamma}{h_{n-1}}\int_{\mathbf{R}}\frac{((y^{2}-t^{2})+t^{2})P_{n}(y)P_{n-1}(y)w(y)}{y-t}dy$
$\displaystyle=$
$\displaystyle\frac{\gamma}{h_{n-1}}\bigg{(}\int_{\mathbf{R}}(y+t)P_{n}(y)P_{n-1}(y)w(y)dy+t^{2}\int_{\mathbf{R}}\frac{P_{n}(y)P_{n-1}(y)w(y)}{y-t}dy\bigg{)}$
$\displaystyle=$
$\displaystyle\frac{\gamma}{h_{n-1}}\bigg{(}\beta_{n}h_{n-1}+t^{2}\int_{\mathbf{R}}\frac{P_{n}(y)P_{n-1}(y)w(y)}{y-t}dy\bigg{)}$
$\displaystyle=$ $\displaystyle\gamma\beta_{n}+\frac{\gamma
t^{2}}{h_{n-1}}\int_{\textbf{R}}\frac{P_{n}(y)P_{n-1}(y)w(y)}{y-t}dy,$ (3.10)

where we used (1.5). Substituting (3), (3), (3) and (3) into (3) and (3), the
theorem is established. ∎

Putting (3.1) and (3.2) into ($S_{1}$), and comparing the constants and
coefficients of $\frac{1}{z}$, respectively, we have

$2\alpha_{n}=t+R_{n}(t),$ (3.11)

and

$r_{n+1}(t)+r_{n}(t)=\gamma+(t-\alpha_{n})R_{n}(t).$ (3.12)

Carrying out a similar calculation with ($S_{2}$) and ($S_{2}^{\prime}$), they
give rise to another three identities:

$1-r_{n}(t)+r_{n+1}(t)=2\beta_{n+1}-2\beta_{n},$ (3.13)
$(t-\alpha_{n})(r_{n+1}(t)-r_{n}(t))=\beta_{n+1}R_{n+1}(t)-\beta_{n}R_{n-1}(t)$
(3.14)

and

$\sum\limits_{j=0}^{n-1}R_{j}(t)=2\beta_{n}\left(R_{n-1}(t)+R_{n}(t)\right)-tr_{n}(t).$
(3.15)

A telescopic sum of (3.13) gives

$n+r_{n}(t)=2\beta_{n}.$ (3.16)

Multiplying $R_{n}(t)$ on both sides of (3.14),

$(t-\alpha_{n})R_{n}(t)(r_{n+1}(t)-r_{n}(t))=\beta_{n+1}R_{n+1}(t)R_{n}(t)-\beta_{n}R_{n-1}(t)R_{n}(t),$

then eliminating $(t-\alpha_{n})R_{n}(t)$ by (3.12), it yields

$(r_{n+1}^{2}(t)-r_{n}^{2}(t)-\gamma)(r_{n+1}(t)-r_{n}(t))=\beta_{n+1}R_{n+1}(t)R_{n}(t)-\beta_{n}R_{n-1}(t)R_{n}(t).$

A telescopic sum of above equation,

$r_{n}^{2}(t)-\gamma r_{n}(t)=\beta_{n}R_{n-1}(t)R_{n}(t).$ (3.17)

######  Remark.

(3.11), (3.12), (3.13), (3.14), (3.15), (3.16) and (3.17) are generated from
($S_{1}$), ($S_{2}$) and ($S_{2}^{\prime}$), which are significant for the
next discussions.

##  4 The $t$ dependance

Mention again that the weight (1.2) depends on $t$, it means all of the
quantities considered in this paper can be viewed as functions in $t$. In this
section, we will investigate their dependance with respect to their parameter.

Taking a derivative with respect to $t$ in (1.4) when $i=j=n$,

$h_{n}(t,\gamma)=\int_{\mathbf{R}}P_{n}^{2}(z,t,\gamma)w(z.t,\gamma)dz=\int_{\mathbf{R}}P_{n}^{2}(z,t,\gamma)e^{-z^{2}+tz}|z-t|^{\gamma}(A+B\theta(z-t))dz,$

we find

$\displaystyle\frac{\partial h_{n}(t,\gamma)}{\partial t}$
$\displaystyle=\int_{\mathbf{R}}P_{n}^{2}(z,t,\gamma)w(z,t,\gamma)zdz+\int_{\mathbf{R}}\frac{\gamma
P_{n}^{2}(z,t,\gamma)w(z,t,\gamma)}{t-z}dz$
$\displaystyle=\alpha_{n}h_{n}-\gamma\int_{\mathbf{R}}\frac{P_{n}^{2}(z,t,\gamma)w(z,t,\gamma)}{z-t}dz$
$\displaystyle=\alpha_{n}h_{n}-R_{n}(t)h_{n},$ (4.1)

where

$\frac{\partial(|z-t|^{\gamma})}{\partial
t}=\frac{\partial(|t-z|^{\gamma})}{\partial
t}=\bigg{(}(t-z)^{\gamma}-(z-t)^{\gamma}\bigg{)}\delta(t-z)+\frac{\gamma|t-z|^{\gamma}}{t-z}$

and the definition of $R_{n}(t)$ are used. Straightforward,

$\frac{d\ln
h_{n}(t)}{dt}=\frac{1}{h_{n}(t)}\frac{dh_{n}(t)}{dt}=\alpha_{n}-R_{n}(t)=\frac{t}{2}-\frac{R_{n}(t)}{2},$
(4.2)

where $\alpha_{n}$ is eliminated by (3.11).

It follows

$\displaystyle\frac{d\ln\beta_{n}}{dt}$
$\displaystyle=\frac{1}{h_{n}(t)}\frac{dh_{n}(t)}{dt}-\frac{1}{h_{n-1}(t)}\frac{dh_{n-1}(t)}{dt}$
$\displaystyle=\frac{1}{2}(R_{n-1}(t)-R_{n}(t)),$

where we used (1.8), and we can pose the derivative of $\beta_{n}$ by the
difference between $R_{n-1}(t)$ and $R_{n}(t)$,

$\frac{d\beta_{n}}{dt}=\frac{\beta_{n}}{2}(R_{n-1}(t)-R_{n}(t)).$

On the other hand, similarly steps, taking a derivative of

$\int_{\mathbf{R}}P_{n}(z,t,\gamma)P_{n-1}(z,t,\gamma)w(z,t,\gamma)dz=0$

with respect to $t$ and using the definition of $r_{n}(t)$, it produces

$\displaystyle 0=$ $\displaystyle\int_{\mathbf{R}}\frac{\partial
P_{n}(z,t,\gamma)}{\partial
t}P_{n-1}(z,t,\gamma)w(z,t,\gamma)dz+\int_{\mathbf{R}}P_{n}(z,t,\gamma)P_{n-1}(z,t,\gamma)w(z,t,\gamma)zdz$
$\displaystyle+\int_{\mathbf{R}}P_{n}(z,t,\gamma)P_{n-1}(z,t,\gamma)e^{-z^{2}+tz}\frac{\partial(|z-t|^{\gamma})}{\partial
t}(A+B\theta(z-t))dz$
$\displaystyle+\int_{\mathbf{R}}P_{n}(z,t,\gamma)P_{n-1}(z,t,\gamma)e^{-z^{2}+tz}|z-t|^{\gamma}\frac{\partial(A+B\theta(z-t))}{\partial
t}dz$ $\displaystyle=$ $\displaystyle
h_{n-1}\frac{d\textbf{p}(n,t)}{dt}+\beta_{n}h_{n-1}-\gamma\int_{\mathbf{R}}\frac{P_{n}(z)P_{n-1}(z)w(z)}{z-t}dz$
$\displaystyle=$ $\displaystyle
h_{n-1}\frac{d\textbf{p}(n,t)}{dt}+\beta_{n}h_{n-1}-r_{n}(t)h_{n-1},$

simplify,

$\frac{d\textbf{p}(n,t)}{dt}=r_{n}(t)-\beta_{n}.$ (4.3)

Differential (1.7) and applying (4.3),

$\displaystyle\alpha_{n}^{\prime}(t)$
$\displaystyle=r_{n}(t)-\beta_{n}-r_{n+1}(t)+\beta_{n+1}$
$\displaystyle=\frac{1}{2}(r_{n}(t)-r_{n+1}(t)+1),$

since (3.16). We summarize in the following theorem.

######  Theorem 4.1.

The recurrence coefficients $\alpha_{n}$ and $\beta_{n}$ are expressed in
terms of $r_{n}(t)$, $r_{n+1}(t)$, $R_{n-1}(t)$ and $R_{n}(t)$ as follows:

$\beta_{n}^{\prime}(t)=\frac{\beta_{n}}{2}\left(R_{n-1}(t)-R_{n}(t)\right),$
(4.4)

and

$\displaystyle\alpha_{n}^{\prime}(t)$
$\displaystyle=\frac{1}{2}(r_{n}(t)-r_{n+1}(t)+1).$ (4.5)

######  Theorem 4.2.

The auxiliary quantities $r_{n}(t)$ and $R_{n}(t)$ satisfy the coupled Riccati
equations

$r_{n}^{\prime}(t)=\frac{r_{n}^{2}(t)-\gamma
r_{n}(t)}{R_{n}(t)}-\frac{(r_{n}(t)+n)R_{n}(t)}{2},$ (4.6)
$R_{n}^{\prime}(t)=2r_{n}(t)-\frac{1}{2}(t-R_{n}(t))R_{n}(t)-\gamma.$ (4.7)

###### Proof.

Taking a derivative of (3.16) with respect to $t$,

$2\beta_{n}^{\prime}(t)=r_{n}^{\prime}(t),$ (4.8)

eliminating $\beta_{n}^{\prime}(t)$ by (4.4),

$r_{n}^{\prime}(t)=\beta_{n}R_{n-1}(t)-\beta_{n}R_{n}(t),$

then eliminating $\beta_{n}R_{n-1}(t)$ by (3.17), and eliminating another
$\beta_{n}$ by (3.16), respectively, we arrive (4.6).

Combining (3.12) and (3.11), and eliminating $\alpha_{n}$,

$r_{n+1}(t)+r_{n}(t)=\gamma+\frac{1}{2}(t-R_{n}(t))R_{n}(t).$ (4.9)

Substituting (4.9) into (4.5), and eliminating $r_{n+1}(t)$,

$\alpha_{n}^{\prime}(t)=r_{n}(t)-\frac{1}{4}(t-R_{n}(t))R_{n}(t)+\frac{1-\gamma}{2}.$
(4.10)

We note the derivative on both sides of (3.11),

$2\alpha_{n}^{\prime}(t)=1+R_{n}^{\prime}(t).$ (4.11)

A simple computation with combining (4.10) and (4.11) and eliminating
$\alpha_{n}^{\prime}(t)$, (4.7) is confirmed. ∎

##  5 $\sigma-$Form

In this section, we introduce two auxiliary quantities $\sigma_{n}(t)$ and
$\hat{\sigma}_{n}(t)$, and obtain their second order non-linear differential
or difference equations.
Define a quantity allied to the Hankel determinant,

$\sigma_{n}(t):=\frac{d}{dt}\ln
D_{n}(t)=\frac{d}{dt}\sum\limits_{j=0}^{n-1}\ln
h_{j}(t)=\sum\limits_{j=0}^{n-1}\frac{d}{dt}\left(\ln h_{j}(t)\right).$

Due to (4.2), (3.11) and (1.9),

$\displaystyle\sigma_{n}(t)$
$\displaystyle=\frac{nt}{2}-\frac{1}{2}\sum\limits_{j=0}^{n-1}R_{j}(t)=nt-\sum\limits_{j=0}^{n-1}\alpha_{j}=\textbf{p}(n,t,\gamma)+nt.$
(5.1)

######  Theorem 5.1.

The quantity $\sigma_{n}(t)$ satisfies

$(\sigma_{n}^{\prime\prime}(t))^{2}=\frac{1}{4}(t\sigma_{n}^{\prime}(t)-\sigma_{n}(t))^{2}-4(\sigma_{n}^{\prime}(t)-\frac{n}{2})\sigma_{n}^{\prime}(t)(\sigma_{n}^{\prime}(t)-\frac{n+\gamma}{2}),$
(5.2)

which is a particular Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé IV
equation [15].

###### Proof.

In view of (5.1),

$nt-2\sigma_{n}(t)=\sum\limits_{j=0}^{n-1}R_{j}(t).$ (5.3)

Combining (3.15) and above equation,

$nt-2\sigma_{n}(t)=2\beta_{n}R_{n-1}(t)+2\beta_{n}R_{n}(t)-tr_{n}(t),$

then eliminating $\beta_{n}R_{n-1}(t)$ and $\beta_{n}$ by (3.17) and (3.16),
respectively,

$(n+r_{n}(t))R_{n}(t)+\frac{2(r_{n}^{2}(t)-\gamma
r_{n}(t))}{R_{n}(t)}=tr_{n}(t)-2\sigma_{n}(t)+nt.$ (5.4)

On account of (4.6),

$\frac{2(r_{n}^{2}(t)-\gamma
r_{n}(t))}{R_{n}(t)}-(n+r_{n}(t))R_{n}(t)=2r_{n}^{\prime}(t).$ (5.5)

The sum of (5.4) and (5.5) gives

$\frac{4(r_{n}^{2}(t)-\gamma
r_{n}(t))}{R_{n}(t)}=tr_{n}(t)-2\sigma_{n}(t)+nt+2r_{n}^{\prime}(t);$ (5.6)

oppositely, (5.4) minus (5.5),

$2(n+r_{n}(t))R_{n}(t)=tr_{n}(t)-2\sigma_{n}(t)+nt-2r_{n}^{\prime}(t).$ (5.7)

Then the product of (5.6) and (5.7) is

$8r_{n}(t)(r_{n}(t)+n)(r_{n}(t)-\gamma)=(tr_{n}(t)-2\sigma_{n}(t)+nt)^{2}-4(r_{n}^{\prime}(t))^{2}.$
(5.8)

Taking the derivative of (5.1) with respect to $t$ gives to

$\sigma_{n}^{\prime}(t)=\textbf{p}^{\prime}(n,t,\gamma)+n,$ (5.9)

and eliminating $\beta_{n}$ between (3.16) and (4.3),

$\textbf{p}^{\prime}(n,t,\gamma)=\frac{r_{n}(t)}{2}-\frac{n}{2}.$ (5.10)

Inserting (5.10) into (5.9), and eliminating
$\textbf{p}^{\prime}(n,t,\gamma)$,

$\sigma_{n}^{\prime}(t)=\frac{r_{n}(t)}{2}+\frac{n}{2}.$ (5.11)

Some simplifications after substituting (5.11) into (5.8), we get (5.2)
immediately. ∎

######  Remark 4.

$\sigma_{n}(t)$ satisfies the continuous $\sigma-$form of the Painlevé IV,
which theorem 5.1 is coincident with Tracy and Widom [20].

######  Proposition 5.2.

We have

$\displaystyle\sigma_{n}(t)=$ $\displaystyle\frac{d}{dt}\ln D_{n}(t)$
$\displaystyle=$
$\displaystyle-\frac{(R_{n}^{\prime}(t))^{2}}{4R_{n}(t)}+\frac{R_{n}^{3}(t)}{16}-\frac{t}{8}R_{n}^{2}(t)+\bigg{(}\frac{t^{2}}{16}-\frac{n}{2}-\frac{\gamma}{4}\bigg{)}R_{n}(t)+\frac{(2n+\gamma)t}{4}$
$\displaystyle+\frac{\gamma^{2}}{4R_{n}(t)}$ (5.12)

###### Proof.

For (3.15), eliminating $\beta_{n}R_{n-1}(t)$ and $\beta_{n}$ by (3.17) and
(3.16),

$\sum\limits_{j=0}^{n-1}R_{j}(t)=\frac{2(r_{n}^{2}(t)-\gamma
r_{n}(t))}{R_{n}(t)}+(n+r_{n}(t))R_{n}(t)-tr_{n}(t).$ (5.13)

Substituting (5.13) into (5.3),

$\displaystyle\sigma_{n}(t)=\frac{nt}{2}-\frac{r_{n}^{2}(t)-\gamma
r_{n}(t)}{R_{n}(t)}-\frac{(n+r_{n}(t))R_{n}(t)}{2}+\frac{tr_{n}(t)}{2}.$
(5.14)

Combining (4.7) and (5.14), and eliminating $r_{n}(t)$, the proof is
completed.

∎

Also define the other quantity

$\hat{\sigma}_{n}(t):=-\sum\limits_{j=0}^{n-1}R_{j}(t).$ (5.15)

######  Theorem 5.3.

A second-order difference equation satisfied by $\hat{\sigma}_{n}(t)$,

$\displaystyle
2(n\hat{\sigma}_{n-1}+\hat{\sigma}_{n}-n\hat{\sigma}_{n+1})^{2}-2\gamma(n\hat{\sigma}_{n-1}+\hat{\sigma}_{n}-n\hat{\sigma}_{n+1})(t-\hat{\sigma}_{n-1}+\hat{\sigma}_{n+1})$
$\displaystyle=$
$\displaystyle(\hat{\sigma}_{n-1}-\hat{\sigma}_{n})(\hat{\sigma}_{n}-\hat{\sigma}_{n+1})(\hat{\sigma}_{n}+nt)(\hat{\sigma}_{n+1}-\hat{\sigma}_{n-1}+t),$
(5.16)

which is a discrete $\sigma$-form equation.

###### Proof.

From (5.15),

$R_{n}(t)=\hat{\sigma}_{n}(t)-\hat{\sigma}_{n+1}(t),$ (5.17)

we can rewrite (3.15) with the aid of (3.16) and (5.17), i.e.

$(n+r_{n}(t))(\hat{\sigma}_{n-1}(t)-\hat{\sigma}_{n+1}(t))-tr_{n}(t)=-\hat{\sigma}_{n}(t),$

solving $r_{n}(t)$,

$r_{n}(t)=\frac{n(\hat{\sigma}_{n-1}(t)-\hat{\sigma}_{n+1}(t))+\hat{\sigma}_{n}(t)}{t-\hat{\sigma}_{n-1}(t)+\hat{\sigma}_{n+1}(t)}.$
(5.18)

Similarly steps to recall (3.17) by using (3.16) and (5.17),

$r_{n}^{2}(t)-\gamma
r_{n}(t)=\frac{1}{2}(n+r_{n}(t))(\hat{\sigma}_{n-1}(t)-\hat{\sigma}_{n}(t))(\hat{\sigma}_{n}(t)-\hat{\sigma}_{n+1}(t)).$
(5.19)

Inserting (5.18) into (5.19) and simplify, it yields (5.3). ∎

######  Remark.

When $\gamma=0$, (5.3) is a discrete $\sigma$-form of Painlevé IV equation,
which corresponds to Min and Chen [14].

##  6 Painlevé IV and asymptotic behaviors

######  Theorem 6.1.

The quantity $R_{n}(t)$ satisfies the following second order differential
equation:

$R_{n}^{\prime\prime}(t)=\frac{(R_{n}^{\prime}(t))^{2}}{2R_{n}(t)}+\frac{3}{8}R_{n}^{3}(t)-\frac{t}{2}R_{n}^{2}(t)+\frac{1}{8}\left(t^{2}-8n-4-4\gamma\right)R_{n}(t)-\frac{\gamma^{2}}{2R_{n}(t)},$
(6.1)

which is a particular Painlevé IV equation. Moreover, let
$\widetilde{R}_{n}:=\frac{1}{2}R_{n}(t)$ and $u:=-\frac{1}{4}t$, then

$\widetilde{R}_{n}^{\prime\prime}(u)=\frac{(\widetilde{R}_{n}^{\prime}(u))^{2}}{2\widetilde{R}_{n}(u)}+\frac{3}{2}\widetilde{R}_{n}^{3}(u)+4u\widetilde{R}_{n}^{2}(u)+2\left(u^{2}-\theta_{1}\right)\widetilde{R}_{n}(u)+\frac{\theta_{2}}{\widetilde{R}_{n}(u)},$

which satisfies the Painlevé IV equation[11] with
$\theta_{1}=\frac{2n+1+\gamma}{4}$, $\theta_{2}=-\frac{\gamma^{2}}{8}$.

###### Proof.

Solving for $r_{n}(t)$ from (4.7),

$r_{n}(t)=\frac{1}{2}R_{n}^{\prime}(t)+\frac{1}{4}(t-R_{n}(t))R_{n}(t)+\frac{\gamma}{2},$

and substituting the solution into (4.6), simplify, we find (6.1).
Let $\widetilde{R}_{n}:=\frac{1}{2}R_{n}(t)$ and $u:=-\frac{1}{4}t$, it is to
see that $\widetilde{R}_{n}(u)$ takes the form

$\widetilde{R}_{n}^{\prime\prime}(u)=\frac{(\widetilde{R}_{n}^{\prime}(u))^{2}}{2\widetilde{R}_{n}(u)}+\frac{3}{2}\widetilde{R}_{n}^{3}(u)+4u\widetilde{R}_{n}^{2}(u)+2\left(u^{2}-\frac{n}{2}-\frac{1}{4}-\frac{\gamma}{4}\right)\widetilde{R}_{n}(u)-\frac{\gamma^{2}}{8\widetilde{R}_{n}(u)}.$

∎

######  Remark 5.

A second-order differential equation satisfied by $r_{n}(t)$, which is related
to the Chazy equation [12, 14], can also be obtained by solving for $R_{n}(t)$
from (4.6) and substituting the result of $R_{n}(t)$ into (4.7). We do not
write it down due to its too complicated.

Disregard the derivative in (6.1) and get a quartic equation

$\frac{3}{8}R_{n}^{4}(t)-\frac{t}{2}R_{n}^{3}(t)+\frac{1}{8}(t^{2}-8n-4-4\gamma)R_{n}^{2}(t)-\frac{\gamma^{2}}{2}=0.$
(6.2)

If $\gamma=0$ for (6.2), $X_{n}(t)$ satisfies a quadratic equation,

$3X_{n}^{2}(t)-4tX_{n}(t)+t^{2}-8n-4=0,$

with the solutions

$X_{n}(t)=\frac{2t\pm\sqrt{t^{2}+24n+12}}{3}.$

Due to the definition of weight (1.2), we separate two parts to discuss the
asymptotic behaviors of $R_{n}(t)$.
(i)If $B>0$, we choose

$X_{n}(t)=\frac{2t+\sqrt{t^{2}+24n+12}}{3},$

as $n\rightarrow\infty$,

$X_{n}(t)=\frac{2\sqrt{6}}{3}n^{\frac{1}{2}}+\frac{2t}{3}+\frac{12+t^{2}}{12\sqrt{6}}n^{-\frac{1}{2}}-\frac{(12+t^{2})^{2}}{1152\sqrt{6}}n^{-\frac{3}{2}}+\mathcal{O}(n^{-\frac{5}{2}}).$

Hence, we assume the expansion of $R_{n}(t)$, as $n\rightarrow\infty$,

$R_{n}(t)=\sum\limits_{j=0}^{n-1}d_{j}(t)n^{\frac{1-j}{2}}.$ (6.3)

Substituting (6.3) into (6.1), we obtain

$\displaystyle d_{0}(t)$
$\displaystyle=\frac{2\sqrt{6}}{3},~{}~{}~{}~{}d_{1}(t)=\frac{2t}{3},~{}~{}~{}~{}d_{2}(t)=\frac{\sqrt{6}(t^{2}+12\gamma+12)}{72},~{}~{}~{}~{}d_{3}(t)=0,$
$\displaystyle d_{4}(t)$
$\displaystyle=\frac{\sqrt{6}(288\gamma^{2}-24t^{2}\gamma-288\gamma-t^{4}-24t^{2}-240)}{6912},~{}~{}~{}~{}d_{5}(t)=\frac{(2-9\gamma^{2})t}{72},$

which means

$\displaystyle R_{n}(t)=$
$\displaystyle\frac{2\sqrt{6}}{3}n^{\frac{1}{2}}+\frac{2t}{3}+\frac{\sqrt{6}(t^{2}+12\gamma+12)}{72}n^{-\frac{1}{2}}$
$\displaystyle+\frac{\sqrt{6}(288\gamma^{2}-24t^{2}\gamma-288\gamma-t^{4}-24t^{2}-240)}{6912}n^{-\frac{3}{2}}+\frac{(2-9\gamma^{2})t}{72}n^{-2}$
$\displaystyle+\mathcal{O}(n^{-\frac{5}{2}}).$

(ii)If $B<0$, we choose

$X_{n}(t)=\frac{2t-\sqrt{t^{2}+24n+12}}{3},$

as $n\rightarrow\infty$,

$X_{n}(t)=-\frac{2\sqrt{6}}{3}n^{\frac{1}{2}}+\frac{2t}{3}-\frac{12+t^{2}}{12\sqrt{6}}n^{-\frac{1}{2}}+\frac{(12+t^{2})^{2}}{1152\sqrt{6}}n^{-\frac{3}{2}}+\mathcal{O}(n^{-\frac{5}{2}}).$

Similarly, we assume the expansion of $R_{n}(t)$, as $n\rightarrow\infty$,

$R_{n}(t)=\sum\limits_{j=0}^{n-1}\widetilde{d}_{j}(t)n^{\frac{1-j}{2}}.$ (6.4)

Substituting (6.4) into (6.1), we obtain

$\displaystyle\widetilde{d}_{0}(t)$
$\displaystyle=-\frac{2\sqrt{6}}{3},~{}~{}~{}~{}\widetilde{d}_{1}(t)=\frac{2t}{3},~{}~{}~{}~{}\widetilde{d}_{2}(t)=-\frac{\sqrt{6}(t^{2}+12\gamma+12)}{72},~{}~{}~{}~{}\widetilde{d}_{3}(t)=0,$
$\displaystyle\widetilde{d}_{4}(t)$
$\displaystyle=-\frac{\sqrt{6}(288\gamma^{2}-24t^{2}\gamma-288\gamma-t^{4}-24t^{2}-240)}{6912},~{}~{}~{}~{}\widetilde{d}_{5}(t)=\frac{(2-9\gamma^{2})t}{72},$

which means

$\displaystyle R_{n}(t)=$
$\displaystyle-\frac{2\sqrt{6}}{3}n^{\frac{1}{2}}+\frac{2t}{3}-\frac{\sqrt{6}(t^{2}+12\gamma+12)}{72}n^{-\frac{1}{2}}$
$\displaystyle-\frac{\sqrt{6}(288\gamma^{2}-24t^{2}\gamma-288\gamma-t^{4}-24t^{2}-240)}{6912}n^{-\frac{3}{2}}+\frac{(2-9\gamma^{2})t}{72}n^{-2}$
$\displaystyle+\mathcal{O}(n^{-\frac{5}{2}}).$

Straightforward, we summarize the following item.

######  Theorem 6.2.

As $n\rightarrow\infty$, the quantity $R_{n}(t)$ has the asymptotic
expansions:

  * •

if $B>0$

$\displaystyle R_{n}(t)=$
$\displaystyle\frac{2\sqrt{6}}{3}n^{\frac{1}{2}}+\frac{2t}{3}+\frac{\sqrt{6}(t^{2}+12\gamma+12)}{72}n^{-\frac{1}{2}}$
$\displaystyle+\frac{\sqrt{6}(288\gamma^{2}-24t^{2}\gamma-288\gamma-t^{4}-24t^{2}-240)}{6912}n^{-\frac{3}{2}}+\frac{(2-9\gamma^{2})t}{72}n^{-2}$
$\displaystyle+\mathcal{O}(n^{-\frac{5}{2}});$ (6.5)

  * •

if $B<0$

$\displaystyle R_{n}(t)=$
$\displaystyle-\frac{2\sqrt{6}}{3}n^{\frac{1}{2}}+\frac{2t}{3}-\frac{\sqrt{6}(t^{2}+12\gamma+12)}{72}n^{-\frac{1}{2}}$
$\displaystyle-\frac{\sqrt{6}(288\gamma^{2}-24t^{2}\gamma-288\gamma-t^{4}-24t^{2}-240)}{6912}n^{-\frac{3}{2}}+\frac{(2-9\gamma^{2})t}{72}n^{-2}$
$\displaystyle+\mathcal{O}(n^{-\frac{5}{2}}).$ (6.6)

By direct computations, we arrive at the following expansion formula for the
scaled Hankel determinant

######  Theorem 6.3.

For fixed $s>0$, $\gamma>-1$, we obtain

  * •

if $B>0$, as $n\rightarrow\infty$,

$\displaystyle\ln\frac{D_{n}(s)}{D_{n}(0)}=$
$\displaystyle-\frac{2\sqrt{6}}{9}sn^{\frac{3}{2}}+\frac{s^{2}n}{12}-\frac{\sqrt{6}(s^{3}+36\gamma
s)n^{\frac{1}{2}}}{216}+\frac{s^{4}}{864}+\frac{s^{2}\gamma}{24}$
$\displaystyle+\mathcal{O}(n^{-\frac{1}{2}})$ (6.7)

  * •

if $B<0$, as $n\rightarrow\infty$,

$\displaystyle\ln\frac{D_{n}(s)}{D_{n}(0)}=$
$\displaystyle\frac{2\sqrt{6}}{9}sn^{\frac{3}{2}}+\frac{s^{2}n}{12}+\frac{\sqrt{6}(s^{3}+36\gamma
s)n^{\frac{1}{2}}}{216}+\frac{s^{4}}{864}+\frac{s^{2}\gamma}{24}$
$\displaystyle+\mathcal{O}(n^{-\frac{1}{2}}).$ (6.8)

###### Proof.

Putting (• ‣ 6.2) into (5.2), the logarithmic derivative of the Hankel
determinant $D_{n}(t)$ for $B>0$ becomes,

$\displaystyle\frac{d}{dt}\ln D_{n}(t)=$
$\displaystyle-\frac{2\sqrt{6}}{9}n^{\frac{3}{2}}+\frac{nt}{6}-\frac{\sqrt{6}(t^{2}+12\gamma)n^{\frac{1}{2}}}{72}+\frac{t^{3}}{216}+\frac{t\gamma}{12}$
$\displaystyle+\mathcal{O}(n^{-\frac{1}{2}}).$ (6.9)

Then, integration on both sides of equation (6) from 0 to $s$,

$\int_{0}^{s}\frac{d}{dt}\ln D_{n}(t)dt=\ln D_{n}(s)-\ln D_{n}(0),$

we obtain (• ‣ 6.3). The proof is similar for $B<0$. ∎

Once the asymptotic behaviors for $R_{n}(t)$ is obtained, we also consider the
large $n$ behavior of the monic orthogonal polynomials $P_{n}(z)$.

######  Theorem 6.4.

As $n\rightarrow\infty$ and $t\rightarrow\infty$, $P_{n}(z)$ satisfies the
second order differential-difference equation ,

$P_{n}^{\prime\prime}(z)+\left(\frac{3}{z}-2z+t\right)P_{n}^{\prime}(z)+\bigg{(}\left(\frac{2t^{2}}{3}+\frac{\gamma}{12}\right)\frac{n^{2}\gamma}{z^{6}}+\frac{2n^{2}\gamma
t}{3z^{5}}+\frac{2n^{2}\gamma}{3z^{4}}\bigg{)}P_{n}(z)=0.$ (6.10)

###### Proof.

Substituting (3.1) and (3.2) into (2.17) and (2.18), $Q_{n}(z)$ and $T_{n}(z)$
can be expressed in terms of $r_{n}(t)$, $R_{n}(t)$ and $\beta_{n}$. It should
be pointed out that relationship among $R_{n}(t)$, $r_{n}(t)$, $\alpha_{n}$
and $\beta_{n}$ are shown by (3.11), (3.16) and (4.7). The coefficients of
(2.16) are given in terms of $R_{n}(t)$. Sending $n\rightarrow\infty$,
$t\rightarrow\infty$, and combining the asymptotic value (• ‣ 6.2) or (• ‣
6.2), we obtain (6.10). ∎

######  Remark 6.

When $\gamma=0$, (6.10) satisfies the bi-confluent Heun equation. More details
of the Heun equation, see [5, 10, 16, 18, 21, 19].

##  7 Acknowledgements

D. Wang, M. Zhu and Y. Chen would like to give thanks the Science and
Technology Development Fund of the Macau SAR for providing FDCT 023/ 2017/A1.
They would also like to thank the University of Macau for MYRG 2018-00125 FST.

##  8 Availability of data

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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# Contractility-driven cell motility against a viscoelastic resistance

Tapas Singha  Pierre Sens<EMAIL_ADDRESS>Institut Curie, PSL Research
University, CNRS UMR 168; F-75005 Paris, France

###### Abstract

We study a model of contraction-based cell motility inside a microchannel to
investigate the regulation of cell polarization and motion by the mechanical
resistance of the environment. A positive feedback between the asymmetry of
the acto-myosin cortex density and cell motion gives rise to a spontaneous
symmetry breaking beyond a threshold contractility that depends on the
resistance of extracellular medium. In highly viscous environments, we predict
bistability under moderate contractility, so that symmetry breaking needs to
be activated. In a viscoelastic environment, we find periodic oscillations in
cortex density and velocity polarization. At the boundary between viscous and
viscoelastic environments, the cell may either cross into the viscoelastic
medium, bounce back into the viscous medium, or become trapped at the
boundary. The different scenarios defined different phase diagram that are
confirmed by numerical simulations.

Cell motility is a fundamental cellular mechanism crucial to many biological
processes, from morphogenesis to cancer progression. Cells move using a
combination of protrusive forces generated by the polymerisation of polarised
actin filaments and contractile forces, often concentrated in the cell cortex,
a thin layer of actin and myosin motors underlying the cell membrane Bray
(1992); Rafelski and Theriot (2004). Contractility-driven motility is
particularly relevant for cells in three-dimensional environments made by
other cells or the extracellular matrix Doyle  _et al._ (2009, 2013);
Poincloux  _et al._ (2011). This phenotype is reproduced when cells are
confined in microchannels Liu  _et al._ (2015); Ruprecht  _et al._ (2015);
Bergert  _et al._ (2015), a popular setup to study how cell responds to
geometrical constraints Vargas  _et al._ (2014); *Rolli10POne; *Irimia09;
*Irimia07LabChip; *Maiuri12.

The cell cortex is a thin contractile layer of cytoskeleton of thickness
$\simeq 200nm$ Clark  _et al._ (2013), composed of multiple proteins: actin,
myosin, cross-linkers, nucleators, etc. Biro  _et al._ (2013). Theoretical
models based on the hydrodynamic theory of active gels Marchetti  _et al._
(2013) have been developed to described its active properties Salbreux  _et
al._ (2009); Mayer  _et al._ (2010); Bois  _et al._ (2011); Joanny  _et al._
(2013); Turlier  _et al._ (2014), and applied to contractility-based cell
motility. In one class of model Bergert  _et al._ (2015), a predefined cell
polarisation imposes a gradient of contractility which yields a cortical flow
toward the more contractile region (the rear) of the cell. Friction between
the flow and the environment generates traction forces that can move the cell.
A second class of model Hawkins  _et al._ (2011); Callan-Jones and Voituriez
(2013); Recho  _et al._ (2013); Ruprecht  _et al._ (2015) invokes an important
feature of active contractile gels: the existence of a positive feedback loop
between contractility gradients and cortical flowsBois  _et al._ (2011). As
the flow advects contractile elements toward more contractile regions of the
cortex, the homogeneous state may be unstable, resulting in spontaneous cell
polarisation and translocation without externally-imposed polarisation Hawkins
_et al._ (2011); Callan-Jones and Voituriez (2013); Recho  _et al._ (2013);
Ruprecht  _et al._ (2015).

The mechanical properties of the extra-cellular environment is known to affect
the way cells can polarise and move H. Yu and and Weaver (2011); *wolf:2011;
*gu:2014. Theoretical models predict that a cell with an imposed polarisation
can move in a microchannel filled with a viscous fluid provided the friction
between cell and channel wall exceeds a threshold proportional to viscous
dragBergert  _et al._ (2015). Here, we investigate how the cell’s ability to
exhibit spontaneous polarisation and motion depends on the mechanical
resistance of the environment. We find that even in the limit of infinite
friction, the motile steady state of broken symmetry only exists below a
threshold of viscous drag coefficient. This could have strong physiological
implications, as extracellular environments are quite often viscous. We
further show that cortical dynamics qualitatively affects the ability of the
cell to move in complex environments, such as viscoelastic environments, and
mechanical boundaries between different extracellular media.

Figure 1: No-slip cell motion in a microchannel. Top: Sketch of the model:
motion is the consequence of spontaneous symmetry breaking leading to a cortex
denser at the cell back and sparser at the cell front. This results in a
difference of cortical tension that drive cortical flow. Cortex recycling (net
depolymerisation at the back and net polymerisation at the front) transfers
cortical matter forward, while the stress difference pushes the cytosol
forward. Along the lateral surface in contact with the walls, the cortex is
immobile with respect to the laboratory frame. The forward force
$\Delta\gamma$ is balanced by a resistive force ${\cal R}$ which is either
viscous, viscoelastic or the boundary between viscous and viscoelastic.

Model. We study a minimal model for the spontaneous polarisation and motion of
a incompressible cell of fixed geometry confined in a circular channel
(Fig.1), wherein gradients of contractility in the cytoskeleton cortex create
cortical flow and generate friction forces with the channel walls. We neglect
the friction between the cortical flow and the medium inside the channel at
the cell edges, such that both the cortical tension $\gamma$ and the cortex
curvature are constant at the two cell edges. Global force balance at the cell
scale demands that $\Delta\gamma={\cal R}[U]$, where
$\Delta\gamma=\gamma_{b}-\gamma_{f}$ is the tension difference between the
cell edges (denoted back (b) and front (f)) and ${\cal R}$ is the resistive
force (divided by the perimeter of the cell cross section) which depends on
the cell velocity $U$. As detailed in the Supplementary Information (S.I.), it
can stem from dissipation in the extracellular medium, in the cytoplasm, or
from the detachment dynamics of adhesion molecules Décavé  _et al._ (2002);
Ward and Hammer (1993); DiMilla  _et al._ (1991).

Following previous works Bois  _et al._ (2011); Recho  _et al._ (2013);
Ruprecht  _et al._ (2015), we describe a simplified version of the model based
on the one dimensional (1D) active fluid formalism, where the cortex is
described by a single density field, $\rho(x,t)$ that accounts for all protein
populations, and a velocity field $v(x,t)$ describing cortical flow. The
equation for density conservation reads

$\frac{\partial\rho(x,t)}{\partial t}+\frac{\partial(\rho v)}{\partial
x}=D\frac{\partial^{2}\rho}{\partial x^{2}}-k_{d}\left(\rho-\rho_{0}\right)$
(1)

where $\rho_{0}$ is the reference (homeostatic) cortex density, $D$ is a
diffusion constant and $k_{d}$ is the cortex recycling rate.

Various forms of cortical stress tensor have been proposed in the literature
Salbreux  _et al._ (2009); Mayer  _et al._ (2010); Bois  _et al._ (2011);
Joanny  _et al._ (2013); Turlier  _et al._ (2014); Hawkins  _et al._ (2011);
Callan-Jones and Voituriez (2013); Recho  _et al._ (2013); Ruprecht  _et al._
(2015); Bergert  _et al._ (2015), and the specific formalism does not
fundamentally alter the phenomenology given that the stress tensor $\gamma$
includes contractile and viscous terms, and higher order compressibility terms
to ensure that teh density remains finite Joanny  _et al._ (2013). Here we
adopt the following expression for a 1D compressible contractile fluid:

$\gamma(x,t)=\zeta\,\left(\rho(x,t)-\frac{\rho^{2}(x,t)}{2\rho^{*}}\right)+\eta\frac{\partial}{\partial
x}v(x,t).$ (2)

where $\zeta$ is the contractility, $\eta$ is the viscosity, and the non-
linear saturating term is chosen to be quadratic with a characteristic density
$\rho^{*}$. The force balance of the cortex can be written as Bois  _et al._
(2011); Hawkins  _et al._ (2011); Callan-Jones and Voituriez (2013); Recho
_et al._ (2013); Ruprecht  _et al._ (2015); Bergert  _et al._ (2015)

$\frac{\partial\gamma(x,t)}{\partial x}=\Gamma\,v(x,t)$ (3)

where $\Gamma$ is the friction coefficient. In the 1D model, the cell edges
are of length $\ell$, fixed by the channel width, and the cell length in
contact with the channel wall is denoted by $L$. In what follows, we neglect
lateral friction at the cell edges ($\Gamma=0$ for $x\notin[\ell,L+\ell]$), an
approximation valid for long cells (see S.I.), and assume infinite friction
with the walls ($\Gamma\rightarrow\infty$ for $\ell\leq x\leq L+\ell$), so
that in the reference frame of the moving cell, the cortex velocity in contact
with the walls is $v(x)=-U={\rm constant}$.

Viscous resistance: symmetry breaking and steady states. Spontaneous cortical
instability is driven by a mechanical feedback between contraction and flow,
while recycling, diffusion and viscous dissipation stabilise an homogeneous
cortex. Linear stability analysis (LSA) (detailed in the S.I.) shows that, in
the absence of friction, a uniform isolated slab of cortex of length $\ell$ is
linearly unstable for $\bar{\zeta}>1+(\pi\ell_{D})^{2}$ where
$\bar{\zeta}=\zeta(1-\rho_{0}/\rho^{*})/(\eta k_{d})$ and
$\ell^{2}_{D}=D/\ell^{2}k_{d}$. Upon loss of stability, the cortical
heterogeneities eventually saturate and coarsen due to non-linear terms Bois
_et al._ (2011).

Under high friction with the channel walls, cell with broken symmetry moves in
a caterpillar fashion, with an immobile lateral cortex (for $\ell\leq x\leq
L+\ell$) and a net rearward flux of cortical material balanced by
polymerisation of the front ($x\leq\ell$, $\rho(x)<\rho_{0}$) and
depolymerisation of the rear ($x\geq L+\ell$, $\rho(x)>\rho_{0}$) (Fig.1). The
steady-state solution of Eqs.(1,2,3) can be obtained numerically in the moving
reference frame with appropriate boundary conditions: $v(0)=v(L+2\ell)=0$,
$v(\ell)=v(L+\ell)=-U$, $\partial_{x}\rho(0)=\partial_{x}\rho(L+2\ell)=0$ and
continuity of the flux at $x=\ell$ and $x=L+\ell$. Example of steady-state
density and velocity profiles for different cell velocity are presented in
Fig.2(A) and (B), showing that the density gradient increase with cell
velocity.

The cortical tension at each end of the cell is obtained from Eq.(2).
Enforcing global force balance then lead to a self-consistent determination of
the steady-state cell velocity against a viscous environment using

$\Delta\gamma[U]=\mathcal{R}=\beta\,U.$ (4)

where $\beta$ is an effective viscous drag and $U$ is the velocity of the
cell. The variation of $\Delta\gamma$ with the cell velocity is shown as an
inset in Fig.2(C). Beyond a critical contractility, it shows a linear rise for
small velocities, followed by a maximum and a decrease to negative values for
high velocities. The self-consistent cell velocity solution of Eq.(4) is
presented in Fig.2C, showing that the motile steady-state only exist below a
finite value of the viscous resistance $\beta$.

The critical value of contractility that gives rise to a motile state can be
obtained from a LSA of the homogeneous immobile state, detailed in the S.I.
Briefly, Eqs.(1,2,3) are linearised considering a small perturbation about
homogeneous density $\rho_{0}$ of a static cell ($U=0$) :
$\rho(x,t)=\rho_{0}+\delta\rho(x,t)$ and $v(x,t)=\delta v(x,t)$. In the cell
edges we have: $\frac{\partial\delta\rho}{\partial t}+\rho_{0}\frac{\partial
v}{\partial x}=D\frac{\partial^{2}\delta\rho}{\partial
x^{2}}-k_{d}\delta\rho$, and the local force balance equation is
$\zeta(1-\rho_{0}/\rho^{*})\frac{\partial\delta\rho}{\partial
x}+\eta\frac{\partial^{2}\delta v}{\partial x^{2}}=0$ and in the middle part:
$\frac{\partial\delta\rho}{\partial t}=D\frac{\partial^{2}\delta\rho}{\partial
x^{2}}-k_{d}\,\delta\rho$ and $\delta v=-U$, with appropriate boundary
conditions.

Figure 2: Viscous environment. A,B. steady-state density (A) and cortex
velocity (B) profiles. C. Variation of the steady-state cell velocity with the
viscous resistance $\beta$ (inset is a graphical representation of the force
balance Eq.(4)). D. Viscous phase diagram showing the region where the cell
spontaneously polarises and moves(teal). The threshold contractility (black
line) - is drawn from the LSA. Above the red horizontal line is a region of
bistability, showing the coexistence of a immobile state with a symmetric
heterogeneous cortex, and a motile state with asymmetric cortex, which is
stable even in the absence of motion (when $\beta\rightarrow\infty)$ (Eq.(6)).
With $\ell_{D}^{2}=0.25$ and $\rho^{*}=(3/2)\rho_{0}$.

Linear stability is investigated by expanding perturbations in density and
velocity using eigenmodes that satisfy the boundary conditions. For very long
cells (results for arbitrary length are discussed in the S.I.), these take the
form:
$\delta\rho(x,t)=\sum_{n=1}^{\infty}\rho_{n}e^{\alpha_{n}t}[\cos{(q_{n}\,x)}+A_{n}]$
and $\delta
v(x,t)=\sum_{n=1}^{\infty}v_{n}e^{\alpha_{n}t}[\sin{(q_{n}\,x)}+B_{n}x]$ at
the cell edges and
$\delta\rho(x,t)=\sum_{n=1}^{\infty}b_{n}\,e^{\alpha_{n}t}e^{-p_{n}(x-l)}$ and
$\delta v(x,t)=-U$ for the region in contact with the walls, where $n$ is the
mode index and $\alpha_{n}$ is the growth rate which is to be determined.

The linear stability analysis predicts the existence of a contractility
threshold $\overline{\zeta^{*}}$ beyond which there exists at least one
unstable mode (see S.I.). The relation of the threshold contractility with the
viscous resistance is:

$\tan\left(\frac{\sqrt{\overline{\zeta^{*}}-1}}{\ell_{D}}\right)=\frac{\sqrt{\overline{\zeta^{*}}-1}}{\overline{\zeta^{*}}-1-\frac{\overline{\zeta^{*}}\,\frac{\bar{\beta}}{2}\ell_{D}}{\overline{\zeta^{*}}-(1+\frac{\bar{\beta}}{2})}}.$
(5)

with $\bar{\beta}=\beta\ell/\eta$. The numerical solution of this equation is
shown as a function of the viscous resistance $\beta$ in the motility phase
diagram of Fig.2(D) (black curve). It separates the parameter region where the
homogeneous state is stable and no cell motion is expected ($\zeta<\zeta^{*}$)
and the region where the cell spontaneously polarises and moves
($\zeta>\zeta^{*}$). The asymptotic solutions of Eq.(5) for low and high
resistance are:

$\overline{\zeta^{*}}\xrightarrow[\beta\rightarrow
0]{}1+\left(\frac{\pi}{2}\ell_{D}\right)^{2}\qquad\overline{\zeta^{*}}\xrightarrow[\beta\rightarrow\infty]{}1+\left(a\pi\ell_{D}\right)^{2}$
(6)

with $1<a<3/2$. The first limit corresponds to the threshold obtained for an
isolated slab of twice the length of one cell edge. The second limit (red line
in Fig.2(D)) is in fact relevant for any $\beta$ value. Above this threshold,
cortex heterogeneities can develop and reach a steady state independently at
the two cell edges. The two profiles may be symmetrical with respect to the
cell center, in which case the cell does not move, or non-symmetrical, which
allows for cell motion with a velocity dependent upon $\beta$. The steady-
state is thus bistable and the cell velocity in the motile state decreases
continuously to zero as $\beta$ increases to infinity. This highly contractile
behaviour is discussed in the S.I.

Viscoelastic resistance. While physiological extracellular media often appear
rather viscous at long times, their short-time response is generally elastic.
One may expect the interplay between the intrinsic dynamics of the cortex and
the relaxation dynamics of the medium to give rise to interesting features. We
investigate this by considering that the channel is filled by a simple Maxwell
viscoelastic (VE) material of stiffess $k$ and relaxation time $\tau$, for
which the global force balance reads:

$\frac{d\Delta\gamma}{dt}+\frac{\Delta\gamma}{\tau}=k\,U.$ (7)

The LSA may be performed using the same eigenmodes as before, but the presence
of a time derivative in Eq.(7) results in qualitative differences, as the
perturbation growth rate ($\alpha_{n}$) may be complex. The lowest mode
($n=1$) is always the first to become unstable, and its growth rate satisfies
$\alpha_{1}=\bar{\zeta}-1-q^{2}$, where $q=\ell_{D}q_{1}$ satisfies

$\tan\left(\frac{q}{\ell_{D}}\right)=\frac{R(q)\,q\,\sqrt{\overline{\zeta}-q^{2}}}{R(q)\,q^{2}-\frac{\bar{k}\,\bar{\tau}\overline{\zeta}\ell_{D}}{2q^{2}}\sqrt{\overline{\zeta}-q^{2}}}$
(8)

where
$R(q)=(1+\frac{\bar{k}\,\bar{\tau}}{2}-\bar{\tau})+\bar{\tau}\,\overline{\zeta}\,\left(1-\frac{k}{2q^{2}}\right)-\bar{\tau}q^{2}$,
with $\bar{\tau}=k_{d}\tau$ and $\bar{k}=k\ell/(\eta k_{d})$. Solving both the
real and imaginary parts of Eq.(8), with $q_{1}$ a complex number and
requiring the condition for marginal stability $Re[\alpha_{1}]=0$ fully
determines the critical contractility $\zeta^{**}$ for instability in a VE
environment.

The numerical solutions of Eq.(8) is shown in the viscoelastic phase diagram
of Fig.3. For fast VE relaxation ($k_{d}\tau$ smaller than a threshold shown
in Fig.3B), $\alpha_{1}$ is real and the VE threshold contractility is
identical to the viscous threshold with a viscous resistance $\beta=k\tau$.
Physically, this is because the viso-elastic relaxation rate is faster than
the rate at which concentration heterogeneities develop, so that the short-
time elastic response has no influence. Mathematically, Eq.(5) is strictly
equivalent to Eq.(8) when its solution is real. Motile cells then display
steady motions equivalent to the viscous case. On the other hand, for slow VE
relaxation, the short time elastic force resisting cell motion drives the
reversal of polarity. The unstable growth rate is then complex and leads in
permanent oscillations of both cortical asymmetry and cell position, resulting
in a non-linear oscillatory motion.

Figure 3: Viscoelastic resistance. A. Critical VE contractility threshold
solution of Eq.(8) for different values of $k$. The kink in the curves
indicates the VE relaxation time beyond which eigenmodes are complex, leading
to oscillatory motion. B. Value of the relaxation time at which the short-time
elastic response influences motion (kink) as a function of $k$. C. Phase
diagram of cell motion. The critical contractility in viscous environment is
shown as a dashed line. Oscillatory motion is observed for long VE relaxation
time. In this regime, the transition to oscillatory motion is captured by the
linear stability analysis, but the boundary between steady and oscillatory
motion is inherently non-linear and requires numerical resolution of the full
set of spatio-temporal equations (Eqs.(1,2,7) - dots indicate numerical
results). Inset: examples of non-linear periodic variation of the cell
velocity, position and edge density difference in the oscillatory regime.
Parameters are $\rho^{*}=3/2$, $\ell_{D}^{2}=0.25$ and $\overline{k}=1$ for
the bottom panel.

Viscoelastic boundary. Cells migrating through physiological tissues encounter
mechanically heterogeneous environments. An idealised version of this can be
studied by analysing the behaviour of a motile cell encountering a
viscoelastic boundary. This situation involves the non-linear response of the
cell and must be studied numerically. Examples of the cell trajectory at the
boundary are shown in Fig.4(A) and the phase diagram for the long-time cell
behaviour in the ($\tau,\zeta$) space is shown in Fig.4(B). For fast VE
relaxation, the behaviour corresponds to that at the boundary between two
viscous media with different viscous resistance and can be understood from the
viscous phase diagram (Fig.2). The cell is able to penetrate if its
contractility is above the viscous threshold of the medium it encounters, or
gets trapped at the boundary otherwise. In the latter case, the dynamics
evolution toward the symmetric state is over-damped and no reversal of
symmetry and escape from the boundary is possible without noise. For slower VE
relaxation time, the short-time elastic nature of the boundary leads to the
existence of a bouncing state. The cell movement and polarisation may be
reversed if the elastic restoring force is slow to relax, in which case the
cell changes direction and bounces off the boundary back into the viscous
medium. This bouncing state becomes prominent as $\tau$ increases, making it
unlikely for a cell to penetrate a viscoelastic medium with a relaxation time
much slower than $1/k_{d}$.

Figure 4: Cell penetrating a viscoelastic medium. (left) Example for cell
trajectories at the boundary between a viscous fluid (pink: $\beta_{0}=0.5$)
and a visco-elastic fluid (green: $k=1$), showing the three different
behaviours observed for different VE relaxation times: penetration, trapping
and bouncing. (right) Phase diagram for the three types of behaviour varying
the cell contractility and the VE relaxation time.

Discussion. The mechanical properties of the environment plays a crucial role
in regulating the ability of cells to invade new tissues H. Yu and and Weaver
(2011); *wolf:2011; *gu:2014. While cellular mechano-sensing involves complex
molecular pathways Discher  _et al._ (2005), pure mechanical response does
have a direct effect on the cell polarisation. We use a simple 1D model of a
very long cell in a confined geometry to study how the mechanical resistance
of the environment influences contractility-driven cell motility. A positive
feedback between local density variations and flows in the cellular cortex
leads to spontaneous symmetry breaking and cell polarisation. Friction between
the cortical flow and the confining walls generates a propulsion force that
allows cell motion. We restrict ourselves to the limit of strong friction, and
use linear stability analysis to predict a variety of cell behaviours
depending on the mechanical resistance of the external medium, including
spontaneous motion, bistability where steady motion needs to be activated, and
oscillatory behaviour. Numerical simulations support these predictions, and
allow us to compute the non-linear cell behaviour at the boundary between two
elastically distinct media. These result show that a cell may not easily
penetrate a viscoelastic medium with a relaxation time slower than the typical
turnover time of the cellular cortex.

Our results are expressed in terms of dimensionless parameters: primarily the
cell contractility $\bar{\zeta}\propto\zeta\rho_{0}/(\eta k_{d})$, the
normalised friction $\bar{\beta}\equiv\beta l/\eta$ and relaxation time
$\bar{\tau}\equiv k_{d}\tau$ of the extracellular medium. Using typical values
of the parameters (cortical tension: $\zeta\rho_{0}=10^{-4}$ N/m, cortex
viscosity: $\eta=10^{-3}$ Pa.m.s and recycling rate: $k_{d}\simeq 0.01-0.1$
s-1 Salbreux  _et al._ (2012); Fritzsche  _et al._ (2014); Saha  _et al._
(2016)) one obtains a value of the dimensionless contractility
$\overline{\zeta}\simeq 1-10$, well within the range of spontaneous symmetry
breaking. The effective viscoelastic parameters $\beta$ and $k$ are related to
the viscosity and Young’s modulus of the extracellular medium in complex ways,
that depend on the boundary conditions at the channel walls and may involve
the channel radius and length (see S.I.) Bergert  _et al._ (2015). With
$\beta\sim$ viscosity $\simeq 10^{3}-10^{4}$ Pa.s Stirbat  _et al._ (2013) and
Young’s modulus for soft tissues $\simeq 0.1-1$ kPa Gu  _et al._ (2014), one
finds $\bar{\beta}\simeq 1-10$ and $\bar{\tau}\simeq 0.1-10$. Thus the
expected parameter values are within the regions of interest of the phase
diagrams (Figs.2,3,4) establishing the physiological relevance of our results.

###### Acknowledgements.

We acknowledge funding from: INSERM ITMO cancer 20CR110-00

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