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\title{On the Randomness of Modular Inverse Mappings}
\author{Jonathan Bober}

%\newtheorem{twdef}{definition}[section]
%\newtheorem{involution}{definition}[section]
%\newtheorem{signed}{definition}[section]
%\newtheorem{fixedpoint}{definition}[section]

\newtheorem{defi}{Definition}
\newtheorem{conj}{Conjecture}

\begin{document}
\maketitle

\begin{abstract}
In this paper the randomness of the inverse mapping of a finite field is investigated
in the context of random permutations.
The function $x \rightarrow x^{-1} \mod p$, $p$ a prime, is studied as a permutation
of the integers from $1$ to $p - 1$.  Specifically, the function is its own inverse
and is signed, so it forms a signed involution.
Baik, Deift, and Johansson [1] and Baik and Rains [3] have given results concerning
the distribution length of the longest increasing subsequence of a random permutation
and of certain subsets of random permutations (such as a signed involution). These distributions
are either the same as or are expressed in terms of the Tracy-Widom distributions for the largest
eigenvalue of a random matrix from certain matrix ensembles. Calculations suggest that
the same is true of the permutations from the inverse mapping, and, specifically, 
it is conjectured that, looked at in the right way, the distribution of the length 
of the longest increasing subsequence of the inverse mapping is asymptotically the same
as the case of a random fixed point free signed involution.
\end{abstract}


\section{Introduction}
Arithmetic of integers modulo a prime forms the finite field $\mathbb{Z}/p\mathbb{Z}$.  In
this field the question arises of whether certain arithmetic operations
exhibit properties of "randomness". 


Evidence already exists for some "randomness" of this function.
The Hooley $R^*$ conjecture surmises that the partial Kloosterman sum
\[
\left|\sum_{A \le x \le A + N} e\left(\frac{cx^{-1}}{p}\right)\right|
\ll N^{1/2}p^\epsilon, \epsilon > 0
\]
where $e(x) = e^{2\pi ix}$ and $i = \sqrt{-1}$.

This is known true for $N > \sqrt{p}$, by Weil's bound on Kloosterman
sums [5]. Of course, the full sum
\[
\left|Kl\left(0,1,p)\right)\right| = 
\left|\sum_{1 \le x < p} e\left(\frac{x^{-1}}{p}\right)\right|
 = \left|\sum_{1 \le x < p} e\left(\frac{x}{p}\right)\right|
 = 1,
 \]

because it is the sum of the p pth-roots of 1, excluding the root 1.  Intuitively, if $x \rightarrow
x^{-1}$ is a random mapping, then this sum should take on small values
over a large enough interval, as is partially known.  Though it is not the focus of
this paper, it would be instructive and interesting to test Hooley's
conjecture to examine whether it may hold up.

Another way in which this "randomness" can be investigated is through
permutations.  The mapping $x \rightarrow x^{-1}$ is one-to-one on the set
$\mathbb{Z}/p\mathbb{Z} - \{0\}$ and the ordering $1^{-1}, 2^{-1}, ..., (p-1)^{-1}$
defines a permutation of the set ${1...(p-1)}$. This permutation is an almost
fixed point free signed involution.

\begin{defi}[Involution]
A permutation $\pi(x)$ is an involution if it is the product of
disjoint 2-cycles, or equivalently, if $\pi(\pi(x)) = x$.
\end{defi}
\begin{defi}[Signed permutation]
A permutation $\pi_n(x)$ of $n-1$ elements is signed if $P(-x) = -P(x)$, 
where the arithmetic inverse is defined as normal in the ring $\mathbb{Z}/n\mathbb{Z}$, 
where $x + (-x) = 0\mod n$.
\end{defi}
\begin{defi}[Fixed and negated points]
A fixed point in a permutation $\pi(x) $is an $x$ for which $\pi(x) = x$.  
A negated point is an $x$ for which $\pi(x) = -x$. 
\end{defi}

The permutation on the set $\{1, 2, 3, 4, ..., p-1\}$ defined by $x \rightarrow
x^{-1} \mod p$, $p$ prime, is a signed involution because
$(x^{-1})^{-1} = x$ and $(-x)^{-1} = -x^{-1}$.  It is almost fixed point free
because the equation $x^2 = 1$ can have at most 2 solutions in $\mathbb{Z}/p\mathbb{Z}$,
and it has the two solutions $\pm 1$.  The two fixed points will have no
effect on the asymptotic statistics of these permutations.

\section{Random permutations and the Tracy-Widom distributions}

Let $\pi$ be a random permutation of $n$ elements. We say that 
$\pi(i_1), \pi(i_2), ... \pi(i_k)$ is an increasing subsequence of $\pi$
with length $k$ if $\pi(i_1) < \pi(i_2) < ... < \pi(i_k)$ and $i_1 < 
i_2 < ... < i_k$. For example, if $\pi$ maps $1\; 2\; 3\; 4\; 5\; 6\; 7$ to 
$3\; 5\; 1\; 6\; 7\; 2\; 4$, than the longest increasing subsequence of $\pi$ is
$3\; 5\; 6\; 7$, with length $4$.

The asymptotics of the distribution of the length of the longest
increasing subsequence for a random permutation have been much computed
and studied and have been fully solved by Baik, Deift and Johanson in [1].
Also, Baik and Rains have solved the asymptotics for various types of 
permutations [3].  Some of these results are restated below.

\subsection{Tracy-Widom distribution functions}
Let $u(x)$ be the solution to the Painleve II equation \[u_{xx} = 2u^3 + xu,\]
with the condition that $u(x)\rightarrow -Ai(x)$ as $x \rightarrow +\infty$,
where $Ai(x)$ is the Airy function.  The existence and uniqueness of the
function is established in [4].  Define \[v(x) := \int_\infty^x (u(s))^2 ds \].

\begin{defi}[Tracy-Widom Distribution] Define 
\[
\begin{array}{lcl}
F(x) &:=& exp\left(\frac{1}{2} \right) \int_x^\infty v(s)ds, \\
E(x) &:=& exp\left(\frac{1}{2} \right) \int_x^\infty u(s)ds, 
\end{array}
\]
and then set
\[
\begin{array}{lcl}
F_2(x) &:=& F(x)^2, \\
F_1(x) &:=& F(x)E(x), \\
F_4(x) &:=& F(x)[E(x)^{-1} + E(x)]/2
\end{array}
\]
\end{defi}

\begin{figure}[h]
\caption{Tracy-Widom distribution functions}
\begin{center}
\includegraphics[scale = 1]{twgraphs.eps}
\end{center}
\end{figure}

It is interesting to note that these distributions are not centered around
0.  Each of them has negative mean.  In both the random matrix model and 
random permutation model, these distributions arise in the exploration 
of deviation around an asymptotic mean.  In both cases, however, the asymptotic
mean used for the distribution is just a first order approximation, 
and the true asymptotic mean is fully solved.  Once the true mean is obtained,
it seems that it would make more sense to center the distributions around this mean,
but this was not done, and thus these distributions are off center.

Tracy and Widom have shown that $F_1(x), F_2(x)$ and $F_4(x)$ are the
asymptotic distribution functions for largest eigenvalues of the Gaussian matrix
ensembles for $\beta = 1, 2, 4$ (suitably rescaled.) Moreover, in [1] Baik, 
Deift and Johanson showed that for a random permutation, the distribution of 
\[ \frac{L_n - 2\sqrt{n}}{n^{\frac{1}{6}}} \] also converges to $F_2(x)$.  In [3],
Baik and Rains give results for various types of random permutations, all of which
are also related to the three Tracy-Widom distribution functions.  

\subsection{Random Permutations}
Let $S_n$ be the set of all permutations of $n$ elements, and define $fp(P)$ the number
of fixed points of the permutation $P(x)$ and $np(P)$ the number of negated
points. Define:
\[
\begin{array}{lcl}
S^\symmU_n &=& S_n \\
S^\symmO_{n,m} &=& \{P(x) \in S_{2n + m} : P(P(x)) = x, fp(P) = m \}\\
S^\symmS_{n,m} &=& \{P(x) \in S_{2n + m} : P(x) = -P^{-1}(x), np(P) = m\}\\
S^\symmUU_n &=& \{P(x) \in S_{2n} : P(-x) = -P(x) \}\\
S^\symmu_{n, m_+, m_-} &=& \{P(x) \in S_{4n + 2m_+ + 2m_-} : P(-x) = -P(x), P(P(x)) = x,\\
	& & fp(P) =  2m_+, np(P) = 2m_-\}
\end{array}
\]

and the random variables 
\[
L^\symmU_n, L^\symmO_{n,m}, L^\symmS_{n,m}, L^\symmUU_n, L^\symmu_{n, m_+, m_-}
\]
as the length of the longest increasing subsequence of a random
permutation chosen uniformly from the corresponding set above.

In language closer to english, $S^\symmO_{n,m}$ is the set of involutions of $2n+m$ 
elements with $m$ fixed points, $S^\symmUU_n$ is the set of 
signed permutations of $2n$ elements, and $S^\symmu_{n, m_+, m_-}$ is the 
set of signed involutions of $4n+2m_+ + 2m_-$ elemtents with $2m_+$ fixed points and $2m_-$ negated points.
If the notation seems a bit odd, it is because it is a bit odd, but it is
taken from the context of [3], in which it makes more sense.

For each of the random variables, $L^\symmg$ with $m, n \le \sqrt{N}$ we have 
\[
\lim_{N \to \infty}\frac{E(L^\symmg)}{\sqrt{N}} = 2
\]

and for each we have an asymptotic distribution, with fixed $x, \alpha, \beta$
\[
\begin{array}{lcl}
\lim_{n\to\infty}\Pr\left(\frac{L^\symmU_n - 2\sqrt{N}}{N^{1/6}} \le x\right) &=& F_2(x), \\
\lim_{n\to\infty}\Pr\left(\frac{L^\symmUU_{n,m} - 2\sqrt{N}}{2^{2/3}N^{1/6}} \le x\right) &=& F_2(x)^2, \\
\lim_{n\to\infty}\Pr\left(\frac{L^\symmO_{n,[\sqrt{2n}\alpha]} - 2\sqrt{N}}{N^{1/6}} \le x\right) &=& 
\left\{ \begin{array}{rll}
	F_4(x) & \mbox{if} & 0 \le \alpha < 1; \\
	F_1(x) & \mbox{if} & \alpha = 1; \\
	0 & \mbox{if} & \alpha > 1.
	\end{array}\right.
	\\
\lim_{n\to\infty}\Pr\left(\frac{L^\symmS_{n, [\sqrt{2n}\beta]} - 2\sqrt{N}}{N^{1/6}} \le x\right) &=& F_1(x), \beta \ge 0,\\
\lim_{n\to\infty}\Pr\left(\frac{L^\symmu_{n, [\sqrt{n}\alpha],[\sqrt{n}\beta]} - 2\sqrt{N}}{2^{2/3}N^{1/6}} \le x\right) &=& 
\left\{ \begin{array}{rll}
	F_2(x) & \mbox{if} & 0 \le \alpha < 1, \beta \ge 0; \\
	F_1(x)^2 & \mbox{if} & \alpha = 1, \beta \ge 0; \\
	0 & \mbox{if} & \alpha > 1, \beta \ge 0. \\
	\end{array}\right.
	\\
\end{array}
\]
\subsection{Relation to Modular Inverse Mapping}
Because these distributions arise in the context of random permutations,
we investigate whether they might arise in the permutation defined by 
$x \rightarrow x^{-1}$.  Specifically, because $x \rightarrow x^{-1}$
defines a fixed point free involution, we investigate whether the
asymptotic distribution of the longest increasing subsequence of the
involution is the function $F_2(x)$, and whether the distribution scales
the same way as for a fixed point free involution.

Let $L_p$ be the length of the longest increasing subsequence
of the permutation $\pi_p(x)$ defined by $x \rightarrow x^{-1} \mod p$, and define
\[
S_{N,K} = \{ L_p(x) : N - K(\log(N))^{1+\epsilon} \le p \le N + K(log(N))^{1+\epsilon} \}
\]
for $\epsilon = 1$ or $2$, for example.  Note that $N$ need not be 
prime in this definition.

By the Prime Number Theorem, this set has approximately $(log(N))^\epsilon$ elements, and thus
as $N \rightarrow \infty, |S_{N,K}| \rightarrow \infty$.  Additionally,
we have, for fixed $K$
\[
\lim_{N \rightarrow \infty} \frac{K(log(N))^{1+\epsilon}}{\sqrt{N}} = 0
\]
so that, if the mean is asymptotic to $2\sqrt{N}$ then, in this set, $L_p$
will not change too much, and also, if the permutation behaves randomly 
will see that the mean of each set, $\mu(N)$ (just one function for the mean
is defined, as it will change very little for reasonable intervals, and not at
all in the limit as $N \rightarrow \infty$) will be asymptotic to $2\sqrt N$

Additionally, we look at the standard deviation of each set around an $N$.  Define 
\[
\sigma(N) = \sqrt{\frac{1}{|S_{N,K}|}\sum_{L(p) \in S_{N,K}}(L_p - \mu(p))^2}
\]

and now we hope to see that $\sigma(N)$ is aymptotic to $2^{2/3}N^{1/6}$.

Also, various comparisons are made between computed distributions and asymptotic distributions.

\section{Computations and Results}
\subsection{Mean and Deviation}
For $\epsilon = 1, K = 1000$, we have the following for $\mu(N)$, from 1000000 to 4500000

\begin{figure}[hbtf]
\caption{$2N^{1/2}$ and $\mu(N)$}
\begin{center}
\includegraphics[scale = 1]{meannew3.ps}
\end{center}
\end{figure}

\FloatBarrier


$\mu(N)$ remains just below $2\sqrt N$ over the entire interval, suggesting
that it is asymptotic to $2\sqrt N$, and it indeed appears to be $O(N^{1/6})$ away 
from $2\sqrt N$, as is shown by looking at the deviation compared to $2^{2/3}N^{1/6}$.
\begin{figure}[hbtf]\caption{Standard Deviation}
\begin{center}
\includegraphics[scale = 1]{stddev3.ps}
\end{center}
\end{figure}

\FloatBarrier


Both the mean and the standard deviation seem to be well within what would
be expected for random involutions.

\subsection{Comparisons to Random Permutations}
Now for an even larger interval we examine the random variable we examine
the set $\hat{S}_N = \{\chi(p) = L_p - \mu(p)\}$ for $p$ in an interval about $N$.
After obtaining the standard deviation of this set, we examine the function
\[
Q_N(x) = \Pr\left(\frac{\chi(p)}{\sigma(S_N)} \le x\right)
\]

and differentiate to obtain the normalized probability density function $\frac{dQ_N}{dx}$
with mean $0$ and standard deviation $1$.
\begin{center}
\includegraphics[scale = 1]{normalized.ps}
\end{center}

Compared to the normalized derivative of the Tracy-Widom distribution function $F_2(x)$,
a possible relation is seen.
\begin{center}
\includegraphics[scale = 1]{twandnormal.ps}
\end{center}

Of course, this visual itself is not too strong of evidence that the two functions
are the same asymptotically.  In fact, when normalized to have mean $0$ and standard
deviation $1$, all three Tracy-Widom distribution functions are virtually inditinguishable
to the eye.  It is the normalization factors above that provide stronger evidence of
correlation, and thus it is instructive to look at the function 
\[
\hat{Q}_N(x) = \Pr\left(\frac{L_p - 2\sqrt{p}}{2^{2/3}p^{1/6}} \le x\right), L_p \in S_{N,K}
\]
in an interval around $N$ as compared to $F_2(x)$ as defined originally.

\begin{figure}
\caption{$f_2(x)$ and $d\hat{Q}_N(x)/dx$(dotted)}
\begin{center}
\includegraphics[scale = 1]{unnormal.ps}
\end{center}
\end{figure}

\FloatBarrier

Figure 4 shows a graph for $N = 3000000$ and a rather large interval.  The graph of $f_2(x)$ is a bit to the right of the
computed graph.  This result shows what was shown in part in both the mean and
standard deviation plots above, and a bit more.  First, the mean is significantly 
less than $2\sqrt{N}$, though asymptotically similar, and,
moreso, is less than the mean of the Tracy-Widom function.  Second, the standard
deviation is significantly less than $2^{2/3}p^{1/6}$, though again asymptotically
similar.  The fully normalized graph is a more natural graph to make, as the mean and the
deviation about that mean are separate issues, but there is still something
instructive in the graphs that do not have mean $0$, and they allow more in the
way of visual comparison.

We have the following statistics of the probablity density function \\
$d\hat{Q}_N(x)/dx$
for intervals $[a,b]$:

\[\begin{array}{ccccc}
  \mbox{a} & \mbox{b} & \mbox{Mean} & \mbox{Variance} & \mbox{Std Dev}\\ \hline
     0 & 1000000 & -2.02129 & .860861 & .927826 \\
     1000000 & 2000000 & -2.04177 & .871930 & .933772 \\
     2000000 & 3000000 & -2.03890 & .884904 & .940694 \\
     3000000 & 4000000 & -2.04323 & .887816 & .942240 \\
     4000000 & 5000000 & -2.04980 & .873988 & .934873 \\ \hline
     \mbox{All computed values} && -2.03687 & .876913  & .936436\\ \hline
     \mbox{Possible aysmptotic expectation} && -1.77109 & .813189 & .901770 \\ \hline
  \end{array}
\]

\section{Conclusions and the Future}
This data is by no means fully conclusive, and without theoretical conclusions,
more computed data is necessary to obtain a better understanding of this problem.
However, it seems safe to make the following conjectures:
\begin{conj}
\[
\lim_{N \rightarrow \infty} \frac{\mu(N)}{\sqrt{N}} = 2
\]
\end{conj}

and (slightly less safe)
\begin{conj}
\[
\lim_{N \rightarrow \infty} \frac{\sigma(N)}{N^{1/6}} = 2^{2/3}
\]
\end{conj}

which are the same as the values of the limits for a random fixed point free signed
involution.

Also, it seems plausible that
\[
\lim_{N \rightarrow \infty} \hat{Q}_N(x) = F_2(x)
\]
but it seems that current data is not nearly enough to conjucture on
whether this statement is true or false.  Current data suggests that
this is not true, but that a rescaling by simply subtracting a value
other than $2\sqrt{N}$ may make the statement true.  In truth, I am 
not willing to make a guess either way, but will guess that these
permutations are indeed related to random permutations somehow.

In order to make a better comparison between inverse mapping permutations
and random permutations, it might be good to look at the length of the
longest increasing subsequence after the first is removed, and then remove
that and continue.  This is equivalent to building what are called Young
tableaux, which for random permutations are also related to the eigenvalues
for random matrices [2].

Also, while asymptotics are known for fixed point free involutions, there
is not much numerical data to compare to.  It is possible, though probably not likely,
that convergence to
the Tracy-Widom distribution requires very long permutations and numerical data 
may show that in the range that has been computed for the inverse mappings, they
actually do behave exactly like a random involution.

Other tests can also shine light on the randomness of this mapping.  Hooley's
conjecture should be numerically tested, and also, the distribution of 
$x \rightarrow x^{-1}$, with fixed $p$ can be investigated for a range of $x$ 
less than $p$ to test for randomness in distribution and for correlation in spacings.
Moreover, one can look at the distribution of the inverse of a fixed $x$ for 
varying primes.  Very simple tests on both of these that I did seemed to 
suggest randomness, but the tests were extremely simple, and while they do not
debunk any theories of randomness, neither do they provide evidence.

If anyone continues any of these calculations, please contact me \\
(jwb235@nyu.edu 
or bober@acm.cs.nyu.edu,) as it is quite possible that I will have more 
data available in the future.

\section{Computational methods}
The algorithms used were mostly rather standard and simple, and all
code for computation was written in C++.  The NTL (see http://www.shoup.net) was used 
for its library functions and occasionally for multiprecision arithmetic.
Full source code should hopefully be available from whatever website you
retrieved this paper from (if you did so.)  It will also be available
at http://www.math.nyu.edu/Courses/V63.0393/ and \\ http://acm.cs.nyu.edu/~bober/math/
and will hopefully be well documented and commented, so that it may be
used and modified by anyone who wishes to do so.

Note: The algorithm used to computer the longest increasing subsequence
is not bad but and can be improved upon with a more complicated 
method.

In the next few weeks I hope to program a better algorithm that will be 
faster and will also compute all increasing subsequences.

\section{Acknowledgements}
Looking at the longest increasing subsequence of these permutations was first
suggested to Steven Miller and Peter Sarnak by Jim Propp, who suggested that
such conjectures as made above could probably be made.

Also, much thanks goes to Professor Sarnak, Steve Miller and Alex Barnett, for
running a class with as many instructors as students, and to Brad Weir and Tim
Novikoff, without whom the class probably would not have existed in such a form. 
Discussions with all were extremely useful. And also thanks to Craig Tracy for 
providing Mathematica code for the Tracy-Widom distribution functions.

\section{References}
$\ \ \ \ $[1] J. Baik, P. Deift, and K. Johansson. On the distribution of the length of the
longest increasing subsequence of random permutations. \textit{J. Amer. Math. Soc.}, 12:1119-1178,1999
and e-print math.CO/9810105 \\

[2]J. Baik, P. Deift, and K. Johansson. On the distribution of the length of the
second row of a Young diagram under Plancherel measure. e-print math.CO/9901118 \\

[3]J.Baik and E. M. Rains. Symmetrized random permutations. e-print math.CO/9910019 \\

[4]S. P. Hastings and J. B. McLeod, A boundary value problem associated with the second
Painleve transcendent and the Korteweg de Vries equation, \textit{Arch. Rational Mech. Anal.} 
73,31-51, (1980) \\

[5]D.R. Heath-Brown. "Arithmetic applications of Kloosterman sums", September $2000$\\

\end{document}
