\section{Introduction}

The study of supremum of Gaussian processes is a major area of study in probability and functional analysis as epitomized by the celebrated {\it majorizing measures} theorem of Fernique and Talagrand (see \cite{LedouxT}, \cite{Talagrand05} and references therein). There is by now a rich body of work on obtaining tight estimates and characterizations of the supremum of Gaussian processes with several applications in analysis \cite{Talagrand05}, convex geometry \cite{Pisier99} and more. Recently, in a striking result, Ding, Lee and Peres \cite{DingLP11} used the theory to resolve the Winkler-Zuckerman {\it blanket time} conjectures \cite{WinklerZ96}, indicating the usefulness of Gaussian processes even for the study of combinatorial problems over discrete domains.

Ding, Lee and Peres \cite{DingLP11} used the powerful {\it Dynkin isomorphism theory} and majorizing measures theory to establish a structural connection between the cover time (and blanket time) of a graph $G$ and the supremum of a Gaussian process associated with the Gaussian Free Field on $G$. They then use this connection to resolve the Winkler-Zuckerman blanket time conjectures and to obtain the first deterministic polynomial time constant factor approximation algorithm for computing the cover time of graphs. This latter result resolves an old open question of Aldous and Fill (1994).

Besides showing the relevance of the study of Gaussian processes to discrete combinatorial questions, the work of Ding, Lee and Peres gives evidence that studying Gaussian processes could even be an important algorithmic tool; an aspect seldom investigated in the rich literature on Gaussian processes in probability and functional analysis. Here we address the corresponding computational question directly, which given the importance of Gaussian processes in probability, could be of use elsewhere. In this context, the following question was asked by Lee \cite{leepost} and Ding \cite{Ding11}.

\begin{question}
For every $\epsilon > 0$, is there a deterministic polynomial time algorithm that given a set of vectors $v_1,\ldots,v_m \in \rd$, computes a $(1 + \epsilon)$-factor approximation to $\ex_{X \lfta \N^d}[\sup_i |\inp{v_i,X}|]$\footnote{Throughout, $\N$ denotes the univariate Gaussian distribution with mean $0$ and variance $1$ and for a distribution $\calD$, $X \lfta \calD$ denotes a random variable with distribution $\calD$. By a $\alpha$-factor approximation to a quantity $X$ we mean a number $p$ such that $p \leq X \leq \alpha p$.}.
\end{question}

We remark that Lee \cite{leepost} and \cite{Ding11} actually ask for an approximation to $\ex_{X \lfta \N^d}[\sup_i \inp{v_i, X}]$ (and not the supremum of the absolute value). However, this formulation results in a somewhat artificial asymmetry and for most interesting cases these two are essentially equivalent: if $\ex_{X \lfta \N^d}[\sup_i \inp{v_i, X}] = \omega(\max_i \nmt{v_i})$, then $\ex_{X \lfta \N^d}[\sup_I |\inp{v_i, X}|] = (1+o(1)) \ex_{X \lfta \N^d}[\sup_i \inp{v_i, X}]$\footnote{This follows from standard concentration bounds for supremum of Gaussian processes; we do not elaborate on it here as we ignore this issue.}. We shall overlook this distinction from now on.

There is a simple randomized algorithm for the problem: sample a few Gaussian vectors and output the median supremum value for the sampled vectors. This however, requires $O(d\log d/\epsilon^2)$ random bits. Using Talagrand's majorizing measures theorem, Ding, Lee and Peres give a deterministic polynomial time $O(1)$-factor approximation algorithm for the problem. This approach is inherently limited to not yield a PTAS as the majorizing measures characterization is bound to lose a universal constant factor. Here we give a PTAS for the problem thus resolving the above question.
\begin{theorem}\label{th:main}
  For every $\epsilon > 0$, there is a deterministic algorithm that given a set of vectors $v_1,\ldots,v_m \in \rd$, computes a $(1 + \epsilon)$-factor approximation to $\ex_{x \lfta \N^d}[\sup_i |\inp{v_i,x}|]$ in time $\poly(d) \cdot m^{\tilde{O}(1/\epsilon^2)}$.
\end{theorem}

Our approach is comparatively simpler than the work of Ding, Lee and Peres, using some classical {\it comparison inequalities} in convex geometry

We explain our result on estimating semi-norms with respect to Gaussian measures mentioned in the abstract in \sref{sec:linest}.

We next discuss some applications of our result to computing cover times of graphs as implied by the works of Ding, Lee and Peres \cite{DingLP11} and Ding \cite{Ding11}.
\subsection{Application to Computing Cover Times of Graphs}
The study of random walks on graphs is an important area of research in probability, algorithm design, statistical physics and more. As this is not the main topic of our work, we avoid giving formal definitions and refer the readers to \cite{AldousF}, \cite{Lovasz93} for background information.

Given a graph $G$ on $n$-vertices, the cover time, $\tau_{cov}(G)$, of $G$ is defined as the expected time a random walk on $G$ takes to visit all the vertices in $G$ when starting from the worst possible vertex in $G$. Cover time is a fundamental parameter of graphs and is extensively studied. Algorithmically, there is a simple randomized algorithm for approximating the cover time - simulate a few trials of the random walk on $G$ for $\poly(n)$ steps and output the median cover time. However, without randomness the problem becomes significantly harder. This was one of the motivations of the work of Ding, Lee and Peres \cite{DingLP11} who gave the first deterministic constant factor approximation algorithm for the problem, improving on an earlier work of Kahn, Kim, Lov\'asz and Vu \cite{KahnKLV00} who obtained a deterministic $O((\log \log n)^2)$-factor approximation algorithm. For the simpler case of trees, Feige and Zeitouni \cite{FeigeZ09} gave a FPTAS.

Ding, Lee and Peres also conjectured that the cover time of a graph $G$ (satisfying a certain reasonable technical condition) is asymptotically equivalent to the supremum of an explicitly defined Gaussian process---the Gaussian Free Field on $G$. However, this conjecture though quite interesting on its own, is not enough to give a PTAS for cover time; one still needs a PTAS for computing the supremum of the relevant Gaussian process. Our main result provides this missing piece, thus removing one of the obstacles in their posited strategy to obtain a PTAS for computing the cover time of graphs. Recently, Ding \cite{Ding11} showed the main conjecture of Ding, Lee and Peres to be true for bounded-degree graphs and trees. Thus, combining his result (see Theorem 1.1 in \cite{Ding11}) with \tref{th:main} we get a PTAS for computing cover time on bounded degree graphs with $\tau_{hit}(G) = o(\tau_{cov}(G))$\footnote{The hitting time $\tau_{hit}(G)$ is defined as the maximum over all pairs of vertices $u,v \in G$ of the expected time for a random walk starting at $u$ to reach $v$. See the discussion in \cite{Ding11} for why this is a reasonable condition.}. As mentioned earlier, previously, such algorithms were only known for trees \cite{FeigeZ09}.
\ignore{
\begin{theorem}
  For every $\epsilon > 0$ and $\Delta > 0$ there exists a constant $C_{\Delta,\epsilon}$ such that the following holds. For every graph $G$ with maximum degree at most $\Delta$ and $\tau_{hit}(G) < C_{\Delta,\epsilon}(\tau_{cov}(G))$, there exists a deterministic $n^{O_{\Delta,\epsilon}(1)}$-time algorithm to compute a $(1+\epsilon)$-factor approximation to $\tau_{cov}(G)$.
\end{theorem}}
\section{Outline of Algorithm}
The high level idea of our PTAS is as follows. Fix the set of vectors $V = \{v_1,\ldots,v_m\} \subseteq \R^d$ and $\epsilon > 0$. Without loss of generality suppose that $\max_{v \in V} \nmt{v} = 1$. We first reduce the dimension of $V$ by projecting $V$ onto a space of dimension of $O((\log m)/\epsilon^2)$ \'a la the classical Johnson-Lindenstrauss lemma (JLL). We then give an algorithm that runs in time polynomial in the number of vectors but exponential in the underlying dimension. Our analysis relies on two elegant comparison inequalities in convex geometry---Slepian's lemma  \cite{Slepian62} for the first step and Kanter's lemma \cite{Kanter77} for the second step. We discuss these modular steps below.
\subsection{Dimension Reduction} We project the set of vectors $V\subseteq \R^d$ to $\R^k$ for $k = O((\log m)/\epsilon^2)$ to preserve all pairwise (Euclidean) distances within a $(1+\epsilon)$-factor as in the Johnson-Lindenstrauss lemma (JLL). We then show that the expected supremum of the {\it projected} Gaussian process is within a $(1 + \epsilon)$ factor of the original value. The intuition is that, the supremum of a Gaussian process, though a global property, can be controlled by pairwise correlations between the variables. To quantify this, we use Slepian's lemma, that helps us relate the supremum of two Gaussian processes by comparing pairwise correlations. Finally, observe that using known derandomizations of JLL, the dimension reduction can be done deterministically in time $\poly(d,m,1/\epsilon)$ \cite{DJLL}.

Thus, to obtain a PTAS it would be enough to have a deterministic algorithm to approximate the supremum of a Gaussian process in time exponential in the dimension $k = O((\log m)/\epsilon^2)$. Unfortunately, a naive argument by discretizing the Gaussian measure in $\R^k$ leads to a run-time of at least $k^{O(k)}$; which gives a $m^{O((\log \log m)/\epsilon^2)}$ algorithm. This question was recently addressed by Dadush and Vempala \cite{DadushV12}, who needed a similar sub-routine for their work on computing  {\it M-Ellipsoids} of convex sets and give a deterministic algorithm with a run-time of $(\log k)^{O(k)}$. Combining their algorithm with the dimension reduction step gives a deterministic $m^{O((\log \log \log m)/\epsilon^2)}$ time algorithm for approximating the supremum. We next get rid of this $\omega(1)$ dependence in the exponent.
\subsection{Oblivious Linear Estimators for Semi-Norms}\label{sec:linest}
 We in fact, solve a more general problem by constructing an optimal {\it linear estimator} for semi-norms in Gaussian space.
Let $\phi:\R^k \rgta \R_+$ be a semi-norm, i.e., $\phi$ is homogeneous and satisfies triangle inequality. For normalization purposes, we assume that $1 \leq \ex_{x \lfta \N^k}[\phi(x)]$ and that the Lipschitz constant of $\phi$ is at most $k^{O(1)}$. Note that the supremum function $\phi_V(x) = \sup_{v \in V}|\inp{v,x}|$ satisfies these conditions. Our goal will be to compute a $(1+\epsilon)$-factor approximation to $\ex_{x \lfta \N^k}[\phi(x)]$ in time $2^{O_\epsilon(k)}$.
\ignore{
\begin{theorem}\label{th:epsnet}
  For every $\epsilon > 0$, there exists a distribution $\calD$ on $\R^k$ which can be sampled using $O(k\log(1/\epsilon))$ bits in time $\poly(k,1/\epsilon)$ and space $O(\log k + \log(1/\epsilon))$ such that for every semi-norm $\phi:\R^k \rgta \R_+$,
\[(1-\epsilon) \ex_{x \lfta \calD}[\phi(x)] \leq \ex_{x \lfta \N^k}[\phi(x)] \leq (1+\epsilon)\ex_{x \lfta \calD}[\phi(x)].\]
In particular, there exists a deterministic $(1/\epsilon)^{O(k)}$-time algorithm for computing a $(1+\epsilon)$-factor approximation to $\ex_{X \lfta \N^k}[\phi(X)]$ using only oracle access to $\phi$.
\end{theorem}}
\begin{theorem}\label{th:epsnetintro}
  For every $\epsilon > 0$, there exists a deterministic algorithm running in time $(1/\epsilon)^{O(k)}$ and space $\poly(k,1/\epsilon)$ that computes a $(1+\epsilon)$-factor approximation to $\ex_{X \lfta \N^k}[\phi(X)]$ using only oracle access to $\phi$.
\end{theorem}
Our algorithm has the additional property of being an {\it oblivious linear estimator}: the set of query points does not depend on $\phi$ and the output is a positive weighted sum of the evaluations of $\phi$ on the query points. Further, the construction is essentially optimal as any such oblivious estimator needs to make at least $(1/\epsilon)^{\Omega(k)}$ queries (see \sref{sec:appendix}). In comparison, the previous best bound of Dadush and Vempala \cite{DadushV12} needed $(\log k)^{O(k)}$ queries. We also remark that the query points of our algorithm are essentially the same as that of Dadush and Vempala, however our analysis is quite different and leads to better parameters.

As in the analysis of the dimension reduction step, our analysis of the oblivious estimator relies on a comparison inequality---Kantor's lemma---that allows us to ``lift'' a simple estimator for the univariate case to the multi-dimensional case.

We first construct a symmetric distribution $\mu$ on $\R$ that has a simple {\it piecewise flat graph} and {\it sandwiches} the one-dimensional Gaussian distribution in the following sense. Let $\nu$ be a ``shrinking'' of $\mu$ defined to be the probability density function (pdf) of $(1-\epsilon)x$ for $x \lfta \mu$. Then, for every symmetric interval $I \subseteq \R$, $\mu(I) \leq \N(I) \leq \nu(I)$.

Kantor's lemma \cite{Kanter77} says that for pdf's $\mu,\nu$ as above that are in addition {\it unimodal}, the above relation carries over to the product distributions $\mu^k, \nu^k$: for every symmetric convex set $K \subseteq \R^k$, $\mu^k(K) \leq \N^k(K) \leq \nu^k(K)$. This last inequality immediately implies that semi-norms  cannot {\it distinguish} between $\mu^k$ and $\N^k$: for any semi-norm $\phi$, $\ex_{\mu^k}[\phi(x)] = (1\pm \epsilon)\ex_{\N^k}[\phi(x)]$. We then suitably prune the distribution $\mu^k$ to have small support and prove \tref{th:epsnet}.\\

Our main result, \tref{th:main}, follows by first reducing the dimension as in the previous section and applying \tref{th:epsnet} to the semi-norm $\phi:\R^k \rgta \R_+$, $\phi(x) = \sup_i|\inp{u_i,x}|$ for the projected vectors $\{u_1,\ldots,u_m\}$.

\section{Dimension Reduction}
The use of JLL type random projections for estimating the supremum comes from the following comparison inequality for Gaussian processes. We call a collection of real-valued random variables $\{X_t\}_{t \in T}$ a Gaussian process if every finite linear combination of the variables has a normal distribution with mean zero. For a reference to Slepian's lemma we refer the reader to Corollary 3.14 and the following discussion in \cite{LedouxT}.

\begin{theorem}[Slepian's Lemma \cite{Slepian62}]\label{lm:slepian}
  Let $\{X_t\}_{t \in T}$ and $\{Y_t\}_{t \in T}$ be two Gaussian processes such that for every $s,t \in T$, $\ex[(X_s - X_t)^2] \leq \ex[(Y_s - Y_t)^2]$. Then, $\ex[\sup_t X_t] \leq \ex[\sup_t Y_t]$.
\end{theorem}
We also need a derandomized version of the Johnson-Lindenstrauss Lemma.
\begin{theorem}[\cite{DJLL}]\label{th:djll}
For every $\epsilon > 0$, there exists a deterministic $(d m^2 (\log m + 1/\epsilon)^{O(1)})$-time algorithm that given vectors $v_1,\ldots,v_m \in \R^d$ computes a linear mapping $A:\R^d \rgta \R^k$ for $k = O((\log m)/\epsilon^2)$ such that for every $i,j \in [m]$, $\nmt{v_i - v_j} \leq \nmt{A(v_i) - A(v_j)} \leq (1+\epsilon)\nmt{v_i - v_j}$.
\end{theorem}

Combining the above two theorems immediately implies the following.
\begin{lemma}\label{lm:derandjl}
  For every $\epsilon > 0$, there exists a deterministic $(d m^2 (\log m + 1/\epsilon)^{O(1)})$-time algorithm that given vectors $v_1,\ldots,v_m \in \R^d$ computes a linear mapping $A:\R^d \rgta \R^k$ for $k = O((\log m)/\epsilon^2)$ such that
  \begin{equation}
    \label{eq:jllsup}
    \ex_{x \lfta \N^d}[\sup_i |\inp{v_i,x}|] \leq  \ex_{y \lfta \N^k}[\sup_i |\inp{A(v_i),y}|] \leq (1+\epsilon)  \ex_{x \lfta \N^d}[\sup_i |\inp{v_i,x}|].
  \end{equation}
\end{lemma}
\begin{proof}
  Let $V = \{v_1,\ldots,v_m\} \cup \{-v_1,\ldots,-v_m\}$ and let $\{X_v\}_{v \in V}$ be the Gaussian process where the joint distribution is given by $X_v \equiv \inp{v,x}$ for $x \lfta \N^d$. Then,  $\ex_{x \lfta \N^d}[\sup_i |\inp{v_i,x}|] = \ex[\sup_v X_v]$.

Let $A:\R^d \rgta \R^k$ be the linear mapping as given by \tref{th:djll} applied to $V$. Let $\{Y_v\}_{v \in V}$ be the ``projected'' Gaussian process with joint distribution given by $Y_v \equiv \inp{A(v),y}$ for $y \lfta \N^k$. Then,  $\ex_{y \lfta \N^k}[\sup_i |\inp{v_i,y}|] = \ex[\sup_v Y_v]$.

Finally, observe that for any $u,v \in V$,
\[ \ex[(X_u - X_v)^2] = \nmt{u-v}^2 \leq \nmt{A(u) - A(v)}^2 = \ex[(Y_u - Y_v)^2] \leq (1+\epsilon)^2\ex[(X_u - X_v)^2].\]

Combining the above inequality with Slepian's lemma \lref{lm:slepian} applied to the pairs of processes $\left(\{X_v\}_{v \in V}, \{Y_v\}_{v \in V}\right)$ and $\left(\{Y_v\}_{v \in V}, \{(1+\epsilon)X_v\}_{v \in V}\right)$ it follows that
\[ \ex[\sup_v X_v] \leq \ex[\sup_v Y_v] \leq \ex[\sup_v (1+\epsilon)X_v] = (1+\epsilon)\ex[\sup_v X_v].\]
The lemma now follows.
\end{proof}

\section{Oblivious Estimators for Semi-Norms in Gaussian Space}\label{sec:epsnet}
In the previous section we reduced the problem of computing the supremum of a $d$-dimensional Gaussian process to that of a Gaussian process in $k = O((\log m)/\epsilon^2)$-dimensions. Thus, it suffices to have an algorithm for approximating the supremum of Gaussian processes in time exponential in the dimension. We will give such an algorithm that works more generally for all semi-norms.

Let $\phi:\R^k \rgta \R_+$ be a semi-norm. That is, $\phi$ satisfies the triangle inequality and is homogeneous. For normalization purposes we assume that $1 \leq \ex_{\N^k}[\phi(X)]$ and the Lipschitz constant of $\phi$ is at most $k^{O(1)}$.

\begin{theorem}\label{th:epsnet}
For every $\epsilon > 0$, there exists a set $S \subseteq \R^k$ with $|S| = (1/\epsilon)^{O(k)}$ and a function $p:\R^k \rgta \R_+$ computable in $\poly(k,1/\epsilon)$ time such that the following holds. For every semi-norm $\phi:\R^k \rgta \R_+$,
\[(1 - \epsilon) \left(\sum_{x \in S} p(x) \phi(x)\right) \leq \ex_{X \lfta \N^k}[\phi(X)] \leq (1 + \epsilon) \left(\sum_{x \in S} p(x) \phi(x)\right).\]
Moreover, successive elements of $S$ can be enumerated in $\poly(k,1/\epsilon)$ time and $O(k\log(1/\epsilon))$ space.
\end{theorem}
\tref{th:epsnetintro} follows immediately from the above.
\begin{proof}[Proof of \tref{th:epsnetintro}]
  Follows by enumerating over the set $S$ and computing $\sum_{x \in S} p(x) \phi(x)$ by querying $\phi$ on the points in $S$.
\end{proof}

We now prove \tref{th:epsnet}.
Here and henceforth, let $\gamma$ denote the pdf of the standard univariate Gaussian distribution. Fix $\epsilon > 0$ and let $\delta > 0$ be a parameter to be chosen later. Let $\mu \equiv \mu_{\epsilon,\delta}$ be the pdf which is a piecewise-flat approximator to $\gamma$ obtained by spreading the mass $\gamma$ gives to an interval $I = [i\delta, (i+1)\delta)$ evenly over $I$. Formally, $\mu(z) = \mu(-z)$ and for $z > 0$, $z \in [i\delta,(i+1)\delta)$,
\begin{equation}
  \label{eq:defmu}
  \mu(z) = \frac{\gamma([i\delta,(i+1)\delta))}{\delta}.
\end{equation}
Clearly, $\mu$ defines a symmetric distribution on $\R$. We will show that for $\delta \ll \epsilon$ sufficiently small, semi-norms cannot {\it distinguish} the product distribution $\mu^k$ from $\N^k$:
\begin{lemma}\label{lm:epsnetm}
  Let $\delta = (2\epsilon)^{3/2}$. Then, for every semi-norm $\phi:\R^k \rgta \R$,
$$(1-\epsilon) \ex_{X \lfta \mu^k}[\phi(X)] \leq \ex_{Z \lfta \N^k}[\phi(Z)] \leq \ex_{X \lfta \mu^k}[\phi(X)].$$
\end{lemma}
\newcommand{\hX}{\hat{X}}
We first prove \tref{th:epsnet} assuming the above lemma, whose proof is deferred to the next section.
\begin{proof}[Proof of \tref{th:epsnet}]
Let $\hat{\mu}$ be the symmetric distribution supported on $\delta(\Z + 1/2)$ with pdf defined by
$$\hat{\mu}(\delta(i+1/2)) = \mu([i\delta, (i+1)\delta)),$$ for $i \geq 0$. Further, let $X \lfta \mu^k$, $\hX \lfta \hat{\mu}^k$, $Z \lfta \N^k$.

We claim that $\ex[\phi(\hX)] = (1\pm \epsilon)\ex[\phi(Z)]$. Let $Y$ be uniformly distributed on $[-\delta,\delta]^k$ and observe that random variable $X \equiv \hX + Y$ in law. Therefore,
\begin{multline}
  \ex[\phi(X)] = \ex[\phi(\hX+Y)] = \ex[\phi(\hX)] \pm \ex[\phi(Y)] = \ex[\phi(\hX)] \pm \delta \ex[\phi(Y/\delta)]\\
= \ex[\phi(\hX)] \pm \delta \ex_{Z' \in_u [-1,1]^k}[\phi(Z')] = \ex[\phi(\hX)] \pm \delta \ex[\phi(Z)] \text{ (\lref{lm:cube})}.
\end{multline}
Thus, by \lref{lm:epsnetm},
\begin{equation}
  \label{eq:lm1}
\ex[\phi(\hX)] = (1\pm O(\epsilon)) \ex[\phi(Z)]
\end{equation}
We next prune $\hat{\mu}^k$ to reduce its support. Define $p:\R^k \rgta \R_+$ by $p(x) = \hat{\mu}^k(x)$. Clearly, $p(x)$ being a product distribution is computable in $\poly(k,1/\epsilon)$ time.

Let $S = \left(\delta(\Z + 1/2)\right)^k \cap B_2(3\sqrt{k})$, where $B_2(r) \subseteq \R^k$ denotes the Euclidean ball of radius $r$. As $\phi$ has Lipschitz constant bounded by $k^{O(1)}$, a simple calculation shows that throwing away all points in the support of $\hX$ outside $S$ does not change $\ex[\phi(\hX)]$ much. It is easy to check that for $x \notin S$, $p(x) \leq \exp(-\nmt{x}^2/4)/(2\pi)^{k/2}$. Therefore,
\begin{multline}
  \ex[\phi(\hX)] = \sum_{x} p(x) \phi(x) = \sum_{x \in S}p(x) \phi(x) + \sum_{x \notin S}p(x) \phi(x)\\
= \sum_{x \in S} p(x) \phi(x) \pm \sum_{x \notin S} \frac{\exp(-\nmt{x}^2/4)}{(2\pi)^{k/2}}\cdot ( k^{O(1)} \nmt{x}) = \sum_{x \in S}p(x) \phi(x) \pm o(1).
\end{multline}

From \eref{eq:lm1} and the above equation we get (recall that $\ex[\phi(Z)] \geq 1$)
\[ \ex[\phi(Z)] = (1\pm O(\epsilon)) \left(\sum_{x \in S} p(x) \phi(x)\right),\]
which is what we want to show.

We now reason about the complexity of $S$. First, by a simple covering argument $|S| < (1/\delta)^{O(k)}$:
\[ |S| < \frac{Vol\,(B_2(3\sqrt{k}) + [-\delta,\delta]^k)}{Vol\,([-\delta,\delta]^k)} = (1/\delta)^{O(k)} = (1/\epsilon)^{O(k)},\]
where for sets $A, B \subseteq \R^k$, $A+B$ denotes the Minkowski sum and $Vol$ denotes Lebesgue volume. This size bound almost suffices to prove \tref{th:epsnet} except for the complexity of enumerating elements from $S$. Without loss of generality assume that $R = 3\sqrt{n}/\delta$ is an integer. Then, enumerating elements in $S$ is equivalent to enumerating integer points in the $n$-dimensional ball of radius $R$. This can be accomplished by going through the set of lattice points in the natural lexicographic order, and takes $\poly(k,1/\epsilon)$ time and $O(k\log(1/\epsilon))$ space per point in $S$.
\end{proof}
\section{Proof of \lref{lm:epsnetm}}
Our starting point is the following definition that helps us {\it compare} multivariate distributions when we are only interested in volumes of convex sets. We shall follow the notation of \cite{Ball}.
\begin{definition}
  Given two symmetric pdf's, $f,g$ on $\R^k$, we say that $f$ is less peaked than $g$ ($f \preceq g$) if for every symmetric convex set $K \subseteq \R^k$, $f(K) \leq g(K)$.
\end{definition}
We also need the following elementary facts. The first follows from the unimodality of the Gaussian density and the second from partial integration.
\begin{fact}
  For any $\delta > 0$ and $\mu$ as defined by \eref{eq:defmu}, $\mu$ is less peaked than $\gamma$.
\end{fact}
\begin{fact}\label{fct:peaked}
  Let $f, g$ be distributions on $\R^k$ with $f \preceq g$. Then for any semi-norm $\phi:\R^k \rgta \R$, $\ex_f[\phi(x)] \geq \ex_g[\phi(x)]$.
\end{fact}
\begin{proof}
Observe that for any $t > 0$, $\{x: \phi(x) \leq t\}$ is convex. Let random variables $X \lfta f$, $Y \lfta g$. Then, by partial integration, $\ex[\phi(X)] = \int_0^\infty \phi'(t) \pr[ \phi(X) > t] dt \geq \int_0^\infty \phi'(t) \pr[\phi(Y)> t] dt = \ex[\phi(Y)]$.
\end{proof}
The above statements give us a way to compare the expectations of $\mu$ and $\gamma$ for one-dimensional convex functions. We would now like to do a similar comparison for the product distributions $\mu^k$ and $\gamma^k$. For this we use Kanter's lemma \cite{Kanter77}, which says that the relation $\preceq$ is preserved under tensoring if the individual distributions have the additional property of being {\it unimodal}.

\begin{definition}
  A distribution $f$ on $\R^n$ is unimodal if $f$ can be written as an increasing limit of a sequence of distributions each of which is a finite positively weighted sum of uniform distributions on symmetric convex sets.
\end{definition}

\begin{theorem}[Kanter's Lemma \cite{Kanter77}; cf.~\cite{Ball}]\label{th:kanter}
  Let $\mu_1,\mu_2$ be symmetric distributions on $\R^n$ with $\mu_1 \preceq \mu_2$ and let $\nu$ be a unimodal distribution on $\R^m$. Then, the product distributions $\mu_1 \times \nu$, $\mu_2 \times \nu$ on $\R^n \times \R^m$ satisfy $\mu_1 \times \nu \preceq \mu_2 \times \nu$.
\end{theorem}

We next show that $\mu$ ``sandwiches'' $\gamma$ in the following sense.
\begin{lemma}
  Let $\nu$ be the pdf of the random variable $y = (1-\epsilon)x$ for $x \lfta \mu$. Then, for $\delta \leq (2\epsilon)^{3/2}$, $\mu \preceq \gamma \preceq \nu$.
\end{lemma}
\begin{proof}
As mentioned above, $\mu \preceq \gamma$. We next show that $\gamma \preceq \nu$. Intuitively, $\nu$ is obtained by spreading the mass that $\gamma$ puts on an interval $I = [i\delta, (i+1)\delta)$ evenly on the \emph{smaller} interval $(1-\epsilon)I$. The net effect of this operation is to push the pdf of $\mu$ closer towards the origin and for $\delta$ sufficiently small the inward push from this ``shrinking'' wins over the outward push from going to $\mu$.

Fix an interval $I = [-i \delta(1-\epsilon) - \theta, i\delta(1-\epsilon) + \theta]$ for $0 \leq \theta < \delta(1-\epsilon)$. Then,
\begin{align}\label{eq:cases}
  \nu(I) &= \nu\left(\,[-i\delta(1-\epsilon), i\delta(1-\epsilon)]\,\right) + 2\, \nu\left(\,[i\delta(1-\epsilon), i\delta(1-\epsilon) + \theta]\,\right)\\
&= \gamma\left(\,[-i\delta,i\delta]\,\right) + \frac{2\, \theta \cdot \gamma(\,[i\delta,(i+1)\delta)\,)}{\delta(1-\epsilon)}.
\end{align}
We now consider two cases.

Case 1: $i \geq (1-\epsilon)/\epsilon$ so that $i\delta(1-\epsilon) + \theta \leq i\delta$. Then, from the above equation,
\[ \nu(I) \geq \gamma\left(\,[-i\delta, i \delta]\,\right) \geq \gamma\left(\,[-i\delta(1-\epsilon)-\theta, i\delta(1-\epsilon)+\theta]\,\right) = \gamma(I).\]

Case 2: $i < (1-\epsilon)/\epsilon$. Let $\alpha = (i+1)\delta = \delta/\epsilon$. Then, as $1 - x^2/2 \leq e^{-x^2/2} \leq 1$,
\[ \gamma((i\delta, i\delta + \theta]) \leq \theta \cdot \gamma(0),\;\;\;\; \gamma(\,[i\delta,(i+1)\delta)\,) \geq \delta \cdot \gamma(0) \cdot (1-\alpha^2/2).\]

Therefore,
\begin{align*}
  \nu(I) &= \gamma(I) - 2 \gamma\left(\,(i\delta, i\delta(1-\epsilon) + \theta]\,\right) + \frac{2 \theta \cdot \gamma(\,[i\delta,(i+1)\delta)\,)}{\delta(1-\epsilon)}\\
&\geq \gamma(I) - 2 \gamma\left(\,(i\delta, i\delta + \theta]\,\right) + \frac{2 \theta \cdot \gamma(\,[i\delta,(i+1)\delta)\,)}{\delta(1-\epsilon)}\\
&\geq \gamma(I) - 2 \theta \gamma(0) + \frac{2 \theta \cdot \delta \cdot \gamma(0) \cdot (1-\alpha^2/2)}{\delta(1-\epsilon)}\\
&= \gamma(I) + \frac{2\theta \gamma(0)}{1-\epsilon} \cdot (\epsilon - \alpha^2/2) \geq \gamma(I),
\end{align*}
for $\alpha^2 \leq 2 \epsilon$, i.e., if $\delta \leq (2\epsilon)^{3/2}$.
\ignore{
Let $\delta \ll \epsilon$ be sufficiently small so that the pdf of $\gamma$ is nearly constant in the interval $[0,(i+1)\delta]$, that is $\gamma([\alpha,\beta]) = (\beta-\alpha) \gamma(0) \pm O((\beta-\alpha)^2)$ for $0 < \alpha < \beta < (i+1) \delta$. Then, by \eref{eq:cases}, as $\theta < \delta(1-\epsilon)$,
\begin{align*}
  \nu(I) &\geq \gamma(I) - 2 \gamma\left(\,(i\delta, i\delta(1-\epsilon) + \theta]\,\right) + \frac{2 \theta \cdot \gamma(\,[i\delta,(i+1)\delta)\,)}{\delta(1-\epsilon)}\\
&\geq \gamma(I) - 2 \gamma\left(\,(i\delta, i\delta + \theta]\,\right) + \frac{2 \theta \cdot \gamma(\,[i\delta,(i+1)\delta)\,)}{\delta(1-\epsilon)}\\
&\geq \gamma(I) - 2 \theta \gamma(0) - O(\theta^2) + \frac{2 \theta (\delta \gamma(0) - O(\delta^2))}{\delta(1-\epsilon)}\\
&= \gamma(I) + \frac{2\theta \gamma(0)\epsilon}{1-\epsilon} - O(\theta \,\delta) \geq \gamma(I)
\end{align*}}
\end{proof}
\lref{lm:epsnetm} follows easily from the above two claims.
\begin{proof}[Proof of \lref{lm:epsnetm}]
 Clearly, $\mu,\nu,\gamma$ are unimodal and product of unimodal distributions is unimodal. Thus, from the above lemma and iteratively applying Kanter's lemma we get $\mu^k \preceq \gamma^k \preceq \nu^k$. Therefore, by Fact \ref{fct:peaked}, for any semi-norm $\phi$,
\[ \ex_{\mu^k}[\phi(X)] \geq \ex_{\gamma^k}[\phi(Y)] \geq \ex_{\nu^k}[\phi(X)] = \ex_{\mu^k}[\phi((1-\epsilon)X)] = (1-\epsilon)\ex_{\mu^k}[\phi(X)].\]
\end{proof}
We now prove the auxiliary lemma we used in proof of \tref{th:epsnet}.
\begin{lemma}\label{lm:cube}
  Let $\rho$ be the uniform distribution on $[-1,1]$. Then, $\gamma \preceq \rho$ and for any semi-norm $\phi:\R^k \rgta \R$, $\ex_{\rho^k}[\phi(x)] \leq \ex_{\gamma^k}[\phi(x)]$.
\end{lemma}
\begin{proof}
  It is easy to check that $\gamma \preceq \rho$. Then, by Kanter's lemma $\gamma^k \preceq \rho^k$ and the inequality follows from Fact \ref{fct:peaked}.
\end{proof}
\section{A PTAS for Supremum of Gaussian Processes}
Our main theorem, \tref{th:main}, follows immediately from \lref{lm:derandjl} and \tref{th:epsnetintro} applied to the semi-norm $\phi:\R^k \rgta \R$ defined by $\phi(x) = \sup_{i \leq m} |\inp{A(v_i),x}|$.
\section{Lowerbound for Oblivious Estimators}\label{sec:appendix}
We now show that \tref{th:epsnet} is optimal: any oblivious linear estimator for semi-norms as in the theorem must make at least $(C/\epsilon)^{k}$ queries for some constant $C > 0$.

Let $S \subseteq \R^k$ be the set of query points of an oblivious estimator. That is, there exists a function $f:\R_+^S \rgta \R_+$ such that for any semi-norm $\phi:\R^k \rgta \R_+$, $f((\phi(x): x \in S)) = (1\pm \epsilon) \ex_{Y \lfta \N^k}[\phi(Y)]$. We will assume that $f$ is monotone in the following sense: $f(x_1,\ldots,x_{|S|}) \leq f(y_1,\ldots,y_{|S|})$ if $0 \leq x_i \leq y_i$ for all $i$. This is clearly true for any linear estimator (and also for the median estimator). Without loss of generality suppose that $\epsilon < 1/4$.

\newcommand{\spk}{\mathcal{S}^{k-1}}
The idea is to define a suitable semi-norm based on $S$: define $\phi:\R^k \rgta \R$ by $\phi(x) = \sup_{u \in S}|\inp{u/\nmt{u},x}|$. It is easy to check that for any $v \in S$, $\nmt{v} \leq \phi(v)$. Therefore, the output of the oblivious estimator when querying the Euclidean norm is at most the output of the estimator when querying $\phi$. In particular,
\begin{equation}
  \label{eq:app1}
(1-\epsilon) \ex_{Y \lfta \N^k}[\nmt{Y}] \leq f((\nmt{x}: x \in S)) \leq f((\phi(x): x \in S)) \leq (1+\epsilon) \ex_{Y \lfta \N^k}[\phi(Y)].
\end{equation}
We will argue that the above is possible only if $|S| > (C/\epsilon)^k$. Let $\spk$ denote the unit sphere in $\R^k$. For the remaining argument, we shall view $Y \lfta \N^k$ to be drawn as $Y = R X$, where $X \in \spk$ is uniformly random on the sphere and $R \in \R$ is independent of $X$ and has a Chi-squared distribution with $k$ degrees of freedom. Let $S(\epsilon) = \cup_{u \in S} \{y \in \spk: |\inp{u/\nmt{u},y}| \geq 1 - 4\epsilon\}$.

Now, by a standard volume argument, for any $y \in \spk$, $\pr_X[|\inp{X,y}| \geq 1 - 4\epsilon] < (O(\epsilon))^k$.  Thus, by a union bound, $p = \pr_X[X \in S(\epsilon)] < |S| \cdot (O(\epsilon))^k$. Further, for any $y \in \spk\setminus S(\epsilon)$, $\phi(y) < 1-4\epsilon$. Therefore,
\begin{multline*}
  \ex_{X}[\phi(X)] = \pr[X \notin S(\epsilon)] \cdot \ex[\phi(X) | X \notin S(\epsilon)] + \pr[X \in S(\epsilon)] \cdot \ex[\phi(X) | X \in S(\epsilon)]\leq \\(1-p) (1-4\epsilon) + p.
\end{multline*}
Thus,
\begin{equation}
  \label{eq:app2}
  \ex[\phi(Y)] = \ex[\phi(R X)] = \ex[R] \cdot \ex[\phi(X)] \leq \ex[\nmt{Y}] \cdot ((1-p)(1-4\epsilon) + p).
\end{equation}
Combining Equations \ref{eq:app1} and \ref{eq:app2}, we get
\[ 1 - \epsilon \leq (1+\epsilon)\cdot ((1-p)(1-4\epsilon) + p) < 1-3\epsilon + 2p.\]
As $p < |S| \cdot (O(\epsilon))^k$, the above leads to a contradiction unless $|S| > (C/\epsilon)^{k}$ for some constant $C > 0$.

\bibliographystyle{amsalpha}


\section{Introduction}
\label{intro}
The  development and refinement in recent years of new techniques in materials
growth has made it possible
to fabricate superconducting heterostructures with various materials and
high quality interfaces. These advances, coupled
with the continuing intense level of activity in the study of the nature of
high temperature\cite{agl,dj,har} and other exotic
superconductors,\cite{upt,mae}
has led to renewed interest in
tunneling spectroscopy.

It has been demonstrated\cite{har,hu,tan} that this
technique yields information about
both the magnitude and  the phase of the superconducting pair potential (PP).
This implies that the method  can
provide a systematic way to distinguish among various proposed PP
candidates, including both spin singlet and spin triplet
pairing states.\cite{tan3,sig}
For example, it has been argued that the observed zero bias conductance
peak\cite{hu,tan,xu,wei,alf}
(ZBCP),
attributed to mid-gap surface states, is an indication of
unconventional superconductivity with a sign change of the PP,
as it occurs in pairing with a
$d_{x^2-y^2}$-wave symmetry. Furthermore, the splitting of the ZBCP and
the forming
of a finite bias peak  (FBCP) in the conductance spectrum  has been examined
and interpreted\cite{cov,sauls,ting2} as support for the admixture of
an imaginary
PP component to the dominant $d_{x^2-y^2}$-wave part, leading to a broken
time-reversal symmetry.\cite{shiba,rice}

The same developments, and the ability to make low interfacial
resistance junctions between high spin polarization
ferromagnets and superconductors,
have stimulated significant efforts to study transport in these
structures.\cite{prinz} There have been
various experiments in both
conventional\cite{soul,up,john} and high temperature
superconductors\cite{vas,dong,vas2,chen} (HTSC's),
as well as re-examinations of earlier work\cite{ted,mers} which was
performed generally in the
tunneling limit of strong interfacial barrier.
Theoretical studies of
the effects of spin polarized transport on the current-voltage characteristics
and the conductance in ferromagnet/superconductor (F/S)
junctions have been carried out
 in conventional\cite{been}
and, recently, in high-temperature superconductors.\cite{ting,zv,kash}
The feasibility of nanofabricating F/S structures
has also
generated interest in studying the influence of ferromagnetism on mesoscopic
superconductivity.\cite{vol}

One of the important questions raised by the  possibility of making high
transmissivity F/S junctions was that of studying the influence of Andreev
reflection (AR)\cite{been,and,bru,nat} on spin polarized transport. In AR
an electron, belonging to one of the two spin bands,
incoming from the ferromagnetic region to the F/S
interface will be reflected as a hole  in the opposite spin band.
The splitting of spin bands by the exchange energy in ferromagnetic
materials implies that only a fraction of the incoming incident
majority spin electrons can be Andreev reflected.\cite{been}
This simple argument was used in
previous studies\cite{soul,up,been} to infer that
the effect of spin polarization (exchange energy)
was generally to reduce AR. The sensitivity of AR to
the exchange energy in a ferromagnet was employed\cite{soul,up} to determine
the degree of polarization in various materials.

In this paper we will study the tunneling spectroscopy of F/S junctions.
We will adopt the basic
approach of Ref. \onlinecite{btk} but we will extend and
generalize it to include the
effects of spin polarization,  the presence of an unconventional PP state
(pure or mixed), and the existence of Fermi wavevector
mismatch (FWM)\cite{btk2,dutch}
stemming from the different band widths in the two junction materials.
Our aim in this paper is twofold: Firstly, to investigate and
reveal novel features
in the conductance spectra arising from the interplay of ferromagnetism and
unconventional superconductivity, and secondly, to show the importance of FWM, and
how its inclusion can lead to some unexpected results, even for  F/s-wave
superconductor junctions, where we find, for example, that in some cases
Andreev reflection can be enhanced by spin polarization.

In the next section (Sec. \ref{meth}), we present the  methods we use
to obtain the amplitudes for the  various scattering processes
that occur in the junction when spin polarized
electrons are injected from the F into the S region. We will use these
methods to calculate the conductance
of the F/S junctions. In Sec. \ref{results}, we first give results for
a conventional
($s$-wave) superconductor in the S side,
and then illustrate  the unconventional case of the
pairing potential by considering both pure $d$- and mixed
$d+is$-wave symmetry. In
Sec. \ref{conc},
we summarize our results and discuss future problems.
\section{Methods}
\label{meth}
As explained in the Introduction,
we investigate in this work F/S junctions by extending and generalizing
the techniques previously employed in the study of simpler
cases without spin polarization, or for conventional superconductors.
Thus, we use here the Bogoliubov-de Gennes
(BdG) equations\cite{hu,tan,been,bru,deg} in the ballistic limit.
We consider a geometry where the ferromagnetic material is at $x<0$, and
is described by the
Stoner model. We take the usual approach\cite{been} of assuming a
single particle Hamiltonian
with the exchange energy being therefore of the form
$h({\bf r})=h_0 \Theta(-x)$,  where $\Theta(x)$ is a step
function. The F/S interface is at $x=0$, where there is
interfacial scattering modeled by a potential
$V({\bf r})=H \delta(x)$,\cite{tan,soul,up,btk} and  $H$ is the
variable strength of the
potential barrier. The dimensionless parameter characterizing barrier
strength\cite{btk} is $Z_0\equiv mH/\hbar k_F$, where the
effective mass, $m$, is taken to be equal in the F and S regions.
In the superconducting region, at $x>0$, we
assume\cite{hu,tan,been,btk,arnold}
that there is a pair
potential  $\Delta({\bf k}', {\bf r})=
\Delta({\bf \hat{k}}') \Theta(x)$.
This approximation for the PP becomes more accurate\cite{simple}
in the presence of FWM and allows analytic solution of the BdG equations.
We will denote quantities
pertaining to the S region by primed letters.

 From these considerations, the BdG equations for F/S junction,
in the absence
spin-flip scattering, can be written as\cite{deg}
\begin{eqnarray}
\left[\begin{array}{cc} H_0-\rho_S h & \Delta \\
\Delta^* & -(H_0+\rho_S h)\end{array} \right]
\left[ \begin{array}{c} u_S \\
v_{\overline{S}} \end{array} \right]=\epsilon
\left[ \begin{array}{c} u_S \\
v_{\overline{S}} \end{array} \right],
\label{BdG}
\end{eqnarray}
where $H_0$ is the single particle Hamiltonian and  $\rho_S=\pm1$ for
spin $S=\uparrow, \downarrow$. The exchange energy $h{(\bf r})$ and
the PP $\Delta$ are as defined above. The excitation energy is denoted by
$\epsilon$, and $u_S$, $v_{\overline{S}}$ are the
electronlike quasiparticle (ELQ)
and holelike quasiparticle (HLQ) amplitudes, respectively.
We take
$H_0\equiv -\hbar^2\nabla^2/2m+V({\bf r})-E_F^{F,S}$, where $V({\bf r})$ is
defined above. In  the F region,
we have $E_F^F\equiv E_F = \hbar^2 k_F^2/2m$, so that $E_F$ is the spin
averaged value,
$E_F=(\hbar^2 k_{F\uparrow}^2/2m+\hbar^2 k^2_{F\downarrow}/2m )/2$.
We assume that in general it differs from
the value in the superconductor,
$E_F^S \equiv E'_F = \hbar^2 k'^2_F/2m$. Thus,
we take the Fermi energies to be different in the F and S regions
that is,  we allow for different
band widths, stemming from the different carrier densities
in the two regions. Indeed,
as the results in the next Section will show, the
Fermi wavevector mismatch (FWM) between the two regions
has an important influence on our findings.
We will parameterize the FWM by
the value of $L_0$, $L_0 \equiv k'_F/k_F$ and describe the degree of
spin polarization, related to the exchange energy, by the dimensionless
parameter $X\equiv h_0/E_F$.

The invariance of the Hamiltonian with respect to  translations
parallel to $x=0$ implies conservation\cite{mil} of the
(spin dependent) parallel component of the the wavevector at the junction.
As we shall show, this will be an important
consideration in understanding the possible scattering processes.
An electron
injected from the F side, with  spin $S=\uparrow,
\downarrow$, excitation energy $\epsilon$, and wavevector
${\bf k}^+_S$ (with magnitude
 $k^+_{S}=(2m/\hbar^2)^{1/2} [E_F +\epsilon+\rho_S h_0]^{1/2}$),
at an angle $\theta$ from the interface
normal,  can undergo
four scattering processes \cite{tan,btk} each described by a
different amplitude. Assuming specular reflection at the interface, these can
be characterized as follows:
1) Andreev reflection,  with amplitude that we denote by $a_S$,
as a hole with
spin, $\overline{S}$, belonging to the spin band opposite to that of the
incident electron ($\rho_{\overline{S}}=-\rho_S$), wavevector
${\bf k}_{\overline{S}}^-$
($k^-_{\overline{S}}=(2m/\hbar^2)^{1/2}
[E_F -\epsilon+\rho_{\overline{S}} h_0]^{1/2}$),
and spin dependent angle of reflection $\theta_{\overline{S}}$, generally
different
from $\theta$.\cite{zv} As is the case with the angles
corresponding to the other scattering processes,
$\theta_{\overline{S}}$, as we shall see below,
is determined from  the requirement that the
parallel component of the wavevector is conserved.
Even in the absence of exchange energy ($h_0=0$),
one has that,
for $\epsilon \neq 0$,
$\theta_{\overline{S}}$ (although then spin independent)
is slightly different\cite{kummel} from $\theta$. When $h_0>0$, the
typical situation is, as we
discuss later, that
$|\theta_{\overline{\downarrow}}| <|\theta|
< |\theta_{\overline{\uparrow}}|$.
2) The second process is ordinary
reflection into the F region, characterized by an amplitude
which we call $b_S$, as
an electron with variables $S$, $-{\bf k}^+_{S}$, $-\theta$.
The other two processes are:
3) Transmission  into the S region, with amplitude $c_S$,
as an ELQ with ${\bf k}'^+_S$, and 4) Transmission as a HLQ with
amplitude $d_S$ and wavevector $-{\bf k}'^-_S$. Here the corresponding
wavevector magnitudes
are $k'^{\pm}_{S}=(2m/\hbar^2)^{1/2}
[E'_F\pm(\epsilon^2-|\Delta_{S\pm}|^2)^{1/2}]^{1/2}$. We
denote by $\Delta_{S\pm}
=|\Delta_{S\pm}|\exp(i\phi_{S\pm})$,
the different  PP's  felt by the ELQ and the HLQ, respectively,
as determined by
${\bf k}'^\pm_S$. We see, therefore, that  up to four different energy
scales of the PP are involved for each incident angle $\theta$.
In our considerations, which pertain to the common experimental situation,
\cite{amg,venky}
$E_F, E'_F $ $\gg$ $\max(\epsilon, |\Delta_{S\pm}|)$,
we can employ the Andreev
approximation\cite{tan,been,and,bru} and write
$k^{\pm}_{S} \approx k_{FS}
\equiv(2m/\hbar^2)^{1/2}
[E_F +\rho_S h_0]^{1/2}$,
$k'^{\pm}_{S} \approx k'_F$. It then follows that the appropriate wavevectors
for the transmission of  ELQ's and HLQ's are at
angles $\theta'_S$, $-\theta'_S$,
with the interface normal, respectively.
Within this approximation the components of
the vectors ${\bf k}^{\pm}_{S}$,
${\bf k}'^{\pm}_{S}$
normal and parallel to the interface, can be expressed as
${\bf k}^{\pm}_{S}\equiv(k_{S}, k_{\|S})$, and
${\bf k}'^{\pm}_{S}\equiv(k'_S, k_{\|S})$,
in the F and S regions. From the conservation of $k_{\|S}$, we have then
an analogue of Snell's law
\begin{mathletters}
\begin{equation}
k_{FS}\sin\theta=k_{F\overline{S}}\sin\theta_{\overline{S}},
\label{snella}
\end{equation}
\begin{equation}
k_{FS}\sin\theta=k'_F\sin\theta'_S,
\label{snellb}
\end{equation}
\label{snell}
\end{mathletters}
which has several important implications, including the existence
of critical angles,\cite{crit}  as one encounters in well known phenomena in
the propagation of electromagnetic waves.\cite{jackson}

Using the conservation of $k_{\|S}$,
 the solution to Eq. (\ref{BdG}),
$\Psi_S\equiv(u_S,v_{\overline{S}})^T$, can be expressed in a separable form,
effectively reducing the problem to a one-dimensional one. In the F region
we write
\begin{equation}
\Psi_S({\bf r})\equiv e^{i {\bf k}_{\| S}\cdot {\bf r}}\psi_S(x),
\label{Psif}
\end{equation}
where
\begin{eqnarray}
\psi_S(x)=
e^{i  k_S  x} \left[\begin{array}{c} 1 \\ 0 \end{array} \right]+
a_S e^{i k_{\overline{S}}x} \left[\begin{array}{c} 0 \\ 1 \end{array} \right]
+b_S e^{-i k_S x} \left[\begin{array}{c} 1 \\ 0 \end{array} \right],
\label{psif}
\end{eqnarray}
analogously, in the $S$ region we have\cite{tan}
\begin{equation}
\Psi'_S({\bf r})\equiv e^{i {\bf k}_{\| S}\cdot {\bf r}}\psi'_S(x),
\label{Psis}
\end{equation}
\begin{eqnarray}
\psi'_S(x)=
c_S e^{i k'_S x} \left[\begin{array}{c}
(\epsilon +\Omega_{S+}/2 \epsilon)^{\frac{1}{2}} \\
e^{-i\phi_+} (\epsilon -\Omega_{S+}/2 \epsilon)^{\frac{1}{2}}
 \end{array} \right]
+d_S e^{-i k'_S x} \left[\begin{array}{c}
e^{i\phi_-} (\epsilon -\Omega_{S-}/2 \epsilon)^{\frac{1}{2}} \\
(\epsilon +\Omega_{S-}/2 \epsilon)^{\frac{1}{2}}
\end{array} \right],
\label{psis}
\end{eqnarray}
with $\Omega_{S\pm}\equiv(\epsilon^2-|\Delta_{S\pm}|^2)^{\frac{1}{2}}$,
and the appropriate boundary conditions\cite{tan,btk} at the F/S
interface are
\begin{equation}
\psi_S(0)=\psi'_S(0), \quad
\partial_x \psi_S(0)-\partial_x \psi'_S(0)=\frac{2mH}{\hbar^2}\psi'_S(0).
\label{bc}
\end{equation}

We pause next to discuss some implications of Eq. (\ref{snell})
for the various
scattering processes. In typical realizations of ferromagnet/HTSC structures,
the appropriate FWM corresponds to $L_0 \leq 1$.\cite{amg}
Consider first $L_0=1$, i.e. $E_F=E'_F$. If $X>0$ it follows
that $k_{F\downarrow}<k'_F<k_{F\uparrow}$, for an $S=\downarrow$
incoming electron. Then,
at any incident angle, Eq. (\ref{snell}) is satisfied so that $k_\|$ will be
conserved. In this case  $|\theta|>|\theta'_\downarrow|>|\theta_{\overline
{\downarrow}}|$, and all the corresponding wave vectors are real.
For an $S=\uparrow$ incident electron at angle
$|\theta| > |\sin^{-1}(k'_F/k_{F\uparrow})|$,  a solution of
Eq. (\ref{snellb}) for a real $\theta'_\uparrow$ no longer exist, one
has
a complex $\theta'_{\uparrow}$.\cite{jackson}
The scattering problem does not have a solution with propagating
wavevectors in the S region:
there is total reflection.
The  wavevectors for
ELQ and HLQ have purely imaginary components along the $x$-axis, while
their components parallel to the interface  are real.
This corresponds to  a surface (evanescent) wave, propagating
along the interface and exponentially damped away from
it.\cite{jackson}
An analogous, but physically more interesting, situation occurs for AR in
the particular case where
$|\theta|$ is smaller than the angle of total reflection  and satisfies
$|\theta|>|\sin^{-1}(k_{F\downarrow}/k_{F\uparrow})|$. This regime
corresponds to
$k_{\|\overline{\uparrow}} > k_{F\downarrow}$. In this case it is
Eq. (\ref{snella}) that has no solution for real angles. This means
that Andreev reflection as a {\it propagating} wave is impossible.
 From the condition, which follows from the Andreev approximation,
$k_{\overline{\uparrow}}^2+k_{\|\overline{\uparrow}}^2
\equiv k_{F\downarrow}^2$,  we see that the
component $k_{\overline{\uparrow}}$ along the $x$ axis must
be purely imaginary,\cite{kash} while
$k_{\|\overline{\uparrow}}$ is still real.
With these considerations we then find
\begin{equation}
k_{\overline{\uparrow}}=-i
(k_{F\uparrow}^2 \sin^2\theta-k_{F\downarrow}^2)^{1/2},
\label{kim}
\end{equation}
where we have expressed
$k_{\overline{\uparrow}}$ in terms of quantities which are always real
and which pertain to the F region only. As with total reflection,
there is
propagation only along the interface and an exponential decay away from it.
This case differs from that
of total reflection in that, since the evanescence affects only the
Andreev reflected component, there may still be
transmission across the junction.

The above considerations apply {\it a fortiori} in the
presence of FWM. For example, if we now  consider $L_0<1$, we can see
by inspection of Eq. (\ref{snell}),
that there can also be total reflection for an
$S=\downarrow$ incident electron, when $k_{F\downarrow}>k'_F$. This
condition  would imply the  absence of imaginary
$k_{\overline{\uparrow}}$ for any incident angle and any exchange energy.

Returning now to the basic equations, we see that by solving for
$\psi_S(x)$, $\psi'_S(x)$ in Eq. (\ref{psif}), (\ref{psis}) with
the boundary conditions given by Eq. (\ref{bc}), we can obtain the
amplitudes $a_S$, $b_S$, $c_S$  and $d_S$, $S=\uparrow,\downarrow$.
For each spin, there is a sum rule, related to the conservation of probability,
for the squares  of the absolute
values of the amplitudes.
We can thus, in a way similar  to what was done in
Ref. \onlinecite{btk}, express the various quantities in terms of the
amplitudes $a_S$ and $b_S$ only. These amplitudes are given by
\begin{equation}
a_S=\frac{4 t_S L_S \Gamma_+ e^{-i\phi_{S+}}}
{U_{SS+}U_{\overline{S} S-}-
V_{SS-} V_{\overline{S} S+}
\Gamma_+\Gamma_-e^{i(\phi_{S-}-\phi_{S+})}},
\label{as}
\end{equation}
\begin{equation}
b_S=   \frac
{V_{S S+}U_{\overline{S} S-}-
U_{S S-} V_{\overline{S} S+}
\Gamma_+\Gamma_-e^{i(\phi_{S-}-\phi_{S+})}}
{U_{SS+} U_{\overline{S} S-}-
V_{SS-} V_{\overline{S} S+}
\Gamma_+\Gamma_-e^{i(\phi_{S-}-\phi_{S+})}},
\label{bs}
\end{equation}
where we have introduced the notation
$\Gamma_\pm\equiv
(\epsilon-\Omega_{S\pm})/|\Delta_{S\pm}|$,
$L_S\equiv L_0 \cos\theta'_S/\cos \theta$,
describing FWM,
$t_S\equiv k_S/k_{Fx}=(1+\rho_S X)^{1/2}$,
$t_{\overline{S}}\equiv k_{\overline{S}}/k_{Fx}=
(1-\rho_S X)^{1/2} \cos \theta_{\overline{S}} /\cos \theta$, for
$k_{\overline{S}}$ real, ($-i[(1+X)\sin^2\theta-(1-X)]^{1/2}/\cos\theta$, for
$k_{\overline{\uparrow}}$ imaginary, see Eq. (\ref{kim})).
The other abbreviations are defined as:
$U_{\overline{S} S\pm}\equiv t_{\overline{S}}+w_{S\pm}$,
$V_{S S\pm}\equiv t_{S}-w_{S\pm}$,
 $w_{S\pm}\equiv L_S\pm 2iZ$, $Z\equiv Z_0/\cos\theta$, where
$Z_0\equiv m H/\hbar k_F$  is the interfacial barrier parameter,
as defined above.
The limits  $Z_0 \rightarrow 0$ and
 $Z_0 \rightarrow \infty$ correspond to the extreme cases of a metallic
point contact and the tunnel junction limit, respectively.

Given the above amplitudes, the results for the dimensionless differential
conductance\cite{btk} can  be written down in the standard way
by computing, as a function of the excitation energy arising
from the application of a bias voltage, the ratio of the induced flux
densities across the junction
to the corresponding incident
flux density. One straightforwardly generalizes the
methods used in previous work\cite{tan,btk,aver}
to include now the
effects of unconventional
superconductivity, FWM,  and net spin polarization,
to obtain,
\begin{equation}
G\equiv G_{\uparrow}+G_{\downarrow}=\sum_{S=\uparrow,\downarrow} P_S
(1+\frac{k_{\overline{S}}}{k_S}|a_S|^2-|b_S|^2),
\label{gs}
\end{equation}
where we introduce
the probability $P_S$ of an incident electron having spin $S$,
related to the exchange energy as $P_S=(1+\rho_SX)/2$.\cite{been}
In deriving Eq. (\ref{gs}), care has to be taken to
properly include the flux factors, which
are, at $X>0$,  different for the incident and the Andreev reflected particle.
The ratio of wavevectors in the second term on
the right side of Eq. (\ref{gs}) results from  the incident electron and the AR
hole belonging to different spin bands.
The quantity $k_{\overline{S}}$ in that term
is  real, the case
of imaginary $k_{\overline{S}}$ can only contribute to $G_\uparrow$ indirectly,
by modifying $|b_\uparrow|$.
 It  can be shown\cite{zv}
from the conservation of probability current\cite{btk} that
such a contribution vanishes
for the subgap conductance ($\epsilon<|\Delta_{\uparrow \pm}|$).\cite{total} It
is, furthermore, possible to express
the subgap conductance in terms of the AR amplitude only.\cite{zv}
At $X=0$ we recover the results of  Ref.  \onlinecite{tan}.
The suppression  of  the conductance due to  ordinary reflection
at $X\neq 0$ has the same
form as for the unpolarized case since the magnitude of the normal component of
the wavevectors before and after ordinary reflection remains the same.

We focus in this work (see results in the next Section) on the conductance
spectrum of the charge current as given by Eq. (\ref{gs}),
but the amplitudes $a_S$, $b_S$, given by Eq. (\ref{as}), (\ref{bs})
can be used to calculate many other quantities of interest, such as
current-voltage characteristics, the spin current,
and the spin conductance.\cite{kash} We consider also here
angularly averaged quantities and notice
that  Eq. (\ref{gs}) implies that the conductance
vanishes for $|\theta|$
greater than the angle of total reflection (we recall that this angle
is spin dependent). We define the angularly averaged
(AA) conductance, $\langle G_S\rangle$, as
\begin{equation}
\langle G_S \rangle=\int_{\Omega_S} d\theta \cos \theta G_S(\theta)
/\int_{\Omega_S} d\theta \cos\theta,
\label{ga}
\end{equation}
where $\Omega_S$ is limited by the angle of total reflection
or by experimental setup. This form correctly reduces to
that used in the previously
investigated spin unpolarized situation.\cite{tan} One may choose
a different weight
function in
performing such angular averages, depending
on the specific experimental geometry
and the strengths of the interfacial
scattering.\cite{wei,aver,sauls2}
However, all expressions for angularly averaged results, obtained
from  different averaging methods,
would still have a factor of
$(1+\frac{k_{\overline{S}}}{k_S}|a_S|^2-|b_S|^2)$ in the kernel of
integration, and would merely require numerical integration of the
amplitudes we have already given here.

\section{Results}
\label{results}
\subsection{Conventional pair potentials}
\label{cpp}
We present our results in terms of the dimensionless
differential conductance, plotted  as a function of the
dimensionless energy $E\equiv \epsilon/\Delta_0$. We
concentrate on the region $E\lesssim 1$ since for larger bias
various extrinsic effects, such as heating, tend to dominate the behavior
of the measured conductance.\cite{vas3} While our findings,
and the analytic results from Section \ref{meth}, are valid for any value
of the interfacial scattering, we focus on smaller values of $Z_0$,
$Z_0\leq 1$, where the novel effects of ferromagnetism on Andreev reflection,
and consequently on the conductance, are  more pronounced than in the tunneling
limit, $Z_0 \gg 1$. This regime on which we focus is also that
which is believed to
correspond to several ongoing experiments of F/S structures, where the
samples typically have  small interface resistance.\cite{vas2,chen}
To present numerical results, we choose $E'_F/\Delta_0=12.5$, consistent with
optimally doped
$YBa_2Cu_3O_{7-\delta}$.\cite{kresin,hars} We will include
FWM, as parametrized by the quantity $L_0$ introduced above,
$E_F=E'_F/L_0^2$.

We first give some results for an $s$-wave PP, with a constant energy gap.
This will serve to illustrate the influence of FWM coupled with
that of $Z_0$
within a simpler and more familiar context.
In this case,
for any incident angle, $\theta$, of an injected electron the ELQ and HLQ
feel the same PP with $\Delta_{S\pm}=\Delta_0$, and $\phi_{S\pm}\equiv 0$.
Therefore, the results that we give here for the $s$- wave case and
normal incidence ($\theta=0$),
also correspond to the case of  a PP of the  $d_{x^2-y^2}$ form, with
the  angle $\alpha \in (-\pi/2,\pi/2)$,  between the crystallographic
$a$-axis and the interface normal, set to $\alpha=0$.  This would
represent an F/S interface along the (100) plane.

\begin{figure}[htbp]
\vspace*{-2.0cm} \hspace*{4cm}
\epsfxsize = 3.4 in \epsfbox{wl1.ps}
\vspace*{-0.75cm}
\caption{$G(E)$ (Eq.(\protect{\ref{gs}})) versus
$E\equiv \epsilon/\Delta_0$. Results are for $\theta=0$ (normal incidence).
The  curves are  for $Z_0=0$ (no barrier):
in panel (a) at  exchange energy $X\equiv h_0/E_F=0$ (no spin polarization)
they are (from top to bottom at any $E$)
for the FWM values of $L_0^2=E'_F/E_F=1,1/\sqrt{2},1/2,1/4,1/9,1/16$.
In panel (b) they are for $X=0.866$. Since the curves now cross at $E=1$
they are drawn in different ways for clarity.
For $E>1$ they are in the same order as in panel (a) and
for the same values of $L_0$,
while for $E<1$ they correspond, from top to bottom,
to $L_0^2=1/2,1/\sqrt{2},1,1/9,1/16$. The $L_0^2=1/4$ curve
overlaps with that for $L_0^2=1$ in this range.}\label{l1}
\end{figure}

In Fig. \ref{l1} we show results for $G(E)$, given by Eq. (\ref{gs}), at
$\theta=0$,  and $Z_0=0$
(this limit of no  interfacial barrier was also considered
in Ref.\onlinecite{been}). We plot results for various values of  the FWM
parameter $L_0$. Panel (a) corresponds to no polarization ($X=0$)
and panel (b) to high polarization $X=\sqrt{3}/2\approx 0.866$.
For normal incidence,
we have
$t_S=(1+\rho_S X)^{1/2}$,
$t_{\overline{S}}= (1-\rho_S X)^{1/2}$ (as defined below Eq. (\ref{bs})), and
the subgap conductance can be expressed as
\begin{equation}
G=\frac{32 L^2_0 (1-X^2)^{1/2}}
{|t_\uparrow t_\downarrow+ (t_\uparrow+t_\downarrow)L_0+L^2_0
-(t_\uparrow t_\downarrow- (t_\uparrow+t_\downarrow) L_0+L_0^2)
\Gamma_+ \Gamma_-|^2}.
\label{sub}
\end{equation}
Panel (a) displays results in the absence of exchange energy.
With  increasing
FWM (i.e. decreasing $L_0$), the amplitude at zero bias voltage (AZB) decreases
monotonically. This effect was explained\cite{btk2} in previous work
as resulting from the increase  in a single parameter
$Z_{eff}$, which combined $Z_0$ with the effects of FWM.
Our curves with  FWM ($L_0 < 1$) reduce in the appropriate limits to those
previously found\cite{btk2} with $L_0=1$
and $Z_0 \rightarrow Z_{eff}$, $Z_{eff}>Z_0$. We will see below that this
is not the case at $X\neq 0$.
In panel (b) we give results for high $X$
while keeping the
other parameters at the same values as in panel (a). We notice that the presence of
exchange energy gives rise to non-monotonic behavior in the  AZB. At  low
bias, the conductance can be enhanced with increasing FWM (compare,
for example, the $L_0=1$ and $L_0=1/\sqrt{2}$ results),
and form a zero bias conductance peak (ZBCP.) This behavior is qualitatively
different from that found in the
unpolarized case and the effect of FWM can no longer be reproduced
by simply increasing the interface scattering parameter.
Thus the often implied\cite{soul,up} expectation that the effects of $Z_0$ and $L_0$
could also be
subsumed in a single parameter in the spin polarized case is not fulfilled.
In this panel we have an example of coinciding subgap conductances
for $L_0=1$ and $L_0=1/2$.
The condition for this coincidence to take place at
fixed X can be simply obtained from Eq. (\ref{sub}) as
\begin{equation}
t_\uparrow t_\downarrow/L^2_0=L'^2_0 \quad \Rightarrow \quad
(1-X^2)=L'^2_0 \: (L_0\equiv1),
\label{coincide}
\end{equation}
where $L_0$, $L'_0$ correspond to two different values of FWM for
which the subgap conductances will coincide.

We next look, in the same situation as in the previous figure,
at the effects of the presence of an
interfacial barrier.

\begin{figure}[htbp]
\vspace*{-2.0cm} \hspace*{4cm}
\epsfxsize = 3.4 in \epsfbox{wl2.ps}
\vspace*{-0.75cm}
\caption{$G(E)$ for $\theta=0$ and
interfacial barrier strength $Z_0=1$. All the other  parameters
are taken as in the previous figure. In both panels,
curves from top to bottom correspond  to decreasing values of $L_0$.}
\label{l2}
\end{figure}

In Fig. \ref{l2}, we choose $Z_0=1$,
while keeping all the other parameters the same as in
the corresponding panel of the previous figure. In panel (a)
we show results in the absence of spin
polarization. A finite bias conductance peak
(FBCP) appears at the gap edge. It becomes increasingly narrow with
greater FWM (smaller $L_0$).
Its amplitude
is $2$, independent of $L_0$. In panel (b), at $X=0.866$, the conductance
curves display similar behavior, but  with a reduced FBCP at the gap
edge. From Eqs. (\ref{as}), (\ref{bs}), and (\ref{gs}), the amplitude
of the FBCP in this case is
\begin{equation}
G(E=1)=\frac{4(1-X^2)^{1/2}}{1+(1-X^2)^{1/2}}.
\label{edge}
\end{equation}
An interesting feature of this result is that it depends only
on the exchange energy
(spin polarization) and not on the FWM parameter
or the barrier strength. It can be shown that this property holds
for all angles of incidence. This is in contrast
with the  value of the zero bias conductance which depends,  both for normal
incidence and for other angles,  on the value of the FWM. This dependence
could introduce  difficulties in the accurate determination of spin
polarization from the AZB.\cite{soul,up} The gap edge
value is less susceptible to these problems.

\begin{figure}[htbp]
\vspace*{-3.75cm} \hspace*{4cm}
\epsfxsize = 3.4 in \epsfbox{wl3.ps}
\vspace*{-3.0cm}
\caption{Evolution of the zero bias conductance, $G(E)$  for $\theta=0$.
Results are given  at $Z_0=1$ for $X$ determined from Eq. (\protect\ref{azb})
 and values of $L_0$ as in Fig. \protect{\ref{l1}}. From top to
bottom the curves correspond to
values of $(L_0^2,X)$ given by $(1,0)$, $(1/\sqrt{2},1/\sqrt{2})$,
$(1/2,0.866)$, $(1/4,0.968)$, $(1/9,0.994)$, and $(1/16,0.998)$.}
\label{l3}
\end{figure}

The presence of spin polarized carriers, due to nonvanishing
exchange energy, is usually held\cite{up,been,ting} to result in the suppression
 of Andreev reflection and thus  in a  reduction of the subgap
conductance. A simple explanation,\cite{been} which neglects the effects of FWM,
predicts that the AZB should monotonically decrease with increasing
$X$, because of the reduction of Andreev reflection, when
only a fraction  of injected electrons from the majority spin band
can be reflected as holes belonging to the minority spin band.
This follows from the reduction of the density of states in the
minority spin band with increasing $X$, and eventually causes
the subgap conductance to vanish for a half-metallic ferromagnet when
$X\rightarrow 1$.

\begin{figure}[htbp]
\vspace*{-2.0cm} \hspace*{4cm}
\epsfxsize = 3.4 in \epsfbox{wl4.ps}
\vspace*{-0.75cm}
\caption{$\langle G(E)\rangle$, the $\theta$ averaged
conductance, for  an $s$-wave PP and the same values
of  $X$, $L_0$ as in panels (b) of Figs. \protect{\ref{l1}},
\protect{\ref{l2}}, respectively.
In both panels curves from top to bottom, at
$E=2$, correspond to decreasing $L_0$.}
\label{l4}
\end{figure}

We now proceed to examine whether these findings are
modified when FWM is taken into account.
In Fig. \ref{l3}, which shows results at $Z_0=0$ and normal incidence,
we consider the evolution of the conductance
curves for different values of $X$ and $L_0$ chosen to yield maximum AZB,
$(G(E=0)=2$), starting from the step-like feature at $L_0=1$ and $X=0$
(see Fig. (\ref{l1})).
The condition for maximum AZB at fixed FWM and polarization
can be derived\cite{zv} from Eq. (\ref{sub}) and is
\begin{equation}
k_{\uparrow} k_{\downarrow}=k'^2_F
\quad \Rightarrow \quad(1-X^2)^{1/2}=L_0^2.
\label{azb}
\end{equation}
We have used this equation to determine the optimal value of $X$ for each
value of $L_0$ used
in Figs. \ref{l1}, \ref{l2}. The resulting curves are plotted
in Fig. \ref{l3}.
This figure reveals several interesting features. With the increase
of FWM  and the correspondingly larger
optimal spin polarization (according
to the value of $X$ found from Eq. (\ref{azb})), a ZBCP forms.
This is a novel effect in which the peak arises from a mechanism completely
different
from the one usually put forward, where the ZBCP
is  attributed to  the
presence of unconventional superconductivity. In that case, the ZBCP
is produced by the sign change of the PP and the concomitant
formation of Andreev bound
states.\cite{hu,tan,alf}
Furthermore, if we compare these curves with those in panel (b) of Fig. \ref{l1},
we see that the subgap conductance can increase with increasing
spin polarization at fixed $L_0$. This implies that Andreev
reflection can be enhanced by spin polarization.

We now turn to angular averages (AA).
In Fig. \ref{l4} we show angularly averaged results, obtained
from the  expression  for $\langle G \rangle$,
Eq. (\ref{ga}). The averaged results are no longer equivalent
(as in the previous figures with normal incidence) to the case of
a $d_{x^2-y^2}$ PP with an  F/S interface along the (100) plane: the angular
dependence of the PP would then modify the results.
Each of the two panels shown includes results for the same set of parameter
values
used in panels (b) of Figs. \ref{l1}, and \ref{l2}, respectively.
In panel (a) of the current figure we show how the novel features
previously discussed are largely
preserved after angular averaging. There is still formation
of a ZBCP with increased FWM and the AZB retains its non-monotonic
behavior with $L_0$, as in the case of fixed normal incidence.
The angularly averaged results in panel (b), at $Z_0=1$,
display behavior similar to that found in the $\theta=0$ case,
with the conductance
peak at $E=1$ becoming sharper at increasing FWM.

\subsection{Unconventional pair potentials}
\label{upp}
We next consider an angularly dependent PP, specifically that
for a $d_{x^2-y^2}$ pairing state. With this PP
we have different, spin dependent,  PP's for ELQ's and HLQ's. These
are given respectively by
$\Delta_{S\pm}=\Delta_0\cos(2 \theta'_{S\pm})$, where $\theta'_{S\pm}$
can be expressed as
$\theta'_{S\pm}=\theta'_{S}\mp\alpha$ (we recall  that $\alpha$ is
the angle between the interface normal and the crystallographic $a$-axis,
and $\theta'_S$ is related  to $\theta$ through Eq. (\ref{snell})).

\begin{figure}[htbp]
\vspace*{-2.0cm} \hspace*{4cm}
\epsfxsize = 3.4 in \epsfbox{wl5.ps}
\vspace*{-0.75cm}
\caption{$G(E)$
 for $\theta=\pi/10$, $\alpha=\pi/4$, $Z_0=0$, and $L_0=1$.
In (a) the curves are for
$X=0,0.5,0.7,0.8, 0.866,0.95$,
(top to bottom at $E=0$).
In (b), we plot the
spin resolved conductance, for two values of $X$.
The upper curve at $E>1$ corresponds to
$G_\uparrow$, and  and lower curve to $G_\downarrow$.}
\label{l5}
\end{figure}

In Fig. \ref{l5} we give some of our results
for $d$-wave pairing and $\alpha=\pi/4$ (interface in the (110) plane),
in the absence of both interfacial barrier and FWM and
 at a fixed $\theta=\pi/10$, for various values of $X$.
Panel (a) shows curves  for  the total conductance as it
evolves from  a step-like
feature at $X=0$ to a zero bias conductance dip (ZBCD) for large
spin polarization. The width of the plateau at $X=0$ is determined by a
single energy scale set by the equal magnitudes of the PP's for ELQ and HLQ
in that case, as given by
$\Delta_{S+}=\Delta_{S-}<\Delta_0$, $S=\uparrow,\downarrow$. As the
exchange energy is increased, $k_{F\uparrow}$ and $k_{F\downarrow}$
are no longer equal. As one can see from Eq. (\ref{snellb}), it follows that
$\theta'_\uparrow \neq \theta'_\downarrow$ and thus
$\Delta_{\uparrow\pm} \neq \Delta_{\downarrow\pm}$. These two different energy
scales are responsible for the position of various features, such as the
several finite bias conductance peaks (FBCP's) that are seen.

\begin{figure}[htbp]
\vspace*{-2.0cm} \hspace*{4cm}
\epsfxsize = 3.4 in \epsfbox{wl6.ps}
\vspace*{-0.75cm}
\caption{$G(E)$
 for $\theta=\pi/10$, $\alpha=\pi/6$, $Z_0=0$, and $L_0=1$.
In both panels ordering and values of $X$ for each curve are as in
Fig. \protect{\ref{l5}}.}
\label{l6}
\end{figure}

In panel (b) we show the spin decomposition $G=G_\uparrow+G_\downarrow$,
which better reveals these scales, at two different exchange energies.
At $X=0.5$, the shapes of  $G_\uparrow$, $G_\downarrow$ are only slightly
modified
from those in the unpolarized case.

\begin{figure}[htbp]
\vspace*{-2.0cm} \hspace*{4cm}
\epsfxsize = 3.4 in \epsfbox{wl7.ps}
\vspace*{-0.75cm}
\caption{$G(E)$
 for $\theta=\pi/10$, $Z_0=1$, and $L_0=1$. In both panels
curves (top to bottom at $E=0$) correspond to $X=0,0.5,0.7,0.8,0.9$.
In (a) $\alpha=\pi/4$, and in (b) $\alpha=\pi/6$.}
\label{l7}
\end{figure}

At larger exchange energy, $X=0.866$,
the situation is
very different, as shown in the figure.
We also see, in
panel (b), that as
stated in the previous section, the evanescent wave associated
with the imaginary $k_{\overline{\uparrow}}$
does not contribute to the subgap conductance $G_\uparrow$.

In general, for an arbitrary orientation of the
F/S interface, $\alpha\neq 0,\pi/4$, at a fixed $\theta$, all the four
spin dependent PP's for ELQ and HLQ will have different magnitudes.
There are, therefore, specific features at four different energy scales.
It is only for the particular and atypical (but often
chosen in theoretical work) case of $\alpha=\pi/4$
that these four scales reduce to two.

In Fig. \ref{l6} we show the general behavior by choosing $\alpha=\pi/6$,
while retaining the values of all the other parameters from the previous
figure. One can easily calculate, for example, that at $X=0.5$ the
normalized values
of the PP are, in units of the gap maximum, $\Delta_0$,
$|\Delta_{\uparrow+}|=0.963$, $|\Delta_{\uparrow+}|=0.250$,
$|\Delta_{\downarrow+}|=0.822$, $|\Delta_{\downarrow-}|=0.083$.
These numbers can also be approximately inferred from the spin resolved
results given by the solid lines in panel (b).

\begin{figure}[htbp]
\vspace*{-2.0cm} \hspace*{4cm}
\epsfxsize = 3.4 in \epsfbox{wl8.ps}
\vspace*{-0.75cm}
\caption{$\langle G(E) \rangle$, at $Z_0=1$, and $L_0=1$
for $d_{x^2-y^2}$, and
 $d_{x^2-y^2}+is$ pair potentials. The latter is of the form
$\Delta_{S\pm}=\Delta_0\cos(2 \theta'_{S\pm})+i0.1\Delta_0$.
In panel (a) $\alpha=\pi/4$ and in (b) $\alpha=\pi/6$. From top to
bottom (at $E=0$), the  curves correspond to $X=0,0.5,0.9$,
in both panels and for each pair potential.}
\label{l8}
\end{figure}

We next turn to the case where there is a nonvanishing potential barrier,
choosing for illustration the value $Z_0=1$. In the absence of spin
polarization, the
formation of a ZBCP at finite barrier strength
has been extensively investigated\cite{hu,tan,xu}
and explained by  Andreev bound states
in the context of $d$-wave superconductivity. We will consider here also
the effects of $X$, not included in previous work.
In Fig. \ref{l7} we show results for various values of $X$ at $\alpha=\pi/4$.
(in panel (a)) and $\alpha=\pi/6$ (panel (b)). One can see that
for intermediate values of $X$
the conductance maximum is at finite bias.  Comparing the two panels,
one sees that the AZB
at a fixed $X\neq 0$ is larger for $\alpha=\pi/4$, in agreement with the
results obtained for the
unpolarized case where, at zero bias, the spectral weight is maximal\cite{hu}
for a (110) interface.
For a different choice of incident angle $\theta$ there will be,
if the values of all other parameters are held fixed, a change in the effective
barrier strength for various scattering processes.
We recall
(see below Eq. (\ref{bs})), that $Z=Z_0/\cos\theta$, and with an
increase in $|\theta|$ typically there will be,
as in the unpolarized case,\cite{bag}  a decrease in the amplitude for Andreev
reflection and an increased amplitude for
ordinary reflection.

Results such as those discussed above can be obtained as a function of angle,
and the angular average can then be computed from Eq. (\ref{ga}). We will
combine showing some of these angularly averaged results with a brief study of
another point:
it is straightforward to use the formalism discussed here to examine more
complicated superconducting order parameters. A question that has given
rise to a considerable amount of discussion is that of
whether the superconducting order parameter in high $T_c$ materials
is pure $d$-wave or contains a mixture of $s$ wave as well, with an
imaginary component, so that there would not be, strictly speaking,
gap nodes, but only very deep minima. With this in mind,
the effect of a possible ``imaginary'' PP admixture (for example in a
$d+is$ form) on Andreev bound states has also been recently studied.
\cite{cov,sauls,ting2} We consider this question here, including the
effects of polarization.
In  Fig. \ref{l8}, we illustrate the difference in the
angularly averaged conductance values
obtained for a pure
$d_{x^2-y^2}$ PP and for a mixed  $d_{x^2-y^2}+is$ case. We choose the
particular
form $\Delta_{S\pm}=0.9\Delta_0\cos(2 \theta'_{S\pm})+i 0.1\Delta_0$.
The phase of the PP, $\phi_{S\pm}$, is no longer equal to $\pi$ or $0$ as in
the pure $d$-wave case. We give AA results for several values
of $X$, both for the pure $d$ and the mixed $d+is$ cases.

\begin{figure}[htbp]
\vspace*{-2.0cm} \hspace*{4cm}
\epsfxsize = 3.4 in \epsfbox{wl9.ps}
\vspace*{-0.75cm}
\caption{$G(E)$
 for $\theta=\pi/10$, $Z_0=0$, and $L_0=1/2,L_0=1/3$.
In panel (a) $\alpha=\pi/4$ and in (b) $\alpha=\pi/6$.
 From top to bottom, at $E=0$,  curves correspond to $X=0,0.5,0.8$,
in both panels and for each pairing potential.}
\label{l9}
\end{figure}

The former
represents the angular average of results similar to those
previously displayed.

\begin{figure}[htbp]
\vspace*{-2.0cm} \hspace*{4cm}
\epsfxsize = 3.4 in \epsfbox{wl10.ps}
\vspace*{-0.75cm}
\caption{Conductance curves for $\theta=\pi/10$ at $Z_0=1$ with the same
parameters and ordering as in Fig. \protect{\ref{l9}}.}
\label{l10}
\end{figure}

As in the unpolarized case,\cite{sauls,ting} the $is$
admixture
in the PP is responsible for a FBCP, approximately at
$E=0.1$. The conductance maximum is reduced with increased $X$ and
with departure from  a (110) oriented interface.
Replacing the
 $d_{x^2-y^2}+is$ PP by a ``real'' admixture   $d_{x^2-y^2}+s$
(taking again $0.1\Delta_0$ for the $s$-wave part)  gives
results almost indistinguishable from the  pure $d$-wave for any
value of spin polarization.

To show the effects of FWM on conductance for a pure $d$-wave PP we take
$L_0=1/2,1/3$ and give results at the fixed angle, $\theta=\pi/10$,
previously considered.  In Fig. \ref{l9} we show  curves at $Z_0=0$
and $\alpha=\pi/4$ (panel (a)), and for  $\alpha=\pi/6$ in panel (b).
It is useful to compare this figure to panel (a) in Figs. \ref{l5}, and
\ref{l6}, corresponding to no FWM for $\alpha=\pi/4$ and $\pi/6$, respectively.
In the absence of spin polarization the effect of FWM resembles the influence
of a nonvanishing barrier strength, $Z_0$, and leads to the formation
of a ZBCP,
which becomes increasingly narrow for smaller $L_0$. The effect of moderate
spin polarization ($X \lesssim 0.5$, for comparison with the above
mentioned figures) on the AZB is
rather small for $L_0=1, 1/2$ but it is significantly larger at $L_0=1/3$.
In the next figure, Fig. \ref{l10}, we use $Z_0=1$ and the same parameters as in
the previous
figure, so that the influence of barrier strength can
be gauged.  One sees that in the presence of spin polarization the position
of the conductance maximum depends on FWM. With increasing mismatch,
the FBCP
evolves into a ZBCP. By comparing the curves
corresponding to $L_0=1$ in Fig. \ref{l7} with those
for smaller $L_0$ in Fig. \ref{l10} , it is
interesting to  notice that an effect  similar to that
discussed previously for $s$-wave PP without an interfacial barrier
and at normal incidence is also manifested in other regimes, in that
the conductance maximum can actually be enhanced,
in the spin polarized case (at fixed $X$), by the FWM.
\section{Conclusions}
\label{conc}
In this paper we have studied the conductance spectra of
ferromagnetic/superconductor structures. The expressions for
Andreev reflection and ordinary reflection amplitudes which we have given, allow one to
simply obtain other quantities of interest such as current-voltage
characteristics or conductance spectra for spin current.\cite{kash}
We have developed the appropriate extensions of
the standard approach and approximations used in
the absence of spin polarization. This has enabled us to present analytic
results. Within these approximations, and with the inclusion of FWM,  we have
shown a number of important
qualitative differences from the unpolarized case or from that where
spin polarization is included
in the absence of FWM.

Our considerations may also be important in the interpretation of recent
experiments\cite{soul,up} attempting to use tunneling to
measure the degree of spin polarization in the ferromagnetic
side of the junction, since the experimental determination
of spin polarization in a ferromagnet is a very difficult
and important experimental question in its own right.
As we have shown, the ZBCP is sensitive to both spin polarization and FWM,
while the gap edge amplitude depends only on $X$.
It is then not possible to straightforwardly determine the spin polarization
by using the results for the amplitude of the zero bias conductance unless the
appropriate FWM of the F/S structure is
known and properly taken into account. Furthermore, FWM can not, unlike in the
unpolarized case,\cite{btk2}
be simply described by a rescaled
value of the interfacial barrier strength.

The procedures used here have the advantages of simplicity and of allowing
for analytic solutions. These advantages have enabled us to investigate
widely the relevant parameter space.
We have left  for future work considerations that would have diminished
these advantages. Among these
are the question of the  self consistent treatment of the PP, inclusion
of spin-flip scattering or of a
more realistic band structure, and non-equilibrium transport. However,
we believe that  the the methods we have employed
are sufficient to elucidate the hitherto unappreciated subtleties and
the richness and variety of the phenomena associated with
spin polarized tunneling spectroscopy.

We hope that our work,
as reported here and in Ref.{\onlinecite{zv}, will prompt additional experiments
and theoretical work. In particular,
an important clue about spin polarized transport would be provided by
measurements of the  spin resolved conductance. Indeed, we have
already become aware of two very recent related preprints
leading into these directions, one\cite{si}
on Andreev reflection and spin injection into $s$- and
$d$-wave superconductors, and another\cite{enhance} discussing,
in conventional superconductors,
out of equilibrium enhanced Andreev reflection with spin polarization.
\section{Acknowledgements}
We would like to thank J. Fabian, A.M. Goldman, A.J. Millis, S. Das Sarma,
T. Venkatesan, V.A. Vas'ko, and S. Gasiorowicz for useful
discussions. This work was supported by the US-ONR.