On the Conditional Distribution of a Multivariate Normal given a Transformation - the Linear Case

We show that the orthogonal projection operator onto the range of the adjoint of a linear operator $T$ can be represented as $UT,$ where $U$ is an invertible linear operator. Using this representation we obtain a decomposition of a Normal random vector $Y$ as the sum of a linear transformation of $Y$ that is independent of $TY$ and an affine transformation of $TY$. We then use this decomposition to prove that the conditional distribution of a Normal random vector $Y$ given a linear transformation $\mathcal{T}Y$ is again a multivariate Normal distribution. This result is equivalent to the well-known result that given a $k$-dimensional component of a $n$-dimensional Normal random vector, where $k<n$, the conditional distribution of the remaining $\left(n-k\right)$-dimensional component is a $\left(n-k\right)$-dimensional multivariate Normal distribution, and sets the stage for approximating the conditional distribution of $Y$ given $g\left(Y\right)$, where $g$ is a continuously differentiable vector field.Comment: 2/6/18: Updated the proof of Theorem 4 & added a corollary. arXiv admin note: text overlap with arXiv:1612.0121

A First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle

We study the probability distribution $P(X_N=X,N)$ of the total displacement $X_N$ of an $N$-step run and tumble particle on a line, in presence of a constant nonzero drive $E$. While the central limit theorem predicts a standard Gaussian form for $P(X,N)$ near its peak, we show that for large positive and negative $X$, the distribution exhibits anomalous large deviation forms. For large positive $X$, the associated rate function is nonanalytic at a critical value of the scaled distance from the peak where its first derivative is discontinuous. This signals a first-order dynamical phase transition from a homogeneous `fluid' phase to a `condensed' phase that is dominated by a single large run. A similar first-order transition occurs for negative large fluctuations as well. Numerical simulations are in excellent agreement with our analytical predictions.Comment: 35 pages, 5 figures. An algebraic error in Appendix B of the previous version of the manuscript has been corrected. A new argument for the location $z_c$ of the transition is reported in Appendix B.

Inonu-Wigner Contractions of Kac-Moody Algebras

We discuss In\"on\"u-Wigner contractions of affine Kac-Moody algebras. We show that the Sugawara construction for the contracted affine algebra exists only for a fixed value of the level $k$, which is determined in terms of the dimension of the uncontracted part of the starting Lie algebra, and the quadratic Casimir in the adjoint representation. Further, we discuss contractions of $G/H$ coset spaces, and obtain an affine {\it translation} algebra, which yields a Virasoro algebra (via a GKO construction) with a central charge given by $dim(G/H)$.Comment: 11 pages, IMSc/92-2