On the Lipschitz Constant of the RSK Correspondence

We view the RSK correspondence as associating to each permutation $\pi \in S_n$ a Young diagram $\lambda=\lambda(\pi)$, i.e. a partition of $n$. Suppose now that $\pi$ is left-multiplied by $t$ transpositions, what is the largest number of cells in $\lambda$ that can change as a result? It is natural refer to this question as the search for the Lipschitz constant of the RSK correspondence. We show upper bounds on this Lipschitz constant as a function of $t$. For $t=1$, we give a construction of permutations that achieve this bound exactly. For larger $t$ we construct permutations which come close to matching the upper bound that we prove.Comment: Updated presentation based on comments made by reviewers. Accepted for publication to JCT

Quantum Spectrum Testing

In this work, we study the problem of testing properties of the spectrum of a mixed quantum state. Here one is given $n$ copies of a mixed state $\rho\in\mathbb{C}^{d\times d}$ and the goal is to distinguish whether $\rho$'s spectrum satisfies some property $\mathcal{P}$ or is at least $\epsilon$-far in $\ell_1$-distance from satisfying $\mathcal{P}$. This problem was promoted in the survey of Montanaro and de Wolf under the name of testing unitarily invariant properties of mixed states. It is the natural quantum analogue of the classical problem of testing symmetric properties of probability distributions. Here, the hope is for algorithms with subquadratic copy complexity in the dimension $d$. This is because the "empirical Young diagram (EYD) algorithm" can estimate the spectrum of a mixed state up to $\epsilon$-accuracy using only $\widetilde{O}(d^2/\epsilon^2)$ copies. In this work, we show that given a mixed state $\rho\in\mathbb{C}^{d\times d}$: (i) $\Theta(d/\epsilon^2)$ copies are necessary and sufficient to test whether $\rho$ is the maximally mixed state, i.e., has spectrum $(\frac1d, ..., \frac1d)$; (ii) $\Theta(r^2/\epsilon)$ copies are necessary and sufficient to test with one-sided error whether $\rho$ has rank $r$, i.e., has at most $r$ nonzero eigenvalues; (iii) $\widetilde{\Theta}(r^2/\Delta)$ copies are necessary and sufficient to distinguish whether $\rho$ is maximally mixed on an $r$-dimensional or an $(r+\Delta)$-dimensional subspace; and (iv) The EYD algorithm requires $\Omega(d^2/\epsilon^2)$ copies to estimate the spectrum of $\rho$ up to $\epsilon$-accuracy, nearly matching the known upper bound. In addition, we simplify part of the proof of the upper bound. Our techniques involve the asymptotic representation theory of the symmetric group; in particular Kerov's algebra of polynomial functions on Young diagrams.Comment: 70 pages, 6 figure

Stochastic B\"acklund transformations

How does one introduce randomness into a classical dynamical system in order to produce something which is related to the `corresponding' quantum system? We consider this question from a probabilistic point of view, in the context of some integrable Hamiltonian systems

Viscosity solution of Hamilton-Jacobi equation by a limiting minmax method

For non convex Hamiltonians, the viscosity solution and the more geometric minimax solution of the Hamilton-Jacobi equation do not coincide in general. They are nevertheless related: we show that iterating the minimax procedure during shorter and shorter time intervals one recovers the viscosity solution.Comment: 27 page

Capillarity problems with nonlocal surface tension energies

We explore the possibility of modifying the classical Gauss free energy functional used in capillarity theory by considering surface tension energies of nonlocal type. The corresponding variational principles lead to new equilibrium conditions which are compared to the mean curvature equation and Young's law found in classical capillarity theory. As a special case of this family of problems we recover a nonlocal relative isoperimetric problem of geometric interest.Comment: 37 pages, 4 figure