Moduli spaces of d-connections and difference Painleve equations

We show that difference Painleve equations can be interpreted as isomorphisms of moduli spaces of d-connections on the projective line with given singularity structure. We also derive a new difference equation. It is the most general difference Painleve equation known so far, and it degenerates to both difference Painleve V and classical (differential) Painleve VI equations.Comment: 30 pages (LaTeX

The Ginibre ensemble of real random matrices and its scaling limits

We give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a $2 \times 2$ matrix kernel associated to the ensemble. We apply this result to the real Ginibre ensemble and compute the bulk and edge scaling limits of its correlation functions as the size of the matrices becomes large.Comment: 47 pages, 8 figure

Nonlocal transport in the charge density waves of oo-TaS3_3

We studied the nonlocal transport of a quasi-one dimensional conductor $o$-TaS$_3$. Electric transport phenomena in charge density waves include the thermally-excited quasiparticles, and collective motion of charge density waves (CDW). In spite of its long-range correlation, the collective motion of a CDW does not extend far beyond the electrodes, where phase slippage breaks the correlation. We found that nonlocal voltages appeared in the CDW of $o$-TaS$_3$, both below and above the threshold field for CDW sliding. The temperature dependence of the nonlocal voltage suggests that the observed nonlocal voltage originates from the CDW even below the threshold field. Moreover, our observation of nonlocal voltages in both the pinned and sliding states reveals the existence of a carrier with long-range correlation, in addition to sliding CDWs and thermally-excited quasiparticles.Comment: 8 pages, 4 figure

τ-function of discrete isomonodromy transformations and probability

We introduce the τ-function of a difference rational connection (d-connection) and its isomonodromy transformations. We show that in a continuous limit ourτ-function agrees with the Jimbo–Miwa–Ueno τ-function. We compute the τ-function for the isomonodromy transformations leading to difference Painlevé V and difference Painlevé VI equations. We prove that the gap probability for a wide class of discrete random matrix type models can be viewed as the τ-function for an associated d-connection

Ewens measures on compact groups and hypergeometric kernels

On unitary compact groups the decomposition of a generic element into product of reflections induces a decomposition of the characteristic polynomial into a product of factors. When the group is equipped with the Haar probability measure, these factors become independent random variables with explicit distributions. Beyond the known results on the orthogonal and unitary groups (O(n) and U(n)), we treat the symplectic case. In U(n), this induces a family of probability changes analogous to the biassing in the Ewens sampling formula known for the symmetric group. Then we study the spectral properties of these measures, connected to the pure Fisher-Hartvig symbol on the unit circle. The associated orthogonal polynomials give rise, as $n$ tends to infinity to a limit kernel at the singularity.Comment: New version of the previous paper "Hua-Pickrell measures on general compact groups". The article has been completely re-written (the presentation has changed and some proofs have been simplified). New references added

Quantum complexities of ordered searching, sorting, and element distinctness

We consider the quantum complexities of the following three problems: searching an ordered list, sorting an un-ordered list, and deciding whether the numbers in a list are all distinct. Letting N be the number of elements in the input list, we prove a lower bound of \frac{1}{\pi}(\ln(N)-1) accesses to the list elements for ordered searching, a lower bound of \Omega(N\log{N}) binary comparisons for sorting, and a lower bound of \Omega(\sqrt{N}\log{N}) binary comparisons for element distinctness. The previously best known lower bounds are {1/12}\log_2(N) - O(1) due to Ambainis, \Omega(N), and \Omega(\sqrt{N}), respectively. Our proofs are based on a weighted all-pairs inner product argument. In addition to our lower bound results, we give a quantum algorithm for ordered searching using roughly 0.631 \log_2(N) oracle accesses. Our algorithm uses a quantum routine for traversing through a binary search tree faster than classically, and it is of a nature very different from a faster algorithm due to Farhi, Goldstone, Gutmann, and Sipser.Comment: This new version contains new results. To appear at ICALP '01. Some of the results have previously been presented at QIP '01. This paper subsumes the papers quant-ph/0009091 and quant-ph/000903

Factorizations of Rational Matrix Functions with Application to Discrete Isomonodromic Transformations and Difference Painlev\'e Equations

We study factorizations of rational matrix functions with simple poles on the Riemann sphere. For the quadratic case (two poles) we show, using multiplicative representations of such matrix functions, that a good coordinate system on this space is given by a mix of residue eigenvectors of the matrix and its inverse. Our approach is motivated by the theory of discrete isomonodromic transformations and their relationship with difference Painlev\'e equations. In particular, in these coordinates, basic isomonodromic transformations take the form of the discrete Euler-Lagrange equations. Secondly we show that dPV equations, previously obtained in this context by D. Arinkin and A. Borodin, can be understood as simple relationships between the residues of such matrices and their inverses.Comment: 9 pages; minor typos fixed, journal reference adde

Effects of Particle Shape on Growth Dynamics at Edges of Evaporating Colloidal Drops

We study the influence of particle shape on growth processes at the edges of evaporating drops. Aqueous suspensions of colloidal particles evaporate on glass slides, and convective flows during evaporation carry particles from drop center to drop edge, where they accumulate. The resulting particle deposits grow inhomogeneously from the edge in two-dimensions, and the deposition front, or growth line, varies spatio-temporally. Measurements of the fluctuations of the deposition front during evaporation enable us to identify distinct growth processes that depend strongly on particle shape. Sphere deposition exhibits a classic Poisson like growth process; deposition of slightly anisotropic particles, however, belongs to the Kardar-Parisi-Zhang (KPZ) universality class, and deposition of highly anisotropic ellipsoids appears to belong to a third universality class, characterized by KPZ fluctuations in the presence of quenched disorder