The asymptotics of monotone subsequences of involutions

We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to infinity. The resulting distributions are, depending on the number of fixed points, (1) the Tracy-Widom distributions for the largest eigenvalues of random GOE, GUE, GSE matrices, (2) the normal distribution, or (3) new classes of distributions which interpolate between pairs of the Tracy-Widom distributions. We also consider the second rows of the corresponding Young diagrams. In each case the convergence of moments is also shown. The proof is based on the algebraic work of the authors in \cite{PartI} which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials, and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the Riemann-Hilbert analysis for the orthogonal polynomials by Deift, Johansson and the first author in [BDJ].Comment: LaTex, 65 pages, 8 figures, Abstract and Introduction rewritten. More Comments are added to Sec. 3,5 and 1

Algebraic aspects of increasing subsequences

We present a number of results relating partial Cauchy-Littlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as new proofs of old formulae; relations of these expressions to orthogonal polynomials on the unit circle; and explicit bases for invariant spaces of the classical groups, together with appropriate generalizations of the straightening algorithm.Comment: LaTeX+amsmath+eepic; 52 pages. Expanded introduction, new references, other minor change

Top guns may not fire:Best-shot group contests with group-specific public good prizes

We analyze a group contest in which n groups compete to win a group-specific public good prize. Group sizes can be different and any player may value the prize differently within and across groups. Players exert costly efforts simultaneously and independently. Only the highest effort (the best-shot) within each group represents the group effort that determines the winning group. We fully characterize the set of equilibria and show that in any equilibrium at most one player in each group exerts strictly positive effort. There always exists an equilibrium in which only the highest value player in each active group exerts strictly positive effort. However, perverse equilibria may exist in which the highest value players completely free-ride on others by exerting no effort. We provide conditions under which the set of equilibria can be restricted and discuss contest design implications

The International Urban Energy Balance Comparison Project: Initial Results from Phase 2.

Many urban land surface schemes have been developed, incorporating different assumptions about the features of, and processes occurring at, the surface. Here, the first results from Phase 2 of an international comparison are presented. Evaluation is based on analysis of the last 12 months of a 15 month dataset. In general, the schemes have best overall capability to model net all-wave radiation. The models that perform well for one flux do not necessarily perform well for other fluxes. Generally there is better performance for net all wave radiation than sensible heat flux. The degree of complexity included in the models is outlined, and impacts on model performance are discussed in terms of the data made available to modellers at four successive stages

On the joint distribution of the maximum and its position of the Airy2 process minus a parabola

The maximal point of the Airy2 process minus a parabola is believed to describe the scaling limit of the end-point of the directed polymer in a random medium, which was proved to be true for a few specific cases. Recently two different formulas for the joint distribution of the location and the height of this maximal point were obtained, one by Moreno Flores, Quastel and Remenik, and the other by Schehr. The first formula is given in terms of the Airy function and an associated operator, and the second formula is expressed in terms of the Lax pair equations of the Painleve II equation. We give a direct proof that these two formulas are the same.Comment: 15 pages, no figure, minor revision, to appear in J.Math.Phy

Dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition

The one-dimensional totally asymmetric simple exclusion process (TASEP) is considered. We study the time evolution property of a tagged particle in TASEP with the step-type initial condition. Calculated is the multi-time joint distribution function of its position. Using the relation of the dynamics of TASEP to the Schur process, we show that the function is represented as the Fredholm determinant. We also study the scaling limit. The universality of the largest eigenvalue in the random matrix theory is realized in the limit. When the hopping rates of all particles are the same, it is found that the joint distribution function converges to that of the Airy process after the time at which the particle begins to move. On the other hand, when there are several particles with small hopping rate in front of a tagged particle, the limiting process changes at a certain time from the Airy process to the process of the largest eigenvalue in the Hermitian multi-matrix model with external sources.Comment: 48 pages, 8 figure

Polynuclear growth model, GOE2^2 and random matrix with deterministic source

We present a random matrix interpretation of the distribution functions which have appeared in the study of the one-dimensional polynuclear growth (PNG) model with external sources. It is shown that the distribution, GOE$^2$, which is defined as the square of the GOE Tracy-Widom distribution, can be obtained as the scaled largest eigenvalue distribution of a special case of a random matrix model with a deterministic source, which have been studied in a different context previously. Compared to the original interpretation of the GOE$^2$ as ``the square of GOE'', ours has an advantage that it can also describe the transition from the GUE Tracy-Widom distribution to the GOE$^2$. We further demonstrate that our random matrix interpretation can be obtained naturally by noting the similarity of the topology between a certain non-colliding Brownian motion model and the multi-layer PNG model with an external source. This provides us with a multi-matrix model interpretation of the multi-point height distributions of the PNG model with an external source.Comment: 27pages, 4 figure