Schramm–Loewner evolution (SLE) is a stochastic process that helps classify critical statistical models using one real parameter κ. Numerical study of SLE often involves curves that start and end on the real axis. To reduce numerical errors in studying the critical curves which start from the real axis and end on it, we have used hydrodynamically normalized SLE(κ, ρ) which is a stochastic differential equation governing such curves. In this paper, we directly verify this hypothesis and numerically apply this formalism to the domain wall curves of the Abelian sandpile model (κ = 2) and critical percolation (κ = 6). We observe that this method is more reliable than previously used methods in the literature for analyzing interface loops.

For the quantum Ising model with ferromagnetic random couplings J _{i, j} > 0 and random transverse fields h _i > 0 at zero temperature in finite dimensions d > 1, we consider the lowest order contributions in perturbation theory in ( J _{i, j}/ h _i) to obtain some information on the statistics of various observables in the disordered phase. We find that the two-point correlation scales as , where ξ _typ is the typical correlation length, u is a random variable and ω coincides with the droplet exponent ω _DP( D = d − 1) of the directed polymer with D = ( d − 1) transverse directions. Our main conclusions are as follows. (i) Whenever ω > 0, the quantum model is governed by...

Recently, Chau (2011 Quantum Inform. Comput. 11 721) showed that one can define certain metrics and pseudo-metrics on U( n), the group of all n × n unitary matrices, based on the arguments of the eigenvalues of the unitary matrices. More importantly, these metrics and pseudo-metrics have quantum information theoretical meanings. So it is instructive to study this kind of metrics and pseudo-metrics on U( n). Here we show that any symmetric norm on induces a metric on U( n). Furthermore, using the same technique, we prove an inequality concerning the eigenvalues of a product of two unitary matrices which generalizes a few inequalities obtained earlier by Chau (arXiv:1006.3614v1).

We investigate the properties of two- and three-dimensional non-commutative fermion gases with fixed total z-component of angular momentum, J _z, and at high density for the simplest form of non-commutativity involving constant spatial commutators. Analytic expressions for the entropy and pressure are found. The entropy exhibits non-extensive behaviour, while the pressure reveals the presence of incompressibility in two, but not in three dimensions. Remarkably, for two-dimensional systems close to the incompressible density, the entropy is proportional to the square root of the system size, i.e. for such systems the number of microscopic degrees of freedom is determined by the circumference rather than the area (size) of the system. The absence of incompressibility in three dimensions, and subsequently also the absence of a scaling law for the entropy analogous to the one found in two dimensions, is attributed to the form of the non-commutativity used here,...

We derive asymptotics for the moments as well as the weak limit of the height distribution of watermelons with p branches with wall. This generalizes a famous result of de Bruijn et al (1972 Graph Theory and Computing (New York: Academic) pp 15–22) on the average height of planted plane trees, and results by Fulmek (2007 Electron. J. Combin. 14 R64) and Katori et al (2008 J. Stat. Phys. 131 1067–83) on the expected value and higher moments, respectively, of the height distribution of watermelons with two branches. The asymptotics for the moments depend on the analytic behaviour of certain multidimensional Dirichlet series. In order to obtain this information, we prove a reciprocity relation satisfied by the derivatives of one of Jacobi’s theta functions, which generalizes the well-known reciprocity law for Jacobi’s theta functions.

For the most general off-shell supersymmetric σ-model in projective superspace, we elaborate on its formulation in terms of chiral superfields. A universal (model-independent) expression is obtained for the holomorphic symplectic two-form, which determines the second supersymmetry transformation. This two-form is associated with the two complex structures of the hyperkähler target space, which are complimentary to the one used to realize the target space as a Kähler manifold.

The corner transfer matrix renormalization group method is an efficient method for evaluating physical quantities in statistical mechanical models. It originates from Baxter’s corner transfer matrix equations and method, and was developed by Nishino and Okunishi in 1996. In this paper, we review and adapt this method, previously used for numerical calculations, to derive series expansions. We use this to calculate 92 terms of the partition function of the hard-squares model. We use the resulting series to provide evidence supporting the claim that the method is subexponential in the number of generated terms, and briefly analyse the singularities of the function.

Recently, the present author and Zia (2011 Europhys. Lett. 91 50003) reported on exact results for a far-from-equilibrium system in which two coupled semi-infinite Ising chains at temperatures T _h and T _c, with T _h > T _c, establish a flux of energy across their junction. This paper provides a complete derivation of those results, more explicit expressions for the energy flux, and a more detailed characterization of the system at arbitrary T _c and T _h. We consider the two-point correlation functions and the energy flux F( x) between each spin, located at integer position x, and its associated heat bath. In the T _h → ∞ limit, the flux F( x) decays exponentially into the cold bath (spins with x = 1, 2, …) for all T _c > 0 and transitions into a power-law decay as T _c → 0. We...

We consider Gaussian ensembles of m complex  N × N matrices. We identify an enhanced symmetry in the system and the resultant closed subsector, which is naturally associated with the radial sector of the theory. The density of radial eigenvalues is obtained in the large N limit. It is of the Wigner form only for m = 1. For m ⩾ 2, a new form of the density is obtained.

An integrable coupled short pulse (CSP) equation is proposed for the propagation of ultra-short pulses in optical fibers. Based on two sets of bilinear equations to a two-dimensional Toda lattice linked by a Bäcklund transformation, and an appropriate hodograph transformation, the proposed CSP equation is derived. Meanwhile, its N-soliton solutions are given by the Casorati determinant in a parametric form. The properties of one- and two-soliton solutions are investigated in detail. Same as the short pulse equation, the two-soliton solution turns out to be a breather type if the wave numbers are complex conjugate. We also illustrate an example of soliton–breather interaction.