We consider a random walk on the fully-connected lattice with $N$ sites and study the time evolution of the number of distinct sites $s$ visited by the walker on a subset with $n$ sites. A record value $v$ is obtained for $s$ at a record time $t$ whe
n the walker visits a site of the subset for the first time. The record time $t$ is a partial covering time when $v<n$ and a total covering time when $v=n$. The probability distributions for the number of records $s$, the record value $v$ and the record (covering) time $t$, involving $r$-Stirling numbers, are obtained using generating function techniques. The mean values, variances and skewnesses are deduced from the generating functions. In the scaling limit the probability distributions for $s$ and $v$ lead to the same Gaussian density. The fluctuations of the record time $t$ are also Gaussian at partial covering, when $n-v={mathrm O}(n)$. They are distributed according to the type-I Gumbel extreme-value distribution at total covering, when $v=n$. A discrete sequence of generalized Gumbel distributions, indexed by $n-v$, is obtained at almost total covering, when $n-v={mathrm O}(1)$. These generalized Gumbel distributions are crossing over to the Gaussian distribution when $n-v$ increases.
The probability distribution of the number $s$ of distinct sites visited up to time $t$ by a random walk on the fully-connected lattice with $N$ sites is first obtained by solving the eigenvalue problem associated with the discrete master equation. T
hen, using generating function techniques, we compute the joint probability distribution of $s$ and $r$, where $r$ is the number of sites visited only once up to time $t$. Mean values, variances and covariance are deduced from the generating functions and their finite-size-scaling behaviour is studied. Introducing properly centered and scaled variables $u$ and $v$ for $r$ and $s$ and working in the scaling limit ($ttoinfty$, $Ntoinfty$ with $w=t/N$ fixed) the joint probability density of $u$ and $v$ is shown to be a bivariate Gaussian density. It follows that the fluctuations of $r$ and $s$ around their mean values in a finite-size system are Gaussian in the scaling limit. The same type of finite-size scaling is expected to hold on periodic lattices above the critical dimension $d_{rm c}=2$.
We study the time evolution of the local magnetization in the critical Ising chain in a transverse field after a sudden change of the parameters at a defect. The relaxation of the defect magnetization is algebraic and the corresponding exponent, whic
h is a continuous function of the defect parameters, is calculated exactly. In finite chains the relaxation is oscillating in time and its form is conjectured on the basis of precise numerical calculations.
For a classical system of noninteracting particles we establish recursive integral equations for the density of states on the microcanonical ensemble. The recursion can be either on the number of particles or on the dimension of the system. The solut
ion of the integral equations is particularly simple when the single-particle density of states in one dimension follows a power law. Otherwise it can be obtained using a Laplace transform method. Since the Laplace transform of the microcanonical density of states is the canonical partition function, it factorizes for a system of noninteracting particles and the solution of the problem is straightforward. The results are illustrated on several classical examples.
We consider the Ising model on the Bethe lattice with aperiodic modulation of the couplings, which has been studied numerically in Phys. Rev. E 77, 041113 (2008). Here we present a relevance-irrelevance criterion and solve the critical behavior exact
ly for marginal aperiodic sequences. We present analytical formulae for the continuously varying critical exponents and discuss a relationship with the (surface) critical behavior of the aperiodic quantum Ising chain.
We consider two critical semi-infinite subsystems with different critical exponents and couple them through their surfaces. The critical behavior at the interface, influenced by the critical fluctuations of the two subsystems, can be quite rich. In o
rder to examine the various possibilities, we study a system composed of two coupled Ashkin-Teller models with different four-spin couplings epsilon, on the two sides of the junction. By varying epsilon, some bulk and surface critical exponents of the two subsystems are continuously modified, which in turn changes the interface critical behavior. In particular we study the marginal situation, for which magnetic critical exponents at the interface vary continuously with the strength of the interaction parameter. The behavior expected from scaling arguments is checked by DMRG calculations.
