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Numerical studies involving random copolymers and random surfaces assume self-averaging of thermodynamic and metric properties of the systems to calculate different properties. For the problem of adsorption of a random copolymer, rigorous proofs regarding self-averaging of some properties such as free energy in the thermodynamic limit (n → ∞) exist. This says little about the extent of self-averaging for finite size systems used in numerical studies. For the problem of adsorption of a homopolymer on a random surface, no analytical proofs regarding self-averaging exist. In this work assumptions of self-averaging of thermodynamic and metric properties of a self-avoiding walk model of random copolymer adsorption are tested via multiple Markov chain Monte Carlo method. Numerical evidence is provided in support of self-averaging of energy, heat capacity and the z-component of the self-avoiding walk in different temperature intervals. Self-averaging in energy of a homopolymer interacting with a random surface is also examined.

We investigate how martingale techniques can be used to derive information on the extent of self-averaging of the free energy for some lattice models of finite random copolymers.

We discuss self-averaging of thermodynamic properties in some random lattice models. In particular, we investigate when self-averaging (in an almost sure sense) of the free energy implies self-averaging of the energy and heat capacity, and we discuss the connection between self-averaging in the almost sure sense, and self-averaging in an L p sense. Under quite general conditions we show that the average of the finite size heat capacity converges to the second derivative of the limiting quenched average free energy. We consider the application of these ideas to the problem of adsorption of a random copolymer at a surface, and to some related systems.

We investigate the extent of thermodynamic self-averaging in a coloured self-avoiding walk model of finite random copolymer adsorption. We derive a bound on the extent of self-averaging as a function of the length of the self-avoiding walk.

A constructive proof is given that certain problems in the statistical mechanics of random copolymers are thermodynamically self-averaging. The proof relies on the use of normal numbers, and potentially gives a method for calculating expectations in the quenched averaged ensemble. In particular, we give a new proof that adsorbing random copolymers are self-averaging.

We prove that a lattice polygon model of a self-interacting random ring copolymer is thermodynamically self-averaging. The proof is quite general and applies to any two-body potential linear in the numbers of contacts of different types.

We give a set of conditions under which a system is thermodynamically self-averaging and show that several lattice models of interacting copolymers satisfy these conditions. We prove this result for a general potential which is linear in the numbers of various types of contacts, and show that this includes two potentials which have previously been used in models of random interacting linear copolymers.

We prove the almost sure convergence to zero of the fluctuations of the free energy, resp. local free energies, in a class of disordered mean-field spin systems that generalize the Hopfield model in two ways: 1) Multi-spin interactions are permitted and 2) the random variables ξμi i describing the "patterns" can have arbitrary distributions with mean zero and finite 4+∈-th moments. The number of patterns, M, is allowed to be an arbitrary multiple of the systemsize. This generalizes a previous result of Bovier, Gayrard, and Picco [BGP3] for the standard Hopfield model, and improves a result of Feng and Tirozzi [FT] that required M to be a finite constant. Note that the convergence of the mean of the free energy is not proven.