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Spatial random graphs capture several important properties of real-world networks. We prove quenched results for the continuous-space version of scale-free percolation introduced in [14]. This is an undirected inhomogeneous random graph whose vertices are given by a Poisson point process in$\mathbb{R}^d$. Each vertex is equipped with a random weight, and the probability that two vertices are connected by an edge depends on their weights and on their distance. Under suitable conditions on the parameters of the model, we show that, for almost all realizations of the point process, the degree distributions of all the nodes of the graph follow a power law with the same tail at infinity. We also show that the averaged clustering coefficient of the graph isself-averaging. In particular, it is almost surely equal to the annealed clustering coefficient of one point, which is a strictly positive quantity.

In a disordered system, a quantity is self-averaging when the ratio between its variance for disorder realizations and the square of its mean decreases as the system size increases. Here, we consider a chaotic disordered many-body quantum system and search for a relationship between self-averaging behavior and the properties of the distributions over disorder realizations of various quantities and at different timescales. An exponential distribution, as found for the survival probability at long times, explains its lack of self-averaging, since the mean and the dispersion are equal. Gaussian distributions, however, are obtained for both self-averaging and non-self-averaging quantities. Our studies show also that one can make conclusions about the self-averaging behavior of one quantity based on the distribution of another related quantity. This strategy allows for semianalytical results, and thus circumvents the limitations of numerical scaling analysis, which are restricted to few system sizes., 13 pages, 5 figures

In this paper we give an elementary approach to several results of Chatterjee in arXiv:0907.3381 and arXiv:1404.7178, as well as some generalizations. First, we prove quenched disorder chaos for the bond overlap in the Edwards-Anderson type models with Gaussian disorder. The proof extends to systems at different temperatures and covers a number of other models, such as the mixed $p$-spin model, Sherrington-Kirkpatrick model with multi-dimensional spins and diluted $p$-spin model. Next, we adapt the same idea to prove quenched self-averaging of the bond magnetization for one system and use it to show quenched self-averaging of the site overlap for random field models with positively correlated spins. Finally, we show self-averaging for certain modifications of the random field itself., 17 pages

The study of quenched random systems is facilitated by the idea that the ensemble averages describe the thermal averages for any specific realization of the couplings, provided that the system is large enough. Careful examination suggests that this idea might have a flaw, when the correlation length becomes of the order of the size of the system. We find that certain bounded quantities are not self-averaging when the correlation length becomes of the order of the size of the system. This suggests that the lack of self-averaging, expressed in terms of properly chosen signal-to-noise ratios, may serve to identify phase boundaries. This is demonstrated by using such signal-to-noise ratios to identify the boundary of the ferromagnetic phase of the random field Ising system and compare the findings with more traditional measures.

[1] We study dispersion in heterogeneous porous media for solutes evolving from point-like and extended source distributions in d=2 and d=3 spatial dimensions. The impact of heterogeneity on the dispersion behavior is captured by a stochastic modeling approach that represents the spatially fluctuating flow velocity as a spatial random field. We focus here on the sample-to-sample fluctuations of the dispersion coefficients about their ensemble mean. For finite source sizes, the definition of dispersion coefficients in single realizations is not unique. We consider dispersion measures that describe the extension of the solute distribution, as well as dispersion coefficients that quantify the solute spreading relative to injection points of the partial plumes that constitute the solute distribution. While the ensemble averages of these dispersion quantities may be identical, their fluctuation behavior is found to be different. Using a perturbation approach in the fluctuations of the random flow field, we derive explicit expressions for the temporal evolution of the variances of the dispersion coefficients between realizations. Their evolution is governed by the typical dispersion time over the characteristic heterogeneity scale and the dimensions of the source distribution. We find that the dispersion variance decreases toward zero with time in d=3 spatial dimensions, while in d=2 it converges toward a finite long time value that is independent of the source dimensions.

Methods and treatments of random media have been discussed. Conditions for self-averaging are provided. Main characteristics of random media are introduced.

We consider the random Schrodinger equation as it arises in the paraxial regime for wave propagation in random media. In the white noise limit it becomes the Ito–Schrodinger stochastic partial differential equation which we analyze here in the high frequency regime. We also consider the large lateral diversity limit where the typical width of the propagating beam is large compared to the correlation length of the random medium. We use the Wigner transform of the wave field and show that it becomes deterministic in the large diversity limit when integrated against test functions. This is the self-averaging property of the Wigner transform. It follows easily when the support of the test functions is of the order of the beam width. We also show with a more detailed analysis that the limit is deterministic when the support of the test functions tends to zero but is large compared to the correlation length.

A systematic derivations of self-averaging scaling limits of parabolic waves in terms of the Wigner distribution function is presented. The convergence of the Wigner distribution to one of the six deterministic radiative transfer equations is established. One of the main contributions of this Note is a unified framework for space–time scaling limits that lead to radiative transfer. To cite this article: A.C. Fannjiang, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

We consider several types of scaling limits for the Wigner-Moyal equation of the parabolic waves in random media, the limiting cases of which include the standard radiative transfer limit, the geometrical-optics limit and the white-noise limit. We show under fairly general assumptions on the random refractive index field that sufficient amount of medium diversity (thus excluding the white-noise limit) leads to statistical stability or self-averaging in the sense that the limiting law is deterministic and is governed by one of the 6 different types of transport (Boltzmann or Fokker-Planck) equations depending on the specific scaling involved. We discuss the connection to the statistical stability of time-reversal procedure and the decoherence effect in quantum mechanics.

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 analyze the self-averaging properties of time-reversed solutions of the paraxial wave equation with random coefficients, which we take to be Markovian in the direction of propagation. This allows us to construct an approximate martingale for the phase space Wigner transform of two wave fields. Using a prioriL2-bounds available in the time-reversal setting, we prove that the Wigner transform in the high frequency limit converges in probability to its deterministic limit, which is the solution of a transport equation.

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.

We investigate the long-time asymptotics of the fluctuation SPDE in the Kuramoto synchronization model. We establish the linear behavior for large time and weak disorder of the quenched limit fluctuations of the empirical measure of the particles around its McKean-Vlasov limit. This is carried out through a spectral analysis of the underlying unbounded evolution operator, using arguments of perturbation of self-adjoint operators and analytic semigroups. We state in particular a Jordan decomposition of the evolution operator which is the key point in order to show that the fluctuations of the disordered Kuramoto model are not self-averaging., Comment: 34 pages, 5 figures; V2: introduction simplified, references added and typos corrected

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.

SUMMARY We consider the time evolution of seismic primary arrivals in single realizations of randomly heterogeneous media. Using the Rytov approximation, we construct the Green's function of an initially plane wave propagating in 2-D and 3-D weakly heterogeneous fluids and solids. Our approach is a 2-D and 3-D extension of the dynamic-equivalent medium description of wave propagation in 1-D heterogeneous media, known also as the generalized O’Doherty–Anstey (ODA) formalism. The Green's function is constructed by using averaged logarithmic wavefield attributes and depends on the second-order statistics of the medium heterogeneities. Green's functions constructed in this way describe the primary arrivals in single most probable realizations of seismograms. Similar to the attenuation coefficient and phase increment of transmissivities in one dimension, the logarithmic wavefield attributes in two and three dimensions also demonstrate self-averaging, restricted mainly to the weak fluctuation range, however. We show how to derive the statistical approximations and discuss their limitations. We also show that in the limit of long travel distances (Fraunhofer approximation) the Green's function tends to attain the universal form of a Gaussian pulse. In addition, we compare the outcome of finite difference experiments with the theoretically predicted wavefield and find a good agreement: the statistical approximations presented give a smooth version of the primary arrivals. A statistical analysis of the simulated wavefield allows us to identify the most probable seismograms whose primaries are well predicted by the ODA formalism. In addition, we formulate the traveltime-corrected averaging from first principles. We discuss the relationship between our approach and approaches based on the traveltime-corrected formalism. Strictly speaking, such approaches are not appropriate to describe wavefields in most probable realizations; the generalized O’Doherty–Anstey formalism, however, is.

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.

Standard large deviation estimates or the use of the Hubbard-Stratonovich transformation reduce the analysis of the distribution of the overlap parameters essentially to that of an explicitly known random function $\Phi_{N,\b}$ on $\R^M$. In this article we present a rather careful study of the structure of the minima of this random function related to the retrieval of the stored patterns. We denote by $m^*(\b)$ the modulus of the spontaneous magnetization in the Curie-Weiss model and by $\a$ the ratio between the number of the stored patterns and the system size. We show that there exist strictly positive numbers $0, Comment: 43 pages, uuencoded, Z-compressed Postscript

20 pages. Proceedings for the PSPDE-II conference (Braga, dec 2013).; International audience; We review some recent results on large population behavior of interacting diffusions in a random environment. Emphasis is put on the quenched influence of the environment on the macroscopic behavior of the system (law of large numbers and fluctuations). We address the notion of (non-)self-averaging phenomenon for this class of models. A guiding thread in this survey is the Kuramoto synchronization model which has met in recent years a growing interest in the literature.

To pass from a deterministic dynamics of aggregate quantities to a probabilistic dynamics of a system of microvariables that describe the individual strategies of a population of economic agents, the route is that of Boltzmann's kinetic theory at the half of XIX century (more suitable than that of Gibbs' statistical mechanics), that is the introduction of n "elements" (molecules, agents,...), submitted to some microdynamics, wherefrom to derive the macroscopic behavior. The macrovariate is interpreted as a (time) mean of the average (on all elements) of the individual study-property at time t. The micro-derivation looks unproblematic if means and averages tend to constant values in the limit n -> ?. If this property, defined "self-averaging" in some recent papers by Aoki, holds, it would separate a deterministic result from fluctuations; consequently well defined macroeconomic deterministic relations prevail. However it is easy to show that in most cases in economy this property does not hold, due to long-range correlation existing among economic agents. If individual agents are not independent but exchangeable, also in the limit n -> ? the coefficient of variation of the macrovariable is finite, which tends to a random limit rather than a constant. Finally the term "indistinguishable agent" is criticized, and the alternative "exchangeable agent" is discussed.

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.

Using a high performance computer cluster, we run simulations regarding an open problem about d-dimensional critical branching random walks in a random IID environment The environment is given by the rule that at every site independently, with probability p>0, there is a cookie, completely suppressing the branching of any particle located there. Abstract. The simulations suggest self averaging: the asymptotic survival probability in n steps is the same in the annealed and the quenched case; it is \frac{2}{qn}, where q:=1-p. This particular asymptotics indicates a non-trivial phenomenon: the tail of the survival probability (both in the annealed and the quenched case) is the same as in the case of non-spatial unit time critical branching, where the branching rule is modified: branching only takes place with probability q for every particle at every iteration.

An approach to the definition of infinite-volume Gibbs states for the (quenched) random-field Ising model is considered in the case of a Curie-Weiss ferromagnet. It turns out that these states are random quasi-free measures. They are random convex linear combinations of the free product-measures “shifted” by the corresponding effective mean fields. The conditional self-averaging property of the magnetization related to this randomness is also discussed.

The Pitman-Yor, or Chinese Restaurant Process, is a stochastic process that generates distributions following a power-law with exponents lower than two, as found in a numerous physical, biological, technological and social systems. We discuss its rich behavior with the tools and viewpoint of statistical mechanics. We show that this process invariably gives rise to a condensation, i.e. a distribution dominated by a finite number of classes. We also evaluate thoroughly the finite-size effects, finding that the lack of stationary state and self-averaging of the process creates realization-dependent cutoffs and behavior of the distributions with no equivalent in other statistical mechanical models., Comment: (5pages, 1 figure)

A class of two-parameter Poisson-Dirichlet distributions have non-vanishing coefficient of variation. This phenomenon is also known as non-self averaging stochastic multi-sector endogenous growth model. This model is used to raise questions on the use of means in assessing effectiveness of macroeconomic or other macro-policy effects. The coefficients of variations of the number of total sectors, and of sectors of a given size all remain positive as the model size grows unboundedly.

We establish the averaging property for a queuing process with one server, M(t)/GI/1. It is a new relation between the output flow rate and the input flow rate, crucial in the study of the Poisson Hypothesis. Its implications include the statement that the output flow always possesses more regularity than the input flow., 18 pages, one typo removed

We apply the recently developed critical minimum energy subspace scheme for the investigation of the random-field Ising model. We point out that this method is well suited for the study of this model. The density of states is obtained via the Wang-Landau and broad histogram methods in a unified implementation by employing the N-fold version of the Wang-Landau scheme. The random-fields are obtained from a bimodal distribution ($h_{i}=\pm2$), and the scaling of the specific heat maxima is studied on cubic lattices with sizes ranging from $L=4$ to $L=32$. Observing the finite-size scaling behavior of the maxima of the specific heats we examine the question of saturation of the specific heat. The lack of self-averaging of this quantity is fully illustrated and it is shown that this property may be related to the question mentioned above., 8 pages, 7 figures, extended version with two new figures, version as accepted for publication to Physical Review E

This paper is devoted to a detailed study of a p-spins interaction model with external field, including some sharp bounds on the speed of self averaging of the overlap as well as a central limit theorem for its fluctuations, the thermodynamical limit for the free energy and the definition of an Almeida-Thouless type line. Those results show that the external field dominates the tendency to disorder induced by the increasing level of interaction between spins, and our system will share many of its features with the SK model, which is certainly not the case when the external magnetic field vanishes., Comment: 57 pages

We develop a generalization of the Thouless-Anderson-Palmer (TAP) mean-field approach of disorder physics. which makes the method applicable to the computation of approximate averages in probabilistic models for real data. In contrast to the conventional TAP approach, where the knowledge of the distribution of couplings between the random variables is required, our method adapts to the concrete set of couplings. We show the significance of the approach in two ways: Our approach reproduces replica symmetric results for a wide class of toy models (assuming a nonglassy phase) with given disorder distributions in the thermodynamic limit. On the other hand, simulations on a real data model demonstrate that the method achieves more accurate predictions as compared to conventional TAP approaches.

Using the newly proposed probability-changing cluster (PCC) Monte Carlo algorithm, we simulate the two-dimensional (2D) site-diluted Ising model. Since we can tune the critical point of each random sample automatically with the PCC algorithm, we succeed in studying the sample-dependent ${T}_{c}(L)$ and the sample average of physical quantities at each ${T}_{c}(L)$ systematically. Using the finite-size scaling (FSS) analysis for ${T}_{c}(L),$ we discuss the importance of corrections to FSS both in the strong-dilution and weak-dilution regions. The critical phenomena of the 2D site-diluted Ising model are shown to be controlled by the pure fixed point. The crossover from the percolation fixed point to the pure Ising fixed point with the system size is explicitly demonstrated by the study of the Binder parameter. We also study the distribution of critical temperature ${T}_{c}(L).$ Its variance shows the power-law L dependence, ${L}^{\ensuremath{-}n},$ and the estimate of the exponent n is consistent with the prediction of Aharony and Harris [Phys. Rev. Lett. 77, 3700 (1996)]. Calculating the relative variance of critical magnetization at the sample-dependent ${T}_{c}(L),$ we show that the 2D site-diluted Ising model exhibits weak self-averaging.

The Longest Common Subsequence (LCS) Problem asks for the longest sequence of (non-contiguous) matches between two given strings of characters. Using extensive Monte Carlo simulations, we find a finite size scaling law of the form E(L)/N =C + A/(N^1/2 ln N)+... for the mean LCS length of two random strings of size N over S letters. We provide precise estimates of C for S between 2 and 15. We consider also a related Bernoulli Matching model where the different entries of an N times M array are independently occupied with probability 1/S. In that case we find the expression of the limit of L(N,M)/N as N grows to infinity, as a function of r=M/N. This expression provides a very good approximation for the Random String model, which gets more and more accurate as S increases. The question of the ``universality class'' of the LCS problem is also considered. For the Bernoulli Matching model we find very good agreement with recent scaling predictions of Hwa and Lassig for Needleman-Wunsch sequence alignment. We find however that the variance of the LCS length has a different scaling different in the Random String model, suggesting that long-ranged correlations among the matches are relevant in this model. We finally study the ``ground state'' properties of this problem. We find in particular that the number of solutions typically grows exponentially with N, i.e. this system has a residual entropy at T=0. Also the overlap between two LCSs chosen at random is found to be self averaging and to aproach a definite value q(S), 17 pages, 12 figures. Accepted for publication in EPJ B

The distributions $P(X)$ of singular thermodynamic quantities in an ensemble of quenched random samples of linear size $l$ at the critical point $T_c$ are studied by Monte Carlo in two models. Our results confirm predictions of Aharony and Harris based on Renormalization group considerations. For an Ashkin-Teller model with strong but irrelevant bond randomness we find that the relative squared width, $R_X$, of $P(X)$ is weakly self averaging. $R_X\sim l^{\alpha/\nu}$, where $\alpha$ is the specific heat exponent and $\nu$ is the correlation length exponent of the pure model fixed point governing the transition. For the site dilute Ising model on a cubic lattice, known to be governed by a random fixed point, we find that $R_X$ tends to a universal constant independent of the amount of dilution (no self averaging). However this constant is different for canonical and grand canonical disorder. We study the distribution of the pseudo-critical temperatures $T_c(i,l)$ of the ensemble defined as the temperatures of the maximum susceptibility of each sample. We find that its variance scales as $(\delta T_c(l))^2 \sim l^{-2/\nu}$ and NOT as $\sim l^{-d}. We find that $R_\chi$ is reduced by a factor of $\sim 70$ with respect to $R_\chi (T_c)$ by measuring $\chi$ of each sample at $T_c(i,l)$. We analyze correlations between the magnetization at criticality $m_i(T_c,l)$ and the pseudo-critical temperature $T_c(i,l)$ in terms of a sample independent finite size scaling function of a sample dependent reduced temperature $(T-T_c(i,l))/T_c$. This function is found to be universal and to behave similarly to pure systems., Comment: 31 pages, 17 figures, submitted to Phys. Rev. E

Conditions are given under which one may prove that the stochastic dynamics of on-line learning can be described by the deterministic evolution of a finite set of order parameters in the thermodynamic limit. A global constraint on the average magnitude of the increments in the stochastic process is necessary to ensure self-averaging. In the absence of such a constraint, convergence may only be in probability., Comment: 10 pages

We study the distribution of (1+1)-dimensional self-affine lines resulting from the zero-temperature directed polymer in a random medium or from critical directed percolation problems. The related amplitude ratios depend on whether the finite size scaling statistics is made over finite segments of the infinite objects or over the finite objects of diverging size. As a by-product we obtain the correction to scaling exponents for the finite size scaling of the distribution functions.

We prove a large deviation principle for the finite dimensional marginals of the Gibbs distribution of the macroscopic `overlap'-parameters in the Hopfield model in the case where the number of random patterns, $M$, as a function of the system size $N$ satisfies $\limsup M(N)/N=0$. In this case the rate function (or free energy as a function of the overlap parameters) is independent of the disorder for almost all realization of the patterns and given by an explicit variational formula., 31pp; Plain-TeX, hardcopy available on request from bovier@iaas-berlin.d400.de

We investigate self-averaging properties in the transport of particles through random media. We show rigorously that in the subdiffusive anomalous regime transport coefficients are not self--averaging quantities. These quantities are exactly calculated in the case of directed random walks. In the case of general symmetric random walks a perturbative analysis around the Effective Medium Approximation (EMA) is performed., 4 pages, RevTeX , No figures, submitted to Physical Review E (Rapid Communication)

We consider the Hopfield model with M(N) = αN patterns, where N is the number of neurons. We show that if α is sufficiently small and the temperature sufficiently low, then there exist disjoint Gibbs states for each of the stored patterns, almost surely with respect to the distribution of the random patterns. This solves a problem left open in previous work [BGPl]. The key new ingredient is a self averaging result on the free energy functional. This result has considerable additional interest and some consequences are discussed. A similar result for the free energy of the Sherrington-Kirkpatrick model is also given.

International audience; We show that for the problem of directed polymers on a tree with i.i.d. random complex weights on each bond, three possible phases can exist; the phase of a particular system is determined by the distribution ρ of the random weights. For each of these three phases, we give the expression of the free energy per unit length in the limit of infinitely long polymers. Our proofs require several hypotheses on the distribution ρ, most importantly, that the amplitude and the phase of each complex weight be statistically independent. The main steps of our proofs use bounds on noninteger moments of the partition function and self averaging properties of the free energy. We illustrate our results by some examples and discuss possible generalizations to a larger class of distributions, to Random Energy Models, and to the finite dimensional case. We note that our results are not in agreement with the predictions of a recent replica approach to a similar problem.

We show that in the limitp→ ∞,N → 0,α=p/N → 0 the limit free energy of the Hopfield model equals in probability the Curie-Weiss free energy. We prove also that the free energy of the Hopfield model is self-averaging for any finite ∞.

In our recent paper [Phys. Rev. E 58, 1789 (1998)] we found notable deviations from a power-law decay for the ``magnetization'' of a system of coupled phase oscillators with random interactions claimed by Daido in Phys. Rev. Lett. 68, 1072 (1992). For another long-time property, the Lyaponov exponent, we found that his numerical procedure showed strong time discretization effects and we suspected a similar effect for the algebraic decay. In the Comment to our paper [preceding paper, Phys. Rev. E 61, 2145 (2000)] Daido made clear that the power law behavior was only claimed for the sample averaged magnetization [Z] and he presented new, more accurate numerical results which provide evidence for a power-law decay of this quantity. Our results, however, were obtained for Z itself and not for $[Z].$ In addition, we have taken the intrinsic oscillator frequencies as Gaussian random variables, while Daido in his new and apparently also in his earlier simulations used a deterministic approximation to the Gaussian distribution. Due to the differences in the observed quantity and the model assumptions our and Daido's results may be compatible.

We study minimal vertex covers of trees. Contrarily to the number $N_{vc}(A)$ of minimal vertex covers of the tree $A$, $\log N_{vc}(A)$ is a self-averaging quantity. We show that, for large sizes $n$, $\lim_{n\to +\infty} _n/n= 0.1033252\pm 10^{-7}$. The basic idea is, given a tree, to concentrate on its degenerate vertices, that is those vertices which belong to some minimal vertex cover but not to all of them. Deletion of the other vertices induces a forest of totally degenerate trees. We show that the problem reduces to the computation of the size distribution of this forest, which we perform analytically, and of the average $$ over totally degenerate trees of given size, which we perform numerically.

We present new exact results for a one-dimensional asymmetric disordered hopping model. The lattice is taken infinite from the start and we do not resort to the periodization scheme used by Derrida. An explicit resummation allows for the calculation of the velocityV and the diffusion constantD (which are found to coincide with those given by Derrida) and for demonstrating thatV is indeed a self-averaging quantity; the same property is established forD in the limiting case of a directed walk.