Research in recent years has shown that combining finite‐size scaling theory with the transfer ma... more Research in recent years has shown that combining finite‐size scaling theory with the transfer matrix technique yields a powerful tool for the investigation of critical behavior. In particular, the method has been used to study two‐dimensional statistical mechanical and one‐ ...
We show that two very different temperature regimes exist for problems of the travelling salesman... more We show that two very different temperature regimes exist for problems of the travelling salesman type, and that the annealed approximation is valid for the high-temperature regime. Random-link models are introduced, for which upper and lower bounds on the free energy are obtained in the low-temperature regime. A soluble model is presented, which possesses a phase transition strongly reminiscent of the spin-glass transition. Tome 45
Since we know that mountain shapes are natural fractals, the title would be appropriate for a lec... more Since we know that mountain shapes are natural fractals, the title would be appropriate for a lecture on climbing the Mont Blanc, the Aiguille Verte or any of the peaks we can contemplate from Les Houches. My subject is more down to earth and only consists in a short review of recent work on different types of walks on fractal lattices and on random systems such as percolation clusters. The first part is a reminder of geometrical and physical aspects of dimensionality, and discusses why several dimensions are needed to characterize fractal objects. The second part is devoted to “simple” walks, mainly random walks, and gives a short account of classical diffusion and conduction in random media. Several reviews of these aspects have appeared recently, by ALEXANDER |1|, MITESCU and ROUSSENQ |2|, RAMMAL |3| and AHARONY |4|, and the reader is referred to these papers for deeper discussions and for technical points. The following parts deal with “advanced” walks, where additional rules are specified: this leads to a large variety of problems and to a wealth of new physical effects. The examples include random walks in the presence of traps, biased walks (e.g., dispersion of particles in a flow through a porous medium), selfavoiding walks. For all these problems, the lack of translational invariance of the lattice (fractal or random) introduces new features, and even the simplest situations may provide surprises.
One aim of the study of critical phenomena is the calculation of the exponents which describe the... more One aim of the study of critical phenomena is the calculation of the exponents which describe the singularities of the thermodynamic functions at a second order phase transition point. Most of the models of statistical mechanics are not solvable exactly, therefore these exponents are only known approximatively. The three main approaches which allow to calculate the critical exponents are the series expansions, the Monte Carlo methods and the renormalization group theories. The phenomenological renormalization method which was introduced by NIGHTINGALE [1] is a real space renormalization method. The philosophy of the method is to use the fact that one can calculate exactly the thermodynamic properties of one dimensional systems and then from this information obtain the critical properties of systems in higher dimension. Its main advantages are: 1) the method can be used for a large class of models, 2) it gives satisfactory results which can be improved systematically, 3) only one parameter is renormalized.
Polymers, Liquid Crystals, and Low-Dimensional Solids, 1984
The first and most obvious evolution in the study of defects in ordered systems is toward a much ... more The first and most obvious evolution in the study of defects in ordered systems is toward a much greater diversity, but it is counterbalanced by a powerful effort toward unification. The diversity stems from the experimental study of new materials, such as the liquid heliums, liquid crystals, or two-dimensional systems. These materials contain defects which play an important physical role, and are different from the well-known defects of metals and semiconductors. On the other hand, recent theoretical work based on topological concepts has restored unity by providing deeper understanding and a broader framework that includes many a priori unrelated cases (though not all the old classics!). The topological theory also leads to the prediction of new phenomena and makes contact with other fields of physics where similar concepts are used (e.g., solitons or monopoles may be viewed as defects). Here is a beautiful example of how progress occurs in physics!
Abstract An exact solution is given to the usual model for electro-thermal switching in amorphous... more Abstract An exact solution is given to the usual model for electro-thermal switching in amorphous thin films. The results are compared to previous approximate theories, which are shown to introduce spurious effects and lead to wrong qualitative conclusions.
Journal of Physics A: Mathematical and General, 1987
The authors show that for linear polymers a collapse transition exists on several fractal lattice... more The authors show that for linear polymers a collapse transition exists on several fractal lattices and obtain exact results for the critical exponents at this transition. A 'rod-like' phase is found in some cases at intermediate temperatures, between the swollen phase and the collapsed phase. They introduce infinitesimal recursion relations with correlation function rescaling as the formal limit of a class of fractals, which give a better approximation to Euclidean 2D lattices. The gyration radius exponent at the transition temperature lies in the range nu 1=0.546+or-0.010, in good agreement with a recent transfer matrix calculation. The possible relevance of anisotropy at the collapse transition is discussed.
Photoelectron energy distributions are measured on negative electron affinity (NEA) GaP (100) sur... more Photoelectron energy distributions are measured on negative electron affinity (NEA) GaP (100) surfaces for near-band-gap photon energy as a function of the activation conditions with Cs or Cs and O2. X electrons in NEA GaP have a low escape probability unless Cs-O activation is used and the work function reduced to less than 1 eV. Measured energy distribution curves are
Ground state correlations in the Ising model with random ± J bonds are investigated on a square l... more Ground state correlations in the Ising model with random ± J bonds are investigated on a square lattice, as a function of the concentration of negative bonds. By comparing all the ground‐states of (18×18) samples, we study the size distribution of packets of solitary spins. Large packets, containing more than half the spins, are observed in most samples, even at maximum bond disorder. This finding implies a remarkable rigidity of the random frustrated system at zero temperature.
We study the asymptotic properties of large branched polymers (lattice animals) on a fractal latt... more We study the asymptotic properties of large branched polymers (lattice animals) on a fractal lattice, the two-dimensional modified Sierpinski gasket of base b = 3. The generating function displays an essential singularity of the form G(x) ∼ exp [c(xc - x)-ψ], with ψ = ln(3 - √3)/ln(3 + √3) ≃ 0.1527. This is in contrast to the power law singularity found for periodic lattices and some other fractal lattices such as the standard Sierpinski gasket. This result suggests that similar essential singularities may also exist for polymers in nonhomogeneous media.
Research in recent years has shown that combining finite‐size scaling theory with the transfer ma... more Research in recent years has shown that combining finite‐size scaling theory with the transfer matrix technique yields a powerful tool for the investigation of critical behavior. In particular, the method has been used to study two‐dimensional statistical mechanical and one‐ ...
We show that two very different temperature regimes exist for problems of the travelling salesman... more We show that two very different temperature regimes exist for problems of the travelling salesman type, and that the annealed approximation is valid for the high-temperature regime. Random-link models are introduced, for which upper and lower bounds on the free energy are obtained in the low-temperature regime. A soluble model is presented, which possesses a phase transition strongly reminiscent of the spin-glass transition. Tome 45
Since we know that mountain shapes are natural fractals, the title would be appropriate for a lec... more Since we know that mountain shapes are natural fractals, the title would be appropriate for a lecture on climbing the Mont Blanc, the Aiguille Verte or any of the peaks we can contemplate from Les Houches. My subject is more down to earth and only consists in a short review of recent work on different types of walks on fractal lattices and on random systems such as percolation clusters. The first part is a reminder of geometrical and physical aspects of dimensionality, and discusses why several dimensions are needed to characterize fractal objects. The second part is devoted to “simple” walks, mainly random walks, and gives a short account of classical diffusion and conduction in random media. Several reviews of these aspects have appeared recently, by ALEXANDER |1|, MITESCU and ROUSSENQ |2|, RAMMAL |3| and AHARONY |4|, and the reader is referred to these papers for deeper discussions and for technical points. The following parts deal with “advanced” walks, where additional rules are specified: this leads to a large variety of problems and to a wealth of new physical effects. The examples include random walks in the presence of traps, biased walks (e.g., dispersion of particles in a flow through a porous medium), selfavoiding walks. For all these problems, the lack of translational invariance of the lattice (fractal or random) introduces new features, and even the simplest situations may provide surprises.
One aim of the study of critical phenomena is the calculation of the exponents which describe the... more One aim of the study of critical phenomena is the calculation of the exponents which describe the singularities of the thermodynamic functions at a second order phase transition point. Most of the models of statistical mechanics are not solvable exactly, therefore these exponents are only known approximatively. The three main approaches which allow to calculate the critical exponents are the series expansions, the Monte Carlo methods and the renormalization group theories. The phenomenological renormalization method which was introduced by NIGHTINGALE [1] is a real space renormalization method. The philosophy of the method is to use the fact that one can calculate exactly the thermodynamic properties of one dimensional systems and then from this information obtain the critical properties of systems in higher dimension. Its main advantages are: 1) the method can be used for a large class of models, 2) it gives satisfactory results which can be improved systematically, 3) only one parameter is renormalized.
Polymers, Liquid Crystals, and Low-Dimensional Solids, 1984
The first and most obvious evolution in the study of defects in ordered systems is toward a much ... more The first and most obvious evolution in the study of defects in ordered systems is toward a much greater diversity, but it is counterbalanced by a powerful effort toward unification. The diversity stems from the experimental study of new materials, such as the liquid heliums, liquid crystals, or two-dimensional systems. These materials contain defects which play an important physical role, and are different from the well-known defects of metals and semiconductors. On the other hand, recent theoretical work based on topological concepts has restored unity by providing deeper understanding and a broader framework that includes many a priori unrelated cases (though not all the old classics!). The topological theory also leads to the prediction of new phenomena and makes contact with other fields of physics where similar concepts are used (e.g., solitons or monopoles may be viewed as defects). Here is a beautiful example of how progress occurs in physics!
Abstract An exact solution is given to the usual model for electro-thermal switching in amorphous... more Abstract An exact solution is given to the usual model for electro-thermal switching in amorphous thin films. The results are compared to previous approximate theories, which are shown to introduce spurious effects and lead to wrong qualitative conclusions.
Journal of Physics A: Mathematical and General, 1987
The authors show that for linear polymers a collapse transition exists on several fractal lattice... more The authors show that for linear polymers a collapse transition exists on several fractal lattices and obtain exact results for the critical exponents at this transition. A 'rod-like' phase is found in some cases at intermediate temperatures, between the swollen phase and the collapsed phase. They introduce infinitesimal recursion relations with correlation function rescaling as the formal limit of a class of fractals, which give a better approximation to Euclidean 2D lattices. The gyration radius exponent at the transition temperature lies in the range nu 1=0.546+or-0.010, in good agreement with a recent transfer matrix calculation. The possible relevance of anisotropy at the collapse transition is discussed.
Photoelectron energy distributions are measured on negative electron affinity (NEA) GaP (100) sur... more Photoelectron energy distributions are measured on negative electron affinity (NEA) GaP (100) surfaces for near-band-gap photon energy as a function of the activation conditions with Cs or Cs and O2. X electrons in NEA GaP have a low escape probability unless Cs-O activation is used and the work function reduced to less than 1 eV. Measured energy distribution curves are
Ground state correlations in the Ising model with random ± J bonds are investigated on a square l... more Ground state correlations in the Ising model with random ± J bonds are investigated on a square lattice, as a function of the concentration of negative bonds. By comparing all the ground‐states of (18×18) samples, we study the size distribution of packets of solitary spins. Large packets, containing more than half the spins, are observed in most samples, even at maximum bond disorder. This finding implies a remarkable rigidity of the random frustrated system at zero temperature.
We study the asymptotic properties of large branched polymers (lattice animals) on a fractal latt... more We study the asymptotic properties of large branched polymers (lattice animals) on a fractal lattice, the two-dimensional modified Sierpinski gasket of base b = 3. The generating function displays an essential singularity of the form G(x) ∼ exp [c(xc - x)-ψ], with ψ = ln(3 - √3)/ln(3 + √3) ≃ 0.1527. This is in contrast to the power law singularity found for periodic lattices and some other fractal lattices such as the standard Sierpinski gasket. This result suggests that similar essential singularities may also exist for polymers in nonhomogeneous media.
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Papers by J. Vannimenus