THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.

Theory of Functions

These notes give a summary of techniques used in large deviation theory to study the fluctuations of time-additive quantities, called dynamical observables, defined in the context of Langevin-type equations, which model equilibrium and nonequilibrium processes driven by external forces and noise sources. These fluctuations are described by large deviation functions, obtained by solving a dominant eigenvalue problem similar to the problem of finding the ground state energy of quantum systems. This analogy is used to explain the differences that exist between the fluctuations of equilibrium and nonequilibrium processes. An example involving the Ornstein–Uhlenbeck process is worked out in detail to illustrate these methods. Exercises, at the end of the notes, also complement the theory.

Introduction to dynamical large deviations of Markov processes

Order-disorder phenomena

(2 + 1) [XTpXpH12, CTH11]. + [Zuc11b]. 0 [Fed17]. 1 [BELP15, CAS11, Cor16, Fed17, GDL10, GBL16, Hau16, JV19, KT12, KM19c, Li19, MN14b, Nak17, Pal11, Pan14, RT14, RBS16b, RY12, SS18c, Sug10, dOP18]. 1 + 1 [Sak18, CP15b]. 1/2 [MD10]. 1/f [FDR12]. 1/n [Per17]. 1/|x− y| [MSV10, MSV13]. 13 [DFL17]. 1 ≤ p ≤ ∞ [Dud13]. 1/f [HPF15]. 2 [AB19, ADS19, BF12, BNT13, DSS15, EKD12, Her13, Ily12, Lan10, Li12, Li19, LZ11, Ny13, Ost16, PSS16, ST14, Sch13b, TJ15, WPB15, dWL10]. 2 + 1 [dWL14]. 2.5 [BC15a]. 2R [WLEC17]. 2× 2 [CLTT13]. 3 [BCF19, BLS17, ESPP14, Kar18, SH16, SWKS14, dCCS19]. 3/2 [DK10]. 38 [Cam13]. 4 [BBS14, Zha14]. 4× 4 [LN19a]. 5/2 [DK10, EKD12]. 6 [EC11]. 8 [Zha14]. 90◦ [YM11]. 3 [Afz12]. 1−x [EFO11]. 13 [CDCL18]. 2 [ML15, QR13, ST11c]. 4 [HBB10]. 6 [BCL10a, BCL10b, EFO11]. x [EFO11].

/pdf/a-complete-bibliography-of-the-journal-of-statistical-2bw4esmzh4.pdf

A Complete Bibliography of the Journal of Statistical Physics: 2000{2009

Lecture 2. Renormalisation group in hierarchical models 21 2.1. Massless Gaussian Measure 21 2.2. Finite Range Decompositions 21 2.3. Motivation 23 2.4. The Renormalisation Group and Hierarchical Models 24 2.5. Hierarchical Models 25 2.6. The Formal Infinite Volume Limit and Trivial Fixed Point 28 2.7. Analysis 29 2.8. Expanding Directions, Relevant Operators 29 2.9. Trivial Fixed Point 30 2.10. Appendix. Stable Manifold Theorem 31

/pdf/lectures-on-the-renormalisation-group-3brq01imey.pdf

Lectures on the Renormalisation Group

The Griffiths singularities are fully exhibited for a class of diluted ferromagnetic Ising models defined on the Cayley tree (Bethe lattice). For the deterministic model the Lee-Yang circle theorem is explicitly proven for the magnetization at the origin and it is shown that, in the thermodynamic limit, the Lee-Yang singularities become dense in the entire unit circle for the whole ferromagnetic phase. Smoothness (infinite differentiability) of the quenched magnetizationm at the origin with respect to the external magnetic field is also proven for convenient choices of temperature and disorder. From our analysis we also conclude that the existence of metastable states is impossible for the random models under consideration.

Griffiths singularities in diluted ising models on the cayley tree

Asymptotic Time Decay In Quantum Physics

We give a rigorous proof of power-law falloff in the Kosterlitz-Thouless phase of a two-dimensional Coulomb gas in the sense that there exists a critical inverse temperaturegb and a constant Θ>0 such that for allβ>β and all external charges ξ∈R we have
 
$$G_\xi (x) \leqslant C/|x|^{\Theta \beta \eta ^2 } $$

 , whereG

 ξ
 (x) is the two-point external charges correlation function,η=dist(ξ, Z), and
 
$$C = C(\beta , \bar \eta )< \infty $$

 for
 
$$\bar \eta = dist (\xi , Z - \{ 0\} ) > 0$$

 . In the case of a hard-core or standard Coulomb gas with activityz, we may chooseβ=β(z) such thatβ(z)→24π asz↘0.

Power-law falloff in the kosterlitz-Thouless phase of a two-dimensional lattice Coulomb gas

A hierarchical version of two-dimensional lattice Coulomb gases is investigated. Forβ>β
c=8π there is a locally stable line of fixed points for the renormalization group (“block charges”) transformations. For
 
$$\beta > \bar \beta _c (\beta _c \leqslant \bar \beta _c \leqslant \tfrac{3}{2}\pi \beta _c )$$

 , these fixed points are globally stable. As a consequence we show that there is no screening of external charges for any activity if
 
$$\beta > \bar \beta _c $$

 . Atβ

 c
 a supercritical bifurcation takes place and we investigate the behavior of the model forβ≲β

 c
 to show a weak form of screening.

The Kosterlitz-Thouless Phase Transition in Two-Dimensional Hierarchical Coulomb Gases

We consider two-dimensional Coulomb gases (standard or hard-core) with activityz, and prove that for any β>8π the two-point external charges correlation function exhibits the power-law falloff characteristic of the Kosterlitz-Thouless phase at sufficiently low activity.

Domingos H. U. Marchetti

Papers

Griffiths singularities in diluted ising models on the cayley tree

Asymptotic Time Decay In Quantum Physics

Power-law falloff in the kosterlitz-Thouless phase of a two-dimensional lattice Coulomb gas

The Kosterlitz-Thouless Phase Transition in Two-Dimensional Hierarchical Coulomb Gases

Power-law falloff in two-dimensional Coulomb gases at inverse temperature β>8π