From Asymptotic Series to Self-Similar Approximants

The review presents the development of an approach of constructing approximate solutions to complicated physics problems, starting from asymptotic series, through optimized perturbation theory, to self-similar approximation theory. The close interrelation of underlying ideas of these theories is emphasized. Applications of the developed approach are illustrated by typical examples demonstrating that it combines simplicity with good accuracy.

Limit profiles for reversible Markov chains

In a recent breakthrough, Teyssier (Ann Probab 48(5):2323–2343, 2020) introduced a new method for approximating the distance from equilibrium of a random walk on a group. He used it to study the limit profile for the random transpositions card shuffle. His techniques were restricted to conjugacy-invariant random walks on groups; we derive similar approximation lemmas for random walks on homogeneous spaces and for general reversible Markov chains. We illustrate applications of these lemmas to some famous problems: the k-cycle shuffle, sharpening results of Hough (Probab Theory Relat Fields 165(1–2):447–482, 2016) and Berestycki, Schramm and Zeitouni (Ann Probab 39(5):1815–1843, 2011), the Ehrenfest urn diffusion with many urns, sharpening results of Ceccherini-Silberstein, Scarabotti and Tolli (J Math Sci 141(2):1182–1229, 2007), a Gibbs sampler, which is a fundamental tool in statistical physics, with Binomial prior and hypergeometric posterior, sharpening results of Diaconis, Khare and Saloff-Coste (Stat Sci 23(2):151–178, 2008).

DENSE STABLE RANK AND RUNGE-TYPE APPROXIMATION THEOREMS FOR MAPS

Let $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk ${\mathbb {D}}\subset {{\mathbb C}}$ . We show that the dense stable rank of $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ is $1$ and, using this fact, prove some nonlinear Runge-type approximation theorems for $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for the algebra $H^\infty ({\mathbb {D}})$ .

A note on the Fröhlich dynamics in the strong coupling limit

We revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in Griesemer (Rev Math Phys 29(10):1750030, 2017). In the latter it was shown that the Fröhlich time evolution applied to the initial state $$\varphi _0 \otimes \xi _\alpha $$ φ 0 ⊗ ξ α , where $$\varphi _0$$ φ 0 is the electron ground state of the Pekar energy functional and $$\xi _\alpha $$ ξ α the associated coherent state of the phonons, can be approximated by a global phase for times small compared to $$\alpha ^2$$ α 2 . In the present note we prove that a similar approximation holds for $$t=O(\alpha ^2)$$ t = O ( α 2 ) if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to $$\alpha ^{-2}$$ α - 2 and quadratic in creation and annihilation operators. Our result implies that the electron ground state remains close to its initial state for times of order $$\alpha ^2$$ α 2 , while the phonon fluctuations around the coherent state $$\xi _\alpha $$ ξ α can be described by a time-dependent Bogoliubov transformation.

Distributed Graph Diameter Approximation

We present an algorithm for approximating the diameter of massive weighted undirected graphs on distributed platforms supporting a MapReduce-like abstraction. In order to be efficient in terms of both time and space, our algorithm is based on a decomposition strategy which partitions the graph into disjoint clusters of bounded radius. Theoretically, our algorithm uses linear space and yields a polylogarithmic approximation guarantee; most importantly, for a large family of graphs, it features a round complexity asymptotically smaller than the one exhibited by a natural approximation algorithm based on the state-of-the-art Δ-stepping SSSP algorithm, which is its only practical, linear-space competitor in the distributed setting. We complement our theoretical findings with a proof-of-concept experimental analysis on large benchmark graphs, which suggests that our algorithm may attain substantial improvements in terms of running time compared to the aforementioned competitor, while featuring, in practice, a similar approximation ratio.

Self-similar extrapolation of nonlinear problems from small-variable to large-variable limit

Complicated physical problems are usually solved by resorting to perturbation theory leading to solutions in the form of asymptotic series in powers of small parameters. However, finite, and even large values of the parameters, are often of main physical interest. A method is described for predicting the large-variable behavior of solutions to nonlinear problems from the knowledge of only their small-variable expansions. The method is based on self-similar approximation theory resulting in self-similar factor approximants. The latter can well approximate a large class of functions, rational, irrational, and transcendental. The method is illustrated by several examples from statistical and condensed matter physics, where the self-similar predictions can be compared with the available large-variable behavior. It is shown that the method allows for finding the behavior of solutions at large variables when knowing just a few terms of small-variable expansions. Numerical convergence of approximants is demonstrated.

Strong approximation in h-mass of rectifiable currents under homological constraint

Let {h:\mathbb{R}\to\mathbb{R}_{+}} be a lower semicontinuous subbadditive and even function such that {h(0)=0} and {h(\theta)\geq\alpha|\theta|} for some {\alpha>0} . If {T=\tau(M,\theta,\xi)} is a k-rectifiable chain, its h-mass is defined as \mathbb{M}_{h}(T):=\int_{M}h(\theta)\,d\mathcal{H}^{k}. Given such a rectifiable flat chain T with {\mathbb{M}_{h}(T)<\infty} and {\partial T} polyhedral, we prove that for every {\eta>0} , it decomposes as {T=P+\partial V} with P polyhedral, V rectifiable, {\mathbb{M}_{h}(V)<\eta} and {\mathbb{M}_{h}(P)<\mathbb{M}_{h}(T)+\eta} . In short, we have a polyhedral chain P which strongly approximates T in h-mass and preserves the homological constraint {\partial P=\partial T} . When {h^{\prime}(0^{+})} is well defined and finite, the definition of the h-mass extends as a finite functional on the space of finite mass k-chains (not necessarily rectifiable). We prove in this case a similar approximation result for finite mass k-chains with polyhedral boundary. These results are motivated by the study of approximations of {\mathbb{M}_{h}} by smoother functionals but they also provide explicit formulas for the lower semicontinuous envelope of {T\mapsto\mathbb{M}_{h}(T)+\mathbb{I}_{\partial S}(\partial T)} with respect to the topology of the flat norm.

Describing phase transitions in field theory by self-similar approximants

Self-similar approximation theory is shown to be a powerful tool for describing phase transitions in quantum field theory. Self-similar approximants present the extrapolation of asymptotic series in powers of small variables to the arbitrary values of the latter, including the variables tending to infinity. The approach is illustrated by considering three problems: (i) The influence of the coupling parameter strength on the critical temperature of the O(N)-symmetric multicomponent field theory. (ii) The calculation of critical exponents for the phase transition in the O(N)-symmetric field theory. (iii) The evaluation of deconfinement temperature in quantum chromodynamics. The results are in good agreement with the available numerical calculations, such as Monte Carlo simulations, Padé-Borel summation, and lattice data.

Inapproximability of Rank, Clique, Boolean, and Maximum Induced Matching-Widths under Small Set Expansion Hypothesis

Wu et al. (2014) showed that under the small set expansion hypothesis (SSEH) there is no polynomial time approximation algorithm with any constant approximation factor for several graph width parameters, including tree-width, path-width, and cut-width (Wu et al. 2014). In this paper, we extend this line of research by exploring other graph width parameters: We obtain similar approximation hardness results under the SSEH for rank-width and maximum induced matching-width, while at the same time we show the approximation hardness of carving-width, clique-width, NLC-width, and boolean-width. We also give a simpler proof of the approximation hardness of tree-width, path-width, and cut-widththan that of Wu et al.

Extrapolation of perturbation-theory expansions by self-similar approximants

The problem of extrapolating asymptotic perturbation-theory expansions in powers of a small variable to large values of the variable tending to infinity is investigated. The analysis is based on self-similar approximation theory. Several types of self-similar approximants are considered and their use in different problems of applied mathematics is illustrated. Self-similar approximants are shown to constitute a powerful tool for extrapolating asymptotic expansions of different natures.