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We investigate the RG-time integration of the effective potential in the functional renormalization group in the presence of spontaneous symmetry breaking and its subsequent convexity restoration on the example of a scalar theory in $d=3$.…
We study the renormalization group evolution up to the fixed point of the lattice topological susceptibility in the 2-d O(3) non-linear sigma-model. We start with a discretization of the continuum topological charge by a local charge…
A learning approach for determining which operator from a class of nonlocal operators is optimal for the regularization of an inverse problem is investigated. The considered class of nonlocal operators is motivated by the use of squared…
Motivated by potential applications to partial differential equations, we develop a theory of fine scales of decay rates for operator semigroups. The theory contains, unifies, and extends several notable results in the literature on decay…
We formulate some problems and conjectures about semigroups of rational functions under composition. The considered problems arise in different contexts, but most of them are united by a certain relationship to the concept of amenability.
We study exact renormalization group equations in the framework of the effective average action. We present analytical solutions for the scale dependence of the potential in a variety of models. These solutions display a rich spectrum of…
This is a lecture note on the renormalization group theory for field theory models based on the dimensional regularization method. We discuss the renormalization group approach to fundamental field theoretic models in low dimensions. We…
We study a $\phi^4$-theory at finite temperature in a finite volume. Quantum, thermal and volume fluctuations are treated with the functional renormalisation group. Specifically, we focus on the interplay of temperature and length scales…
We study exact renormalization group equations in the framework of the effective average action. We present analytical approximate solutions for the scale dependence of the potential in a variety of models. These solutions display a rich…
Quantum fields are generally taken to be operator-valued distributions, linear functionals of test functions into an algebra of operators; here the effective dynamics of an interacting quantum field is taken to be nonlinearly modified by…
The functional renormalization group method is used to take into account the vacuum polarization around localized bound states generated by external potential. The application to Atomic Physics leads to improved Hartree-Fock and Kohn-Sham…
Probabilistic graphical models are a key tool in machine learning applications. Computing the partition function, i.e., normalizing constant, is a fundamental task of statistical inference but it is generally computationally intractable,…
We formalize and study a phenomenon called feature collapse that makes precise the intuitive idea that entities playing a similar role in a learning task receive similar representations. As feature collapse requires a notion of task, we…
Nonrelativistic two-body scattering by a short-ranged potential is studied using the renormalisation group. Two fixed points are identified: a trivial one and one describing systems with a bound state at zero energy. The eigenvalues of the…
Recently, the concept of geometric renormalization group provides a good approach for studying the structural symmetry and functional invariance of complex networks. Along this line, we systematically investigate the finite-size scaling of…
We study the anomalous scaling of the structure functions of a scalar field advected by a random Gaussian velocity field, the Kraichnan model, by means of Functional Renormalization Group techniques. We analyze the symmetries of the model…
We investigate uniqueness problems for an entire function that shares two small functions of finite order with their difference operators. In particular, we give a generalization of a result in $[2]$.
We show with several examples that renormalization group (RG) theory can be used to understand singular and reductive perturbation methods in a unified fashion. Amplitude equations describing slow motion dynamics in nonequilibrium phenomena…
A generalization of the Renormalization Group, which describes order-parameter fluctuations in finite systems, is developed in the specific context of percolation. This ``Stochastic Renormalization Group'' (SRG) expresses statistical…
To take full advantage of experimental facilities such as FRIB for applications to nuclear astrophysics, nuclear structure, and explorations of neutrinos and fundamental symmetries, we need a better understanding of the interplay of…