Search

We present the first non-perturbative large-$N$ calculation of the $\mathcal{N}=1$ supersymmetric (SUSY) $\mathrm{SU}(N)$ Yang-Mills gluino condensate, obtained by means of numerical simulations of the lattice-discretized theory, and exploiting large-$N$ twisted volume reduction. We present two different determinations based, respectively, on the Banks-Casher formula and on the Gell-Mann-Oakes-Renner relation, both giving perfectly consistent results. By matching the lattice and the Novikov-Shifman-Vainshtein-Zakharov (NSVZ) regularization schemes, we are able for the first time to compare numerical and analytic computations. Our most accurate determination gives $\Sigma_{\rm RGI} /\Lambda_{\rm NSVZ}^3 = 1.55(57)$, in very good agreement with the $N$-dependence and the value predicted by the weak coupling instanton-based approach $\Sigma_{\rm RGI} /\Lambda_{\rm NSVZ}^3 = 1$., Comment: Main text: 5 pages, 3 figures. Supplemental Material: 4 pages, 3 figures

When dealing with an orbit determination problem, uncertainties naturally arise from intrinsic errors related to observation devices and approximation models. Following the least squares method and applying approximation schemes such as the differential correction, uncertainties can be geometrically summarized in confidence regions and estimated by confidence ellipsoids. We investigate the asymptotic behaviour of the confidence ellipsoids while the number of observations and the time span over which they are performed simultaneously increase. Numerical evidences suggest that, in the chaotic scenario, the uncertainties decay at different rates whether the orbit determination is set up to recover the initial conditions alone or along with a dynamical or kinematical parameter, while in the regular case there is no distinction. We show how to improve some of the results in \cite{maro.bonanno}, providing conditions that imply a non-faster-than-polynomial rate of decay in the chaotic case with the parameter, in accordance with the numerical experiments. We also apply these findings to well known examples of chaotic maps, such as piecewise expanding maps of the unit interval or affine hyperbolic toral transformations. We also discuss the applicability to intermittent maps.

The strong sphaleron rate, i.e., the rate of real time QCD topological transitions, is a key phenomenological quantity, playing a fundamental role in several physical contexts. In heavy-ion collisions, a non-vanishing rate can lead to the so-called Chiral Magnetic Effect. In early-Universe cosmology, instead, it can be related to the rate of thermal production of QCD axions. In this talk, we present the first reliable fully non-perturbative computation of the strong sphaleron rate in $N_f=2+1$ QCD at the physical point by means of lattice simulations, in a range of temperatures going from 200 MeV to 600 MeV. Our strategy is based on the inversion of lattice correlators via a recently-proposed modified version of the Backus-Gilbert method., Comment: 4 pages, 2 figures, contribution to the 2024 QCD session of the 58th Rencontres de Moriond

We simulate $N_f=2+1$ QCD at the physical point combining open and periodic boundary conditions in a parallel tempering framework, following the original proposal by M. Hasenbusch for $2d$ $\mathrm{CP}^{N-1}$ models, which has been recently implemented and widely employed in $4d$ $\mathrm{SU}(N)$ pure Yang-Mills theories too. We show that using this algorithm it is possible to achieve a sizable reduction of the auto-correlation time of the topological charge in dynamical fermions simulations both at zero and finite temperature, allowing to avoid topology freezing down to lattice spacings as fine as $a \sim 0.02$ fm. Therefore, this implementation of the Parallel Tempering on Boundary Conditions algorithm has the potential to substantially push forward the investigation of the QCD vacuum properties by means of lattice simulations., Comment: 26 pages, 7 figures. v2: expanded discussion about PTBC improvement over standard RHMC, improved estimate of tau_int(Q^2) with standard RHMC at finest lattice spacing, final results and conclusions unchanged, matches accepted version on JHEP

We compute the renormalized Dirac spectral density in $N_f = 2+1$ QCD at physical quark masses, temperature $T = 230$ MeV and system size $L_s = 3.4$ fm. To that end, we perform a point-wise continuum limit of the staggered density in lattice QCD with staggered quarks. We find, for the first time, that a clear infrared structure (IR peak) emerges in the density of Dirac operator describing dynamical quarks. Features of this structure are consistent with those previously attributed to the recently-proposed IR phase of thermal QCD. Our results (i) provide solid evidence that these IR features are stable and physical; (ii) improve the upper bound for IR-phase transition temperature $T_{\mathrm{IR}}$ so that the new window is $200 < T_{\mathrm{IR}} < 230\,$MeV; (iii) support non-restoration of anomalous U$_{\mathrm A}$(1) symmetry (chiral limit) below $T \!=\! 230\,$MeV., Comment: 7 pages, 6 figures

This paper explores the domain of meromorphic extension for the dynamical zeta function associated to a class of one-dimensional differentiable parabolic maps featuring an indifferent fixed point. We establish the connection between this domain and the spectrum of the weighted transfer operators of the induced map. Furthermore, we discuss scenarios where meromorphic extensions occur beyond the confines of the natural disc of convergence of the dynamical zeta function.

We investigate the role of topology on the lattice determination of the $\mathrm{SU}(3)$ strong coupling renormalized via gradient flow. To deal with the topological freezing of standard local algorithms, the definition of the coupling is usually projected onto the zero topological sector. However, it is not obvious that this definition is not biased by the loss of ergodicity. We instead avoid the topological freezing using a novel algorithm, the Parallel Tempering on Boundary Conditions. The comparison with a standard algorithm shows that, even in the case where the latter is severely frozen, one obtains the same projected coupling. Moreover, we show that the two definitions of the coupling, projected and non-projected, lead to the same flow of the renormalization scale. Our results imply that projecting the coupling does not affect the determination of the dynamically-generated scale of the theory $\Lambda$, as obtained through the step-scaling method., Comment: v1: 30 pages, 9 figures. v2: 33 pages, 9 figures, revised and expanded text in Sections 1 and 2, expanded discussion about topological quantities, results and conclusions unchanged, matches accepted version on EPJC

Motivated by the recently-established connection between Jarzynski's equality and the theoretical framework of Stochastic Normalizing Flows, we investigate a protocol relying on out-of-equilibrium lattice Monte Carlo simulations to mitigate the infamous computational problem of topological freezing. We test our proposal on $2d$ $\mathrm{CP}^{N-1}$ models and compare our results with those obtained adopting the Parallel Tempering on Boundary Conditions proposed by M. Hasenbusch, obtaining comparable performances. Our work thus sets the stage for future applications combining our Monte Carlo setup with machine learning techniques., Comment: 1+26 pages, 14 figures; v2: 1+27 pages, 14 figures, discussion in section 4 expanded, matches published version

We study the $\theta$-dependence of the string tension and of the lightest glueball mass in four-dimensional $\mathrm{SU}(N)$ Yang-Mills theories. More precisely, we focus on the coefficients parametrizing the $\mathcal{O}(\theta^2)$ dependence of these quantities, which we investigate by means of numerical simulations of the lattice-discretized theory, carried out using imaginary values of the $\theta$ parameter. Topological freezing at large $N$ is avoided using the Parallel Tempering on Boundary Conditions algorithm. We provide controlled continuum extrapolations of such coefficients in the $N=3$ case, and we report the results obtained on two fairly fine lattice spacings for $N=6$., Comment: 22 pages, 5 figures. v2: minor revisions, matches accepted version on JHEP

We compute the sphaleron rate on the lattice. We adopt a novel strategy based on the extraction of the spectral density via a modified version of the Backus-Gilbert method from finite-lattice-spacing and finite-smoothing-radius Euclidean topological charge density correlators. The physical sphaleron rate is computed by performing controlled continuum limit and zero-smoothing extrapolations both in pure gauge and, for the first time, in full QCD.

We investigate, by means of numerical lattice simulations, the $\theta$-dependence of the critical deconfinement temperature of $\mathrm{SU}(N)$ gauge theories at large $N$: $T_c(\theta) = T_c(0)[1-R\theta^2+O(\theta^4)]$, with $R\sim O(1/N^2)$. We follow two different strategies to determine $R$, one based on the calculation of the latent heat of the transition and on the jump of the topological susceptibility at the $\theta=0$ critical point, the other relying on a direct probe of $T_c(\theta)$ by means of imaginary-$\theta$ Monte Carlo simulations. Our results show that $R$ follows the expected large-$N$ scaling., Comment: 18 pages, 17 figures. v2: results for Tc(theta,N) unchanged, added study of N-dependence of top. susc. chi(T=Tc+), matches accepted version on JHEP

We present a proposal for calculating the running of the coupling constant of the $\mathrm{SU}(3)$ pure-gauge theory, which combines the Twisted Gradient Flow (TGF) renormalization scheme with Parallel Tempering on Boundary Conditions (PTBC). The TGF is a gradient flow-based renormalization scheme formulated in an asymmetric lattice with twisted boundary conditions. Combined with step scaling, it has been successfully used to calculate the $\mathrm{SU}(3)$ $\Lambda$ parameter. As with all gradient flow-based schemes, the coupling constant is highly correlated with the topological charge and affected by topology freezing, an issue addressed by projecting the determination of the coupling onto the zero topological sector. As an alternative to the zero-charge projection, we combine TGF with PTBC, by replicating multiple copies of the same lattice, interpolating between periodic and open boundary conditions in a parallel-tempered manner. We present a first exploration of these ideas by analyzing specific ensembles of $\mathrm{SU}(3)$ lattices with and without PTBC., Comment: 11 pages, 5 figures, 40th International Symposium on Lattice Field Theory, LATTICE2023 Fermilab, Batavia, Illinois, USA, from July 31 to August 4, 2023. V2: text unchanged, some previous figures were not correct and have been updated

We study the effects of taste degeneracy on the continuum scaling of the localization properties of the staggered Dirac operator in high-temperature QCD using numerical simulations on the lattice, focusing in particular on the position of the mobility edge separating localized and delocalized modes at the low end of the spectrum. We find that, if the continuum limit is approached at fixed spatial volume, the restoration of taste symmetry leads to sizeable systematic effects on estimates for the mobility edge obtained from spectral statistics, which become larger and larger as the lattice spacing is decreased. Such systematics, however, are found to decrease if the volume is increased at fixed lattice spacing. We argue that spectral statistics estimate correctly the position of the mobility edge in the thermodynamic limit at fixed spacing, and support this with an independent numerical analysis based directly on the properties of the Dirac eigenvectors, that are unaffected by taste degeneracy. We then provide a theoretical argument justifying the exchange of the thermodynamic and continuum limits when studying localization. This allows us to use spectral statistics to determine the position of the mobility edge, and to obtain a controlled continuum extrapolation of the mobility edge for the first time., Comment: Revised and expanded discussion, includes argument for exchange of thermodynamic and continuum limits for the mobility edge; numerical results unchanged; to appear in PRD; 21 pages, 12 figures

We determine the pure-gauge $\mathrm{SU}(3)$ topological susceptibility slope $\chi^\prime$, related to the next-to-leading-order term of the momentum expansion of the topological charge density 2-point correlator, from numerical lattice Monte Carlo simulations. Our strategy consists in performing a double-limit extrapolation: first we take the continuum limit at fixed smoothing radius, then we take the zero-smoothing-radius limit. Our final result is $\chi^\prime = [17.1(2.1)~\mathrm{MeV}]^2$. We also discuss a theoretical argument to predict its value in the large-$N$ limit, which turns out to be remarkably close to the obtained $N=3$ lattice result., Comment: 21 pages, 8 figures. v2: minor changes, matches accepted version on JHEP

We present results for the large-$N$ limit of the chiral condensate computed from twisted reduced models. We followed a two-fold strategy, one constiting in extracting the condensate from the quark-mass dependence of the pion mass, the other consisting in extracting the condensate from the mode number of the Dirac operator., Comment: 11 pages, 5 figures, contribution for the 40th International Symposium on Lattice Field Theory (Lattice 2023), July 31st - August 4th, 2023, Fermi National Accelerator Laboratory

Calculations of topological observables in lattice gauge theories with traditional Monte Carlo algorithms have long been known to be a difficult task, owing to the effects of long autocorrelations times. Several mitigation strategies have been put forward, including the use of open boundary conditions and methods such as parallel tempering. In this contribution we examine a new approach based on out-of-equilibrium Monte Carlo simulations. Starting from thermalized configurations with open boundary conditions on a line defect, periodic boundary conditions are gradually switched on. A sampling of topological observables is then shown to be possible with a specific reweighting-like technique inspired by Jarzynski's equality. We discuss the efficiency of this approach using results obtained for the 2-dimensional $\mathrm{CP}^{N-1}$ models. Furthermore, we outline the implementation of our proposal in the context of Stochastic Normalizing Flows, as they share the same theoretical framework of the non-equilibrium transformations we perform, and can be thought of as their generalization., Comment: 1+8 pages, 6 figures, contribution for the 40th International Symposium on Lattice Field Theory (Lattice 2023), July 31st - August 4th, 2023, Fermi National Accelerator Laboratory

We compute the large-$N$ limit of the QCD chiral condensate on the lattice using twisted reduced models, and performing controlled continuum and chiral extrapolations. We perform two different calculations: one consists in extracting the chiral condensate from the quark mass dependence of the pion mass, and the other consists in extracting the chiral condensate from the behaviour of the mode number of the Wilson-Dirac operator for small eigenvalues. We find consistency between the results of the two methods, giving a joint estimate of $\lim_{N\to\infty}\Sigma(N)/N=[184(13)$ MeV$]^3$ ($\overline{\mathrm{MS}}$, $\mu=2$ GeV, taking the square root of the string tension $\sqrt{\sigma}=440$ MeV to set the scale), in remarkable agreement with the $\mathrm{SU}(3)$ 2-flavor FLAG result., Comment: 25 pages, 11 figures. v2: minor changes, matches accepted version on JHEP

We compute the sphaleron rate on the lattice from the inversion of the Euclidean time correlators of the topological charge density, performing also controlled continuum and zero-smoothing extrapolations. The correlator inversion is performed by means of a recently-proposed modification of the Backus-Gilbert method., Comment: 8 pages, 11 figures, Proceedings of the 26th international conference in HEP (QCD23), 10-14th july 2023, Montpellier-FR

Recently, a variety of quantum algorithms have been devised to estimate thermal averages on a genuine quantum processor. In this paper, we consider the practical implementation of the so-called Quantum-Quantum Metropolis algorithm. As a testbed for this purpose, we simulate a basic system of three frustrated quantum spins and discuss its systematics, also in comparison with the Quantum Metropolis Sampling algorithm., Comment: 12 pages, 9 figures

We compute the chiral condensate of $2+1$ QCD from the mode number of the staggered Dirac operator, performing controlled extrapolations to both the continuum and the chiral limit. We consider also alternative strategies, based on the quark mass dependence of the topological susceptibility and of the pion mass, and obtain consistent results within errors. Results are also consistent with phenomenological expectations and with previous numerical determinations obtained with different lattice discretizations., Comment: 33 pages, 33 figures. v2: improved presentation of the analysis, results unchanged, matches accepted version on JHEP

We compute the sphaleron rate of $N_f=2+1$ QCD at the physical point for a range of temperatures $200$ MeV $\lesssim T \lesssim 600$ MeV. We adopt a strategy recently applied in the quenched case, based on the extraction of the rate via a modified version of the Backus-Gilbert method from finite-lattice-spacing and finite-smoothing-radius Euclidean topological charge density correlators. The physical sphaleron rate is finally computed by performing a continuum limit at fixed physical smoothing radius, followed by a zero-smoothing extrapolation. Dynamical fermions were discretized using the staggered formulation, which is known to yield large lattice artifacts for the topological susceptibility. However, we find them to be rather mild for the sphaleron rate., Comment: Main text: 7 pages, 4 eps figures. Supplementary Material: 9 pages, 10 eps figures. v2: revised text, results unchanged. v3: minor revisions, matches accepted version on Phys. Rev. Lett

We compute the sphaleron rate in quenched QCD for a temperature $T \simeq 1.24~T_c$ from the inversion of the Euclidean lattice time correlator of the topological charge density. We explore and compare two different strategies: one follows a new approach proposed in this study and consists in extracting the rate from finite lattice spacing correlators, and then in taking the continuum limit at fixed smoothing radius followed by a zero-smoothing extrapolation; the other follows the traditional approach of extracting the rate after performing such double extrapolation directly on the correlator. In both cases the rate is obtained from a recently-proposed modification of the standard Backus-Gilbert procedure. The two strategies lead to compatible estimates within errors, which are then compared to previous results in the literature at the same or similar temperatures; the new strategy permits to obtain improved results, in terms of statistical and systematic uncertainties., Comment: 11 pages, 12 eps figures; v2: 16 pages, 18 eps figures, expanded introduction and discussion of results, results unchanged, 2 appendices and references added

We consider a conservative ergodic measure-preserving transformation $T$ of the measure space $(X,\mathcal{B},\mu)$ with $\mu$ a $\sigma$-finite measure and $\mu(X)=\infty$. Given an observable $g:X\to \mathbb{R}$, it is well known from results by Aaronson that in general the asymptotic behaviour of the Birkhoff sums $S_Ng(x):= \sum_{j=1}^N\, (g\circ T^{j-1})(x)$ strongly depends on the point $x\in X$, and that there exists no sequence $(d_N)$ for which $S_Ng(x)/d_N \to 1$ for $\mu$-almost every $x\in X$. In this paper we consider the case $g\not\in L^1(X,\mu)$ assuming that there exists $E\in\mathcal{B}$ with $\mu(E)<\infty$ and $\int_E g\,\mathrm{d}\mu=\infty$ and continue the investigation initiated in previous work by the authors. We show that for transformations $T$ with strong mixing assumptions for the induced map on a finite measure set, the almost sure asymptotic behaviour of $S_Ng(x)$ for an unbounded observable $g$ may be obtained using two methods, adding a number of summands depending on $x$ to $S_Ng$ and trimming. The obtained sums are then asymptotic to a scalar multiple of $N$. The results are applied to a couple of non-regular continued fraction algorithms, the backward (or R\'enyi type) continued fraction and the even-integer continued fraction algorithms, to obtain the almost sure asymptotic behaviour of the sums of the digits of the algorithms., Comment: 21 pages, 1 figure

Phase transitions in a non-perturbative regime can be studied by ab initio Lattice Field Theory methods. The status and future research directions for LFT investigations of Quantum Chromo-Dynamics under extreme conditions are reviewed, including properties of hadrons and of the hypothesized QCD axion as inferred from QCD topology in different phases. We discuss phase transitions in strong interactions in an extended parameter space, and the possibility of model building for Dark Matter and Electro-Weak Symmetry Breaking. Methodological challenges are addressed as well, including new developments in Artificial Intelligence geared towards the identification of different phases and transitions., Comment: 94 pages, 23 figures, GGI workshop "Phase Transitions in Particle Physics" review - v2: minor revisions and typo corrected, matches accepted version on PPNP

We compute the topological susceptibility slope $\chi^\prime$, related to the second moment of the two-point correlator of the topological charge density, of $2d$ $\mathrm{CP}^{N-1}$ models for $N=5,11,21$ and $31$ from lattice Monte Carlo simulations. Our strategy consists in performing a double limit: first, we take the continuum limit of $\chi^\prime$ at fixed smoothing radius in physical units; then, we take the zero-smoothing-radius limit. Since the same strategy can also be applied to $4d$ gauge theories and full QCD, where $\chi^\prime$ plays an intriguing theoretical and phenomenological role, this work constitutes a step towards the lattice investigation of this quantity in such models., Comment: 10 pages, 19 eps figures; v2: few typos corrected, matches published version

We consider the problem of showing that 1 is an eigenvalue for a family of generalised transfer operators of the Farey map. This problem is related to the spectral theory of the modular surface via the Selberg Zeta function and the theory of dynamical zeta functions of maps. After briefly recalling these connections, we show that the problem can be formulated for operators on an appropriate Hilbert space and translated into a linear algebra problem for infinite matrices., Comment: 12 pages

The QCD topological observables are essential inputs to obtain theoretical predictions about axion phenomenology, which are of utmost importance for current and future experimental searches for this particle. Among them, we focus on the topological susceptibility, related to the axion mass. We present lattice results for the topological susceptibility in QCD at high temperatures obtained by discretizing this observable via spectral projectors on eigenmodes of the staggered Dirac operator, and we compare them with those obtained with the standard gluonic definition. The adoption of the spectral discretization is motivated by the large lattice artifacts affecting the standard gluonic susceptibility, related to the choice of non-chiral fermions in the lattice action., Comment: 6 pages, 3 eps figures, Proceedings of the 41st International Conference on High Energy physics, ICHEP2022, 6th-13th July, 2022, Bologna, Italy

Standard local updating algorithms experience a critical slowing down close to the continuum limit, which is particularly severe for topological observables. In practice, the Markov chain tends to remain trapped in a fixed topological sector. This problem further worsens at large $N$, and is known as $\mathit{topological}~\mathit{freezing}$. To mitigate it, we adopt the parallel tempering on boundary conditions proposed by M. Hasenbusch. This algorithm allows to obtain a reduction of the auto-correlation time of the topological charge up to several orders of magnitude. With this strategy we are able to provide the first computation of low-lying glueball masses at large $N$ free of any systematics related to topological freezing., Comment: 9 pages, 8 eps figures, Proceedings of the 39th International Symposium on Lattice Field Theory, LATTICE2022, 8th-13th August, 2022, Rheinische Friedrich-Wilhelms-Universit\"at Bonn, Bonn, Germany

The Ising critical exponents $\eta$, $\nu$ and $\omega$ are determined up to one-per-thousand relative error in the whole range of dimensions $3 \le d < 4$, using numerical conformal-bootstrap techniques. A detailed comparison is made with results by the resummed epsilon-expansion in varying dimension, the analytic bootstrap, Monte Carlo and non-perturbative renormalization-group methods, finding very good overall agreement. Precise conformal field theory data of scaling dimensions and structure constants are obtained as functions of dimension, improving on earlier findings, and providing benchmarks in $3 \le d < 4$., Comment: 43 pages, 44 pdf figures, 7 tables. v2: further data and comments added. v3: published version with better description of boostrap navigator data and resummation techniques

We compute the topological susceptibility of $N_f=2+1$ QCD with physical quark masses in the high-temperature phase, using numerical simulations of the theory discretized on a space-time lattice. More precisely we estimate the topological susceptibility for five temperatures in the range from $\sim200$ MeV up to $\sim600$ MeV, adopting the spectral projectors definition of the topological charge based on the staggered Dirac operator. This strategy turns out to be effective in reducing the large lattice artifacts which affect the standard gluonic definition, making it possible to perform a reliable continuum extrapolation. Our results for the susceptibility in the explored temperature range are found to be partially in tension with previous determinations in the literature., Comment: 33 pages, 33 eps figures, revised version matches published one (minor corrections)

The topological susceptibility of $2d$ $\mathrm{CP}^{N-1}$ models is expected, based on perturbative computations, to develop a divergence in the limit $N \to 2$, where these models reduce to the well-known non-linear $\mathrm{O}(3)$ $\sigma$-model. The divergence is due to the dominance of instantons of arbitrarily small size and its detection by numerical lattice simulations is notoriously difficult, because it is logarithmic in the lattice spacing. We approach the problem from a different perspective, studying the behavior of the model when the volume is fixed in dimensionless lattice units, where perturbative predictions are turned into more easily checkable behaviors. After testing this strategy for $N = 3$ and $4$, we apply it to $N = 2$, adopting at the same time a multicanonic algorithm to overcome the problem of rare topological fluctuations on asymptotically small lattices. Our final results fully confirm, by means of purely non-perturbative methods, the divergence of the topological susceptibility of the $2d$ $\mathrm{CP}^1$ model., Comment: v1: 20 pages, 26 eps figures; v2: 14 pages, 14 eps figures, revised text, conclusions unchanged, matches published version

We construct a Poincar\'e map $\mathcal{P}_h$ for the positive horocycle flow on the modular surface $PSL(2,\mathbb{Z})\backslash \mathbb{H}$, and begin a systematic study of its dynamical properties. In particular we give a complete characterisation of the periodic orbits of $\mathcal{P}_h$, and show that they are equidistributed with respect to the invariant measure of $\mathcal{P}_h$ and that they can be organised in a tree by using the Stern-Brocot tree of rational numbers. In addition we introduce a time-reparameterisation of $\mathcal{P}_h$ which gives an insight into the dynamics of the non-periodic orbits. This paper constitutes a first step in the study of the dynamical properties of the horocycle flow by purely dynamical methods., Comment: 33 pages, 7 figures

In commonly used Monte Carlo algorithms for lattice gauge theories the integrated autocorrelation time of the topological charge is known to be exponentially-growing as the continuum limit is approached. This $\mathit{topological}\,\,\textit{freezing}$, whose severity increases with the size of the gauge group, can result in potentially large systematics. To provide a direct quantification of the latter, we focus on $\mathrm{SU}(6)$ Yang--Mills theory at a lattice spacing for which conventional methods associated to the decorrelation of the topological charge have an unbearable computational cost. We adopt the recently proposed $\mathit{parallel}\,\,\mathit{tempering}\,\,\mathit{on}\,\,\mathit{boundary}\,\,\mathit{conditions}$ algorithm, which has been shown to remove systematic effects related to topological freezing, and compute glueball masses with a typical accuracy of $2-5\%$. We observe no sizeable systematic effect in the mass of the first lowest-lying glueball states, with respect to calculations performed at nearly-frozen topological sector., Comment: 16 pages, 8 eps figures, revised version matches published one

We consider the shift transformation on the space of infinite sequences over a finite alphabet endowed with the invariant product measure, and examine the presence of a \emph{hole} on the space. The holes we study are specified by the sequences that do not contain a given finite word as initial sub-string. The measure of the set of sequences that do not fall into the hole in the first $n$ iterates of the shift is known to decay exponentially with $n$, and its exponential rate is called \emph{escape rate}. In this paper we provide a complete characterization of the holes with maximal escape rate. In particular we show that, contrary to the case of equiprobable symbols, ordering the holes by their escape rate corresponds to neither the order by their measure nor by the length of the shortest periodic orbit they contain. Finally, we adapt our technique to the case of shifts endowed with Markov measures, where preliminary results show that a more intricate situation is to be expected., Comment: 18 pages, 3 figures. Final version for Discrete and Continuous Dynamical Systems

We present preliminary lattice results for the topological susceptibility in high-$T$ $N_f=2+1$ QCD obtained discretizing this observable via spectral projectors on eigenmodes of the staggered operator, and we compare them with those obtained with the standard gluonic definition. The adoption of the spectral discretization is motivated by the large lattice artifacts affecting the continuum scaling of the gluonic susceptibility at high $T$, related to the choice of non-chiral fermions in the action., Comment: 11 pages, 5 eps figures, proceeding for The 38th International Symposium on Lattice Field Theory, LATTICE2021, 26th-30th July, 2021, Zoom/Gather@Massachusetts Institute of Technology

We show that the additive-slow-Farey version of the traditional continued fractions algorithm has a natural interpretation as a method for producing integer partitions of a positive number $n$ into two smaller numbers, with multiplicity. We provide a complete description of how such integer partitions occur and of the conjugation for the corresponding Young shapes via the dynamics of the classical Farey tree. We use the dynamics of the Farey map to get a new formula for $p(2,n)$, the number of ways for partitioning $n$ into two smaller positive integers, with multiplicity. We then do the analogue using the additive-slow-Farey version of the Triangle map (a type of multi-dimensional continued fraction algorithm), giving us a method for producing integer partitions of a positive number $n$ into three smaller numbers, with multiplicity. However different aspects of this generalisations remain unclear., Comment: 38 pages, 1 figure, Section five substantually changed, necessitating the addition of Baalbaki as a co-author. Also a new section six

We consider a conservative ergodic measure-preserving transformation $T$ of a $\sigma$-finite measure space $(X,\mathcal{B},\mu)$ with $\mu(X)=\infty$. Given an observable $f:X\to \mathbb{R}$ we study the almost sure asymptotic behaviour of the Birkhoff sums $S_Nf(x) := \sum_{j=1}^N\, (f\circ T^{j-1})(x)$. In infinite ergodic theory it is well known that the asymptotic behaviour of $S_Nf(x)$ strongly depends on the point $x\in X$, and if $f\in L^1(X,\mu)$, then there exists no real valued sequence $(b(N))$ such that $\lim_{N\to\infty} S_Nf(x)/b(N)=1$ almost surely. In this paper we show that for dynamical systems with strong mixing assumptions for the induced map on a finite measure set, there exists a sequence $(\alpha(N))$ and $m\colon X\times \mathbb{N}\to\mathbb{N}$ such that for $f\in L^1(X,\mu)$ we have $\lim_{N\to\infty} S_{N+m(x,N)}f(x)/\alpha(N)=1$ for $\mu$-a.e. $x\in X$. Moreover if $f\not\in L^1(X,\mu)$ we give conditions on the induced observable such that there exists a sequence $(G(N))$ depending on $f$, for which $\lim_{N\to\infty} S_{N}f(x)/G(N)=1$ holds for $\mu$-a.e. $x\in X$., Comment: 33 pages. We have improved the section on examples

We call internal-wave billiard the dynamical system of a point particle that moves freely inside a planar domain (the table) and is reflected by its boundary according to this rule: reflections are standard Fresnel reflections but with the pretense that the boundary at any collision point is either horizontal or vertical (relative to a predetermined direction representing gravity). These systems are point particle approximations for the motion of internal gravity waves in closed containers, hence the name. For a class of tables similar to rectangular trapezoids, but with the slanted leg replaced by a general curve with downward concavity, we prove that the dynamics has only three asymptotic regimes: (1) there exist a global attractor and a global repellor, which are periodic and might coincide; (2) there exists a beam of periodic trajectories, whose boundary (if any) comprises an attractor and a repellor for all the other trajectories; (3) all trajectories are dense (that is, the system is minimal). Furthermore, in the prominent case where the table is an actual trapezoid, we study the sets in parameter space relative to the three regimes. We prove in particular that the set for (1) has positive measure (giving a rigorous proof of the existence of Arnol'd tongues for internal-wave billiards), whereas the sets for (2) and (3) are non-empty but have measure zero., Comment: Final preprint for Nonlinearity, 29 pages, 11 figures

We simulate $4d$ $SU(N)$ pure-gauge theories at large $N$ using a parallel tempering scheme that combines simulations with open and periodic boundary conditions, implementing the algorithm originally proposed by Martin Hasenbusch for $2d$ $CP^{N-1}$ models. That allows to dramatically suppress the topological freezing suffered from standard local algorithms, reducing the autocorrelation time of $Q^2$ up to two orders of magnitude. Using this algorithm in combination with simulations at non-zero imaginary $\theta$ we are able to refine state-of-the-art results for the large-$N$ behavior of the quartic coefficient of the $\theta$-dependence of the vacuum energy $b_2$, reaching an accuracy comparable with that of the large-$N$ limit of the topological susceptibility., Comment: 11 pages, 9 eps figures, minor changes and few typos corrected

We deal with the orbit determination problem for hyperbolic maps. The problem consists in determining the initial conditions of an orbit and, eventually, other parameters of the model from some observations. We study the behaviour of the confidence region in the case of simultaneous increase of the number of observations and the time span over which they are performed. More precisely, we describe the geometry of the confidence region for the solution, distinguishing whether a parameter is added to the estimate of the initial conditions or not. We prove that the inclusion of a dynamical parameter causes a change in the rate of decay of the uncertainties, as suggested by some known numerical evidences., Comment: Added new section on comparisons with existing numerical results. Some inaccuracies corrected

We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves. We show that it is nevertheless possible that the motion of the particle is chaotic, also under regularity assumptions for the moving boundary. More precisely, we show that there exists a class of functions describing the motion of the boundary for which the billiard map admits invariant probability measures with positive metric entropy. The proof relies on variational techniques based on Aubry-Mather theory.

We present a systematic numerical study of $\theta$-dependence around $\theta=0$ in the small-$N$ limit of $2d$ $CP^{N-1}$ models, aimed at clarifying the possible presence of a divergent topological susceptibility in the continuum limit. We follow a twofold strategy, based on one side on direct simulations for $N = 2$ and $N = 3$ on lattices with correlation lengths up to $O(10^2)$, and on the other side on the small-$N$ extrapolation of results obtained for $N$ up to $9$. Based on that, we provide conclusive evidence for a finite topological susceptibility at $N = 3$, with a continuum estimate $\xi^2 \chi = 0.110(5)$. On the other hand, results obtained for $N = 2$ are still inconclusive: they are consistent with a logarithmically divergent continuum extrapolation, but do not yet exclude a finite continuum value, $\xi^2 \chi \sim 0.4$, with the divergence taking place for $N$ slightly below 2 in this case. Finally, results obtained for the non-quadratic part of $\theta$-dependence, in particular for the so-called $b_2$ coefficient, are consistent with a $\theta$-dependence matching that of the Dilute Instanton Gas Approximation at the point where $\xi^2 \chi$ diverges., Comment: 15 pages, 17 eps figures, minor changes