Searcharxiv⌕ Search

arXiv subjects

Matteo Giordano

Publications and source records attributed to Matteo Giordano.

At least 55 records · Page 3Linked to original sources
arXiv2022

Nonparametric Bayesian inference for reversible multi-dimensional diffusions

We study nonparametric Bayesian models for reversible multi-dimensional diffusions with periodic drift. For continuous observation paths, reversibility is exploited to prove a general posterior contraction rate theorem for the drift gradient vector field under approximation-theoretic conditions on the induced prior for the invariant measure. The general theorem is applied to Gaussian priors and $p$-exponential priors, which are shown to converge to the truth at the minimax optimal rate over Sobolev smoothness classes in any dimension.

math.ST↗
arXiv2022

Exponential reduction of the sign problem at finite density in the 2+1D XY model via contour deformations

We study the 2+1 dimensional XY model at nonzero chemical potential $μ$ on deformed integration manifolds, with the aim of alleviating its sign problem. We investigate several proposals for the deformations, and considerably improve on the severity of the sign problem with respect to standard reweighting approaches. We present numerical evidence that the reduction of the sign problem is exponential both in $μ^2$ and in the spatial volume. We also present a new approach to the optimization procedure based on reweighting, that sensibly reduces its computational cost.

hep-lat↗
arXiv2022

Radius of convergence in lattice QCD at finite $μ_B$ with rooted staggered fermions

In typical statistical mechanical systems the grand canonical partition function at finite volume is proportional to a polynomial of the fugacity $e^{μ/T}$. The zero of this Lee-Yang polynomial closest to the origin determines the radius of convergence of the Taylor expansion of the pressure around $μ=0$. The computationally cheapest formulation of lattice QCD, rooted staggered fermions, with the usual definition of the rooted determinant, does not admit such a Lee-Yang polynomial. We argue that the radius of convergence is then bounded by the spectral gap of the reduced matrix of the unrooted staggered operator. This is a cutoff effect that potentially affects all estimates of the radius of convergence with the standard staggered rooting. We suggest a new definition of the rooted staggered determinant at finite chemical potential that allows for a definition of a Lee-Yang polynomial, and, therefore of the numerical study of Lee-Yang zeros. We also describe an algorithm to determine the Lee-Yang zeros and apply it to configurations generated with the 2-stout improved staggered action at $N_t = 4$. We perform a finite-volume scaling study of the leading Lee-Yang zeros and estimate the radius of convergence of the Taylor expansion extrapolated to an infinite volume. We show that the limiting singularity is not on the real line, thus giving a lower bound on the location of any possible phase transitions at this lattice spacing. In the vicinity of the crossover temperature at zero chemical potential, the radius of convergence turns out to be $μ_B/T \approx 2$ and roughly temperature independent. Our simulations are performed at strange quark chemical potential $μ_s=0$, but the method can be straightforwardly extended to strangeness chemical potential $μ_S=0$ or strangeness neutrality.

hep-lat↗
arXiv2022

Lattice simulations of the QCD chiral transition at real $μ_B$

Most lattice studies of hot and dense QCD matter rely on extrapolation from zero or imaginary chemical potentials. The ill-posedness of numerical analytic continuation puts severe limitations on the reliability of such methods. We studied the QCD chiral transition at finite real baryon density with the more direct sign reweighting approach. We simulate up to a baryochemical potential-temperature ratio of $μ_B/T=2.7$, covering the RHIC Beam Energy Scan range, and penetrating the region where methods based on analytic continuation are unpredictive.This opens up a new window to study QCD matter at finite $μ_B$ from first principles.

hep-lat↗
arXiv2021

New approach to lattice QCD at finite density: reweighting without an overlap problem

Approaches to finite baryon density lattice QCD usually suffer from uncontrolled systematic uncertainties in addition to the well-known sign problem. We test a method - sign reweighting - that works directly at finite chemical potential and is yet free from any such uncontrolled systematics: with this approach the only problem is the sign problem itself. In practice the approach involves the generation of configurations with the positive fermionic weights given by the absolute value of the real part of the quark determinant, and a reweighting by a sign. There are only two sectors, +1 and -1 and as long as the average $\left\langle \pm \right\rangle \neq 0$ (with respect to the positive weight) this discrete reweighting has no overlap problem - unlike reweighting from $μ=0$ - and the results are reliable. We also present results based on this algorithm on the phase diagram of lattice QCD with two different actions: as a first test, we apply the method to calculate the position of the critical endpoint with unimproved staggered fermions at $N_τ=4$; as a second application, we study the phase diagram with 2stout improved staggered fermions at $N_τ=6$. This second one is already a reasonably fine lattice - relevant for phenomenology. We demonstrate that the method penetrates the region of the phase diagram where the Taylor and imaginary chemical potential methods lose predictive power.

hep-lat↗
arXiv2021

Localisation of Dirac modes in finite-temperature $\mathbb{Z}_2$ gauge theory on the lattice

The low-lying Dirac modes become localised at the finite-temperature transition in QCD and in other gauge theories, suggesting a general connection between their localisation and deconfinement. The simplest model where this connection can be tested is $\mathbb{Z}_2$ gauge theory in 2+1 dimensions. We show that in this model the low modes in the staggered Dirac spectrum are delocalised in the confined phase and become localised in the deconfined phase. We also show that localised modes correlate with disorder in the Polyakov loop configuration, in agreement with the "sea/island" picture of localisation, and with negative plaquettes. These results further support the conjecture that localisation and deconfinement are closely related.

hep-lat↗
arXiv2021

Localization properties of Dirac modes at the Roberge-Weiss phase transition

We study the localization properties of the low-lying Dirac eigenmodes in QCD at imaginary chemical potential $\hatμ_I=π$ at temperatures above the Roberge-Weiss transition temperature $T_{\rm RW}$. We find that modes are localized up to a temperature-dependent "mobility edge" and delocalized above it, and that the mobility edge extrapolates to zero at a temperature compatible with $T_{\rm RW}$. This supports the existence of a strong connection between localization of the low Dirac modes and deconfinement, studied here for the first time in a model with a genuine deconfinement transition in the continuum limit in the presence of dynamical fermions.

hep-lat↗
arXiv2021

Localization of Dirac modes in finite-temperature $\mathbb{Z}_2$ gauge theory on the lattice

We study the localization properties of the eigenmodes of the staggered Dirac operator in finite-temperature $\mathbb{Z}_2$ pure gauge theory on the lattice in 2+1 dimensions. We find that the low modes turn from delocalized to localized as the system crosses over from the confined to the deconfined phase in the "physical" sector (positive average Polyakov loop) selected by external fermionic probes, while they remain delocalized in the "unphysical" sector (negative average Polyakov loop). This confirms that the close connection between deconfinement and localization of the low Dirac modes in the physical sector, already observed in other models, holds also in the simplest gauge theory displaying a deconfinement transition. We also observe a clear correlation of localized modes with fluctuations of the Polyakov loop away from the ordered value, as expected according to the "sea/islands" picture of localization, and with clusters of negative plaquettes. A novel finding is the presence of localized modes at the high end of the Dirac spectrum in all phases/sectors of the theory.

hep-lat↗
arXiv2021

Lattice simulations of the QCD chiral transition at real baryon density

State-of-the-art lattice QCD studies of hot and dense strongly interacting matter currently rely on extrapolation from zero or imaginary chemical potentials. The ill-posedness of numerical analytic continuation puts severe limitations on the reliability of such methods. Here we use the more direct sign reweighting method to perform lattice QCD simulation of the QCD chiral transition at finite real baryon density on phenomenologically relevant lattices. This method does not require analytic continuation and avoids the overlap problem associated with generic reweighting schemes, so has only statistical but no uncontrolled systematic uncertainties for a fixed lattice setup. This opens up a new window to study hot and dense strongly interacting matter from first principles. We perform simulations up to a baryochemical potential-temperature ratio of $μ_B/T=2.5$ covering most of the RHIC Beam Energy Scan range in the chemical potential. We also clarify the connection of the approach to the more traditional phase reweighting method.

hep-lat↗
arXiv2021

Localization of Dirac Fermions in Finite-Temperature Gauge Theory

It is by now well established that Dirac fermions coupled to non-Abelian gauge theories can undergo an Anderson-type localization transition. This transition affects eigenmodes in the lowest part of the Dirac spectrum, the ones most relevant to the low-energy physics of these models. Here we review several aspects of this phenomenon, mostly using the tools of lattice gauge theory. In particular, we discuss how the transition is related to the finite-temperature transitions leading to the deconfinement of fermions, as well as to the restoration of chiral symmetry that is spontaneously broken at low temperature. Other topics we touch upon are the universality of the transition, and its connection to topological excitations (instantons) of the gauge field and the associated fermionic zero modes. While the main focus is on Quantum Chromodynamics, we also discuss how the localization transition appears in other related models with different fermionic contents (including the quenched approximation), gauge groups, and in different space-time dimensions. Finally, we offer some speculations about the physical relevance of the localization transition in these models.

hep-lat↗
arXiv2020

Reconfinement, localization and thermal monopoles in $SU(3)$ trace-deformed Yang-Mills theory

We study, by means of numerical lattice simulations, the properties of the reconfinement phase transition taking place in trace deformed $SU(3)$ Yang-Mills theory defined on $\mathbb{R}^3\times S^1$, in which center symmetry is recovered even for small compactification radii. We show, by means of a finite size scaling analysis, that the reconfinement phase transition is first-order, like the usual $SU(3)$ thermal phase transition. We then investigate two different physical phenomena, which are known to characterize the standard confinement/deconfinement phase transition, namely the condensation of thermal magnetic monopoles and the change in the localization properties of the eigenmodes of the Dirac operator. Regarding the latter, we show that the mobility edge signalling the Anderson-like transition in the Dirac spectrum vanishes as one enters the reconfined phase, as it happens in the standard confined phase. Thermal monopoles, instead, show a peculiar behavior: their density decreases going through reconfinement, at odds with the standard thermal theory; nonetheless, they condense at reconfinement, like at the usual confinement transition. The coincidence of monopole condensation and Dirac mode delocalization, even in a framework different from that of the standard confinement transition, suggests the existence of a strict link between them.

hep-lat↗
arXiv2020

New approach to lattice QCD at finite density; results for the critical end point on coarse lattices

All approaches currently used to study finite baryon density lattice QCD suffer from uncontrolled systematic uncertainties in addition to the well-known sign problem. We formulate and test an algorithm, sign reweighting, that works directly at finite $μ= μ_B/3$ and is yet free from any such uncontrolled systematics. With this algorithm the {\em only} problem is the sign problem itself. This approach involves the generation of configurations with the positive fermionic weight $|{\rm Re\; det} D(μ)|$ where $D(μ)$ is the Dirac matrix and the signs ${\rm sign} \; ( {\rm Re\; det} D(μ) ) = \pm 1$ are handled by a discrete reweighting. Hence there are only two sectors, $+1$ and $-1$ and as long as the average $\langle\pm 1\rangle \neq 0$ (with respect to the positive weight) this discrete reweighting by the signs carries no overlap problem and the results are reliable. The approach is tested on $N_t = 4$ lattices with $2+1$ flavors and physical quark masses using the unimproved staggered discretization. By measuring the Fisher (sometimes also called Lee-Yang) zeros in the bare coupling on spatial lattices $L/a = 8, 10, 12$ we conclude that the cross-over present at $μ= 0$ becomes stronger at $μ> 0$ and is consistent with a true phase transition at around $μ_B/T \sim 2.4$.

hep-lat↗
arXiv2020

Towards a reliable lower bound on the location of the critical endpoint

We perform the first direct determination of the position of the leading singularity of the pressure in the complex chemical potential $μ_B$ plane in lattice QCD using numerical simulations with 2-stout improved rooted staggered fermions. This provides a direct determination of the radius of convergence of the Taylor expansion of the pressure that does not rely on a finite-order truncation of the expansion. The analyticity issues in the complex $μ_B$ plane of the grand canonical partition function of QCD with rooted staggered fermions are solved with a careful redefinition of the fermion determinant that makes it a polynomial in the fugacity on any finite lattice, without changing the continuum limit of the observables. By performing a finite volume scaling study at a single coarse lattice spacing, we show that the limiting singularity is not on the real line in the thermodynamic limit, thus showing that the radius of convergence of the Taylor expansion gives a lower bound on the location of a possible phase transition. In the vicinity of the crossover temperature at zero chemical potential, the radius of convergence turns out to be $μ_B/T \approx 2$ and roughly temperature independent.

hep-lat↗
arXiv2020

A reversible allelic partition process and Pitman sampling formula

We introduce a continuous-time Markov chain describing dynamic allelic partitions which extends the branching process construction of the Pitman sampling formula in Pitman (2006) and the birth-and-death process with immigration studied in Karlin and McGregor (1967), in turn related to the celebrated Ewens sampling formula. A biological basis for the scheme is provided in terms of a population of individuals grouped into families, that evolves according to a sequence of births, deaths and immigrations. We investigate the asymptotic behaviour of the chain and show that, as opposed to the birth-and-death process with immigration, this construction maintains in the temporal limit the mutual dependence among the multiplicities. When the death rate exceeds the birth rate, the system is shown to have reversible distribution identified as a mixture of Pitman sampling formulae, with negative binomial mixing distribution on the population size. The population therefore converges to a stationary random configuration, characterised by a finite number of families and individuals.

math.PR↗
arXiv2020

The effect of stout smearing on the phase diagram from multiparameter reweigthing in lattice QCD

The phase diagram and the location of the critical endpoint (CEP) of lattice QCD with unimproved staggered fermions on a $N_t=4$ lattice was determined fifteen years ago with the multiparameter reweighting method by studying Fisher zeros. We first reproduce the old result with an exact algorithm (not known at the time) and with statistics larger by an order of magnitude. As an extension of the old analysis we introduce stout smearing in the fermion action in order to reduce the finite lattice spacing effects. First we show that increasing the smearing parameter $ρ$ the crossover at $μ= 0$ gets weaker, i.e., the leading Fisher zero gets farther away from the real axis. Furthermore as the chemical potential is increased the overlap problem gets severe sooner than in the unimproved case, therefore shrinking the range of applicability of the method. Nevertheless certain qualitative features remain, even after introducing the smearing. Namely, at small chemical potentials the Fisher zeros first get farther away from the real axis and later at around $aμ_q = 0.1 - 0.15$ they start to get closer, i.e., the crossover first gets weaker and later stronger as a function of $μ$. However, because of the more severe overlap problem the CEP is out of reach with the smeared action.

hep-lat↗
arXiv2020

Consistency of Bayesian inference with Gaussian process priors in an elliptic inverse problem

For $\mathcal{O}$ a bounded domain in $\mathbb{R}^d$ and a given smooth function $g:\mathcal{O}\to\mathbb{R}$, we consider the statistical nonlinear inverse problem of recovering the conductivity $f>0$ in the divergence form equation $$ \nabla\cdot(f\nabla u)=g\ \textrm{on}\ \mathcal{O}, \quad u=0\ \textrm{on}\ \partial\mathcal{O}, $$ from $N$ discrete noisy point evaluations of the solution $u=u_f$ on $\mathcal O$. We study the statistical performance of Bayesian nonparametric procedures based on a flexible class of Gaussian (or hierarchical Gaussian) process priors, whose implementation is feasible by MCMC methods. We show that, as the number $N$ of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate $N^{-λ}, λ>0,$ for the reconstruction error of the associated posterior means, in $L^2(\mathcal{O})$-distance.

math.ST↗