arXiv subjects
Publications and source records attributed to Sourav Sarkar.
We show that the law of the KPZ fixed point starting from arbitrary initial condition is absolutely continuous with respect to the law of Brownian motion $B$ on every compact interval. In particular, the Airy$_1$ process is absolutely continuous with respect to $B$ on any compact interval.
The electrical and Hall conductivities in a uniform magnetic field are evaluated for an interacting pion gas using the kinetic theory approach within the ambit of relaxation time approximation (RTA). The in-medium cross sections vis-a-vis the relaxation time for $ππ$ scattering are obtained using a one-loop modified thermal propagator for the exchanged $ρ$ and $σ$ mesons using thermal field theoretic techniques. For higher values of the magnetic field, a monotonic increase of the electrical conductivity with the temperature is observed. However, for a given temperature the conductivity is found to decrease steadily with magnetic field. The Hall conductivity, at lower values of the magnetic field, is found to decrease with temperature more rapidly than the electrical conductivity, whereas at higher values of the magnetic field, a linear increase is seen. Use of the in-medium scattering cross-section is found to produce a significant effect on the temperature dependence of both electrical and Hall conductivities compared to the case where vacuum cross-section is used.
Various thermodynamic quantities and the phase diagram of strongly interacting hot and dense magnetized quark matter are obtained with the $ 2 $-flavour Nambu-Jona-Lasinio model with Polyakov loop considering finite values of the anomalous magnetic moment (AMM) of the quarks. Susceptibilities associated with constituent quark mass and traced Polyakov loop are used to evaluate chiral and deconfinement transition temperatures. It is found that, inclusion of the AMM of the quarks in presence of the background magnetic field results in a substantial decrease in the chiral as well as deconfinement transition temperatures in contrast to an enhancement in the chiral transition temperature in its absence. Using standard techniques of finite temperature field theory, the two point thermo-magnetic mesonic correlation functions in the scalar ($σ$) and neutral pseudoscalar ($π^0$) channels are evaluated to calculate the masses of $σ$ and $ π^0 $ considering the AMM of the quarks.
Dilepton production rate (DPR) from hot and dense quark matter is studied in the presence of an arbitrary external magnetic field using the 2-flavour Nambu--Jona-Lasinio (NJL) model. The anomalous magnetic moment (AMM) of the quarks is taken into consideration while calculating the constituent quark mass as well as the DPR from the thermo-magnetic medium. An infinite number of quark Landau levels is incorporated so that no approximations are made on the strength of the background magnetic field. The analytic structure of the two point vector current correlation function in the complex energy plane reveals that, in addition to the usual Unitary cut, a non-trival Landau cut appears in the physical kinematic domains solely due to the external magnetic field. Moreover, these kinematic domains of the Unitary and Landau cuts are found to be significantly modified due to the AMM of the quarks. With finite AMM of the quarks, for certain values of the external magnetic field, the kinematically forbidden gap between the Unitary and Landau cuts are shown to vanish leading to the generation of a continuous spectrum of dilepton emission over the whole invariant mass region not observed earlier.
In this work the neutral meson properties have been investigated in the presence of thermo-magnetic background using two-flavor Nambu--Jona-Lasinio model. Mass, spectral function and dispersion relations are obtained in the scalar ($σ$) and pseudo-scalar ($π^0$) channels as well as in the vector ($ρ^0$) and axial vector ($a^0_1$) channels. The general Lorentz structures for the vector and axial-vector meson polarization functions have been considered in detail. The ultra-violet divergences appearing in this work have been regularized using a mixed regularization technique where the gamma functions arising in dimensional regularization are replaced with incomplete gamma functions as usually done in the proper time regularization procedure. The meson spectral functions obtained in the presence of a magnetic field possess nontrivial oscillatory structure. Similar to the scalar and pseudo-scalar channel, the spectral functions for each of the modes of $ρ^0$ are observed to overlap with the corresponding modes of its chiral partner $a_1^0$ mesons in the chiral symmetry restored phase. We observe discontinuities in the masses of all the mesonic excitations for a non-zero external magnetic field.
The temperature and density dependence of the relaxation times, thermal conductivity, shear viscosity and bulk viscosity for a hot and dense gas consisting of pions, kaons and nucleons have been evaluated in the kinetic theory approach. The in-medium cross-sections for $ππ$, $πK$ and $πN$ scatterings were obtained by using complete propagators for the exchanged $ρ$, $σ$, $K^*$ and $Δ$ excitations derived using thermal field theoretic techniques. Notable deviations can be observed in the temperature dependence of $η$, $ζ$ and $λ$ when compared with corresponding calculations using vacuum cross-sections usually employed in the literature. The value of the specific shear viscosity $η/s$ is found to be in agreement with available estimates.
Employing a field dependent three-momentum cut-off regularization technique, we study the phase structure and mesonic masses using the $2$-flavour Nambu-Jona Lasinio model at finite temperature and density in presence of arbitrary external magnetic field. This approach is then applied to incorporate the effects of the anomalous magnetic moment(AMM) of quarks on constituent quark mass and thermodynamic observables as a function of temperature/baryonic density. The critical temperature for transition from chiral symmetry broken to the restored phase is observed to decrease with the external magnetic field, which can be classified as inverse magnetic catalysis, while an opposite behaviour is realized in the case of a vanishing magnetic moment, implying magnetic catalysis. These essential features are also reflected in the phase diagram. Furthermore, the properties of the low lying scalar and neutral pseudoscalar mesons are also studied in presence of a hot and dense magnetized medium including AMM of the quarks using random phase approximation. For non-zero values of magnetic field, we notice a sudden jump in the mass of the Goldstone mode at and above the Mott transition temperature which is found to decrease substantially with the increase in magnetic field when the AMM of the quarks are taken into consideration.
The relativistic kinetic theory approach has been employed to study four well-known transport coefficients that characterize heat flow and diffusion for the case of a hot mixture constituting of nucleons and pions. Medium effects on the cross-section for binary collisions ($Nπ$,$ππ$) have been taken into consideration by incorporating self-energy corrections to modify the propagator of the exchanged $Δ$ baryon in $Nπ$ interaction and the $ρ$ and $σ$ meson propagators for the case of $ππ$ interaction. The temperature dependence of the four coefficients have been investigated for several values of the baryon chemical potential.
The one loop self energy of the neutral $ρ$ meson is obtained for the effective $ρππ$ and $ρNN$ interaction at finite temperature and density in the presence of a constant background magnetic field of arbitrary strength. In our approach, the eB-dependent vacuum part of the self energy is extracted by means of dimensional regularization where the ultraviolet divergences corresponding to the pure vacuum self energy manifest as the pole singularities of gamma as well as Hurwitz zeta functions. This improved regularization procedure consistently reproduces the expected results in the vanishing magnetic field limit and can be used quite generally in other self energy calculations dealing with arbitrary magnetic field strength. In presence of the external magnetic field, the general Lorentz structure for the in-medium vector boson self energy is derived which can also be implemented in case of the gauge bosons such as photons and gluons. It has been shown that with vanishing perpendicular momentum of the external particle, essentially two form factors are sufficient to describe the self energy completely. Consequently, two distinct modes are observed in the study of the effective mass, dispersion relations and the spectral function of $ρ^0$ where one of the modes possesses two fold degeneracy. For large baryonic chemical potential, it is observed that the critical magnetic field required to block the $ρ^0\rightarrowπ^+π^-$ decay channel increases significantly with temperature. However, in case of smaller values reaching down to vanishing chemical potential, the critical field follows the opposite trend.
Stochastic point processes with Coulomb interactions arise in various natural examples of statistical mechanics, random matrices and optimization problems. Often such systems due to their natural repulsion exhibit remarkable hyperuniformity properties, that is, the number of points landing in any given region fluctuates at a much smaller scale compared to that of a set of i.i.d. random points. A well known conjecture from physics appearing in the works of Jancovici, Lebowitz, Manificat, Martin, and Yalcin ('80,'83,'93), states that the variance of the number of points landing in a set should grow like the surface area instead of the volume unlike i.i.d. random points. In a recent beautiful work, Chatterjee (2017) gave the first proof of such a result in dimension three for a Coulomb type system, known as the hierarchical Coulomb gas, inspired by Dyson's hierarchical model of the Ising ferromagnet. However the case of dimensions greater than three had remained open. In this paper, we establish the correct fluctuation behavior up to logarithmic factors in all dimensions greater than three, for the hierarchical model. Using similar methods, we also prove sharp variance bounds for smooth linear statistics which were unknown in any dimension bigger than two. A key intermediate step is to obtain precise results about the ground states of such models whose behavior can be interpreted as hierarchical analogues of various crystalline conjectures predicted for energy minimizing systems, and could be of independent interest.
We have calculated the electromagnetic spectral function (SF) vis-à-vis the dilepton rate (DR) by evaluating the one-loop photon polarisation tensor for a strongly magnetised hot and dense medium. The calculation is performed in the ambit of mean field models namely Nambu\textendash Jona-Lasinio (NJL) and its Polyakov loop extended version (PNJL) in the lowest Landau level approximation. These models allow for a dynamical generation of quark mass which, evidently, gets affected in the presence of a magnetised medium. We have shown that the strength of the SF gets boosted because of the presence of dynamical quarks in lieu of the current quarks. It increases as we increase the magnetic field for a given value of temperature in both NJL and PNJL models. This increment is further reflected in the DR which is enhanced as compared to the Born rate for certain values of invariant masses. We also discuss the impact of chemical potential on DR within the present scenario.
Hyperspectral images of land-cover captured by airborne or satellite-mounted sensors provide a rich source of information about the chemical composition of the materials present in a given place. This makes hyperspectral imaging an important tool for earth sciences, land-cover studies, and military and strategic applications. However, the scarcity of labeled training examples and spatial variability of spectral signature are two of the biggest challenges faced by hyperspectral image classification. In order to address these issues, we aim to develop a framework for material-agnostic information retrieval in hyperspectral images based on Positive-Unlabelled (PU) classification. Given a hyperspectral scene, the user labels some positive samples of a material he/she is looking for and our goal is to retrieve all the remaining instances of the query material in the scene. Additionally, we require the system to work equally well for any material in any scene without the user having to disclose the identity of the query material. This material-agnostic nature of the framework provides it with superior generalization abilities. We explore two alternative approaches to solve the hyperspectral image classification problem within this framework. The first approach is an adaptation of non-negative risk estimation based PU learning for hyperspectral data. The second approach is based on one-versus-all positive-negative classification where the negative class is approximately sampled using a novel spectral-spatial retrieval model. We propose two annotator models - uniform and blob - that represent the labelling patterns of a human annotator. We compare the performances of the proposed algorithms for each annotator model on three benchmark hyperspectral image datasets - Indian Pines, Pavia University and Salinas.
Vacuum to nuclear matter phase transition has been studied in presence of constant external background magnetic field with the mean field approximation in Walecka model. The anomalous nucleon magnetic moment has been taken into account using the modified "weak" field expansion of the fermion propagator having non-trivial correction terms for charged as well as for neutral particles. The effect of nucleon magnetic moment is found to favour the magnetic catalysis effect at zero temperature and zero baryon density. However, extending the study to finite temperatures, it is observed that the anomalous nuclear magnetic moment plays a crucial role in characterizing the qualitative behaviour of vacuum to nuclear matter phase transition even in case of the weak external magnetic fields . The critical temperature corresponding to the vacuum to nuclear medium phase transition is observed to decrease with the external magnetic field which can be identified as the inverse magnetic catalysis in Walecka model whereas the opposite behaviour is obtained in case of vanishing magnetic moment indicating magnetic catalysis.
Coalescence of semi-infinite geodesics remains a central question in planar first passage percolation. In this paper we study finer properties of the coalescence structure of finite and semi-infinite geodesics for exactly solvable models of last passage percolation. Consider directed last passage percolation on $\mathbb{Z}^2$ with i.i.d. exponential weights on the vertices. Fix two points $v_1=(0,0)$ and $v_2=(0, \lfloor k^{2/3} \rfloor)$ for some $k>0$, and consider the maximal paths $Γ_1$ and $Γ_2$ starting at $v_1$ and $v_2$ respectively to the point $(n,n)$ for $n\gg k$. Our object of study is the point of coalescence, i.e., the point $v\in Γ_1\cap Γ_2$ with smallest $|v|_1$. We establish that the distance to coalescence $|v|_1$ scales as $k$, by showing the upper tail bound $\mathbb{P}(|v|_1> Rk) \leq R^{-c}$ for some $c>0$. We also consider the problem of coalescence for semi-infinite geodesics. For the almost surely unique semi-infinite geodesics in the direction $(1,1)$ starting from $v_3=(-\lfloor k^{2/3} \rfloor , \lfloor k^{2/3}\rfloor)$ and $v_4=(\lfloor k^{2/3} \rfloor ,- \lfloor k^{2/3}\rfloor)$, we establish the optimal tail estimate $\mathbb{P}(|v|_1> Rk) \asymp R^{-2/3}$, for the point of coalescence $v$. This answers a question left open by Pimentel (Ann. Probab., 2016) who proved the corresponding lower bound.
We study the spectrum of local operators living on a defect in a generic conformal field theory, and their coupling to the local bulk operators. We establish the existence of universal accumulation points in the spectrum at large $s$, $s$ being the charge of the operators under rotations in the space transverse to the defect. Our tools include a formula that inverts the bulk to defect OPE, analogous to the Caron-Huot formula for the four-point function. Analyticity of the formula in $s$ implies that the scaling dimensions of the defect operators are aligned in Regge trajectories $\widehatΔ(s)$. These results require the correlator of two local operators and the defect to be bounded in a certain region, a condition that we do not prove in general. We check our conclusions against examples in perturbation theory and holography, and we make specific predictions concerning the spectrum of defect operators on Wilson lines. We also give an interpretation of the large $s$ spectrum in the spirit of the work of Alday and Maldacena.
Given two vectors in Euclidean space, how unlikely is it that a random vector has a larger inner product with the shorter vector than with the longer one? When the random vector has independent, identically distributed components, we conjecture that this probability is no more than a constant multiple of the ratio of the Euclidean norms of the two given vectors, up to an additive term to allow for the possibility that the longer vector has more arithmetic structure. We give some partial results to support the basic conjecture.
In last passage percolation models, the energy of a path is maximized over all directed paths with given endpoints in a random environment, and the maximizing paths are called geodesics. The geodesics and their energy can be scaled so that transformed geodesics cross unit distance and have fluctuations and scaled energy of unit order. Here we consider Poissonian last passage percolation, a model lying in the KPZ universality class, and refer to scaled geodesics as polymers and their scaled energies as weights. Polymers may be viewed as random functions of the vertical coordinate and, when they are, we show that they have modulus of continuity whose order is at most $t^{2/3}\big(\log t^{-1}\big)^{1/3}$. The power of one-third in the logarithm may be expected to be sharp and in a related problem we show that it is: among polymers in the unit box whose endpoints have vertical separation $t$ (and a horizontal separation of the same order), the maximum transversal fluctuation has order $t^{2/3}\big(\log t^{-1}\big)^{1/3}$. Regarding the orthogonal direction, in which growth occurs, we show that, when one endpoint of the polymer is fixed at $(0,0)$ and the other is varied vertically over $(0,z)$, $z\in [1,2]$, the resulting random weight profile has sharp modulus of continuity of order $t^{1/3}\big(\log t^{-1}\big)^{2/3}$. In this way, we identify exponent pairs of $(2/3,1/3)$ and $(1/3,2/3)$ in power law and polylogarithmic correction, respectively for polymer fluctuation, and polymer weight under vertical endpoint perturbation. The two exponent pairs describe [10,11,9] the fluctuation of the boundary separating two phases in subcritical planar random cluster models.
We study the following one-dimensional model of annihilating particles. Beginning with all sites of $\mathbb{Z}$ uncolored, a blue particle performs simple random walk from $0$ until it reaches a nonzero red or uncolored site, and turns that site blue; then, a red particle performs simple random walk from $0$ until it reaches a nonzero blue or uncolored site, and turns that site red. We prove that after $n$ blue and $n$ red particles alternately perform such walks, the total number of colored sites is of order $n^{1/4}$. The resulting random color configuration, after rescaling by $n^{1/4}$ and taking $n\to \infty$, has an explicit description in terms of alternating extrema of Brownian motion (the global maximum on a certain interval, the global minimum attained after that maximum, etc.).