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Luigi Cantini

Publications and source records attributed to Luigi Cantini.

At least 19 recordsLinked to original sources
arXiv2026

An integrable approach to macroscopic fluctuation theory for the multispecies SSEP

We study the macroscopic fluctuation theory (MFT) of a multispecies generalization of the symmetric simple exclusion process (mSSEP) on the infinite line, in which particles of $N+1$ species -- including, possibly, a vacancy species -- exchange positions at unit rate. Working with the full, redundant set of coarse-grained densities $\bfrho=\{ρ_0,\dots,ρ_N\}$ keeps the relabelling symmetry of the model manifest throughout. We first extend the argument of Derrida and Gerschenfeld to the multispecies setting, showing that the cumulant generating function of the multispecies current between two regions of an arbitrary graph depends on the boundary densities $ \bfrho_L,\bfrho_R$ and on the fugacities $\bflambda$ only through a single scalar variable $ω$. We then formulate the MFT saddle-point equations for the mSSEP on the infinite line and show that they define an integrable system: they are of Landau--Lifshitz type, and a Zakharov--Takhtajan gauge transformation recasts them in AKNS form. Solving the resulting linear scattering problem by the inverse scattering method, we recover the cumulant generating function $F(ω)$ for the multispecies current, as well as the initial and final density profiles conditioned on a prescribed current fluctuation. In particular, we show that $F(ω)$ coincides with the function obtained by Derrida and Gerschenfeld for the single-species SSEP, now derived for an arbitrary number of species directly from the integrable structure of the multispecies MFT equations.

cond-mat.stat-mech↗
arXiv2024

Single impurity in the Totally Asymmetric Simple Exclusion Process

We examine the behavior of a single impurity particle embedded within a Totally Asymmetric Simple Exclusion Process (TASEP). By analyzing the impurity's dynamics, characterized by two arbitrary hopping parameters $ α$ and $β$, we investigate both its macroscopic impact on the system and its individual trajectory, providing new insights into the interaction between the impurity and the TASEP environment. We classify the induced hydrodynamic limit shapes based on the initial densities to the left and right of the impurity, along with the values of the parameters $α$,$β$. We develop a new method that enables the analysis of the impurity's behavior within an arbitrary density field, thereby generalizing the traditional coupling technique used for second-class particles. With this tool, we extend to the impurity case under certain parameter conditions, Ferrari and Kipnis's results on the distribution of the asymptotic speed of a second-class particle within a rarefaction fan.

cond-mat.stat-mech↗
arXiv2023

Steady-state selection in multi-species driven diffusive systems

We introduce a general method to determine the large scale non-equilibrium steady-state properties of one-dimensional multi-species driven diffusive systems with open boundaries, generalizing thus the max-min current principle known for systems with a single type of particles. This method is based on the solution of the Riemann problem of the associated system of conservation laws. We demonstrate that the effective density of a reservoir depends not only on the corresponding boundary hopping rates but also on the dynamics of the entire system, emphasizing the interplay between bulk and reservoirs. We highlight the role of Riemann variables in establishing the phase diagram of such systems. We apply our method to three models of multi-species interacting particle systems and compare the theoretical predictions with numerical simulations.

cond-mat.stat-mech↗
arXiv2022

Hydrodynamic behavior of the 2-TASEP

We address the question of the large scale or hydrodynamic behavior of a 2-species generalization of TASEP (2-TASEP), consisting of two kinds of particles, moving in opposite directions and swapping their positions. We compute the rarefaction and shock solutions of the hydrodynamic equations of the model, showing that these equations form a Temple class system. We solve completely the Riemann problem and compare the theoretical prediction to Monte Carlo simulations.

math-ph↗
arXiv2020

Boundary emptiness formation probabilities in the six-vertex model at $Δ= -\frac12$

We define a new family of overlaps $C_{N,m}$ for the XXZ Hamiltonian on a periodic chain of length $N$. These are equal to the linear sums of the groundstate components, in the canonical basis, wherein $m$ consecutive spins are fixed to the state ${\uparrow}$. We define the boundary emptiness formation probabilities as the ratios $C_{N,m}/C_{N,0}$ of these overlaps. In the associated six-vertex model, they correspond to correlation functions on a semi-infinite cylinder of perimeter $N$. At the combinatorial point $Δ= -\frac12$, we obtain closed-form expressions in terms of simple products of ratios of integers.

math-ph↗
arXiv2019

Integral formulas and antisymmetrization relations for the six-vertex model

We study the relationship between various integral formulas for nonlocal correlation functions of the six-vertex model with domain wall boundary conditions. Specifically, we show how the known representation for the emptiness formation probability can be derived from that for the so-called row configuration probability. A crucial ingredient in the proof is a relation expressing the result of antisymmetrization of some given function with respect to permutations in two sets of its variables in terms of the Izergin-Korepin partition function. This relation generalizes another one obtained by Tracy and Widom in the context of the asymmetric simple exclusion process.

math-ph↗
arXiv2016

Inhomogenous Multispecies TASEP on a ring with spectral parameters

We study an inhomogenous multispecies version of the Totally Asymmetric Simple Exclusion Process (TASEP) on a periodic oriented one dimensional lattice, which depends on two sets of parameters $({\bf τ},{\bf ν})$, attached to the particles. After discussing the Yang-Baxter integrability of our model, we study its (unnormalized) stationary measure. Motivated by the integrability of the model we introduce a further set of spectral parameters ${\bf z}$, attached to the sites of the lattice, and we uncover a remarkable underlying algebraic structure. We provide exact formulas for the stationary measure and prove the factorization of the stationary probability of certain configurations in terms of double Schubert polynomials in $({\bf τ},{\bf ν})$.

math-ph↗
arXiv2016

Matrix product and sum rule for Macdonald polynomials

We present a new, explicit sum formula for symmetric Macdonald polynomials $P_λ$ and show that they can be written as a trace over a product of (infinite dimensional) matrices. These matrices satisfy the Zamolodchikov--Faddeev (ZF) algebra. We construct solutions of the ZF algebra from a rank-reduced version of the Yang--Baxter algebra. As a corollary, we find that the normalization of the stationary measure of the multi-species asymmetric exclusion process is a Macdonald polynomial with all variables set equal to one.

math.RT↗
arXiv2015

Matrix product formula for Macdonald polynomials

We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik--Zamolodchikov equations, which arise by considering representations of the Zamolodchikov--Faddeev and Yang--Baxter algebras in terms of $t$-deformed bosonic operators. These solutions form a basis of the ring of polynomials in $n$ variables, whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalisations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at $q=1$.

math-ph↗
arXiv2015

Asymmetric Simple Exclusion Process with open boundaries and Koornwinder polynomials

In this paper we analyze the steady state of the Asymmetric Simple Exclusion process with open boundaries and second class particles by deforming it through the introduction of spectral parameters. The (unnormalized) probabilities of the particle configurations get promoted to Laurent polynomials in the spectral parameters and are constructed in terms of non-symmetric Koornwinder polynomials. In particular we show that the partition function coincides with a symmetric Macdonald-Koornwinder polynomial. As an outcome we compute the steady current and the average density of first class particles.

math-ph↗
arXiv2012

A one-parameter refinement of the Razumov-Stroganov correspondence

We introduce and prove a one-parameter refinement of the Razumov-Stroganov correspondence. This is achieved for fully-packed loop configurations (FPL) on domains which generalize the square domain, and which are endowed with the gyration operation. We consider one given side of the domain, and FPLs such that the only straight-line tile on this side is black. We show that the enumeration vector associated to such FPLs, weighted according to the position of the straight line and refined according to the link pattern for the black boundary points, is the ground state of the scattering matrix, an integrable one-parameter deformation of the O(1) Dense Loop Model Hamiltonian. We show how the original Razumov-Stroganov correspondence, and a conjecture formulated by Di Francesco in 2004, follow from our results.

math.CO↗
arXiv2011

Finite size emptiness formation probability of the XXZ spin chain at $Δ=-1/2$

In this paper we compute the Emptiness Formation Probability of a (twisted-) periodic XXZ spin chain of finite length at $Δ=-1/2$, thus proving the formulae conjectured by Razumov and Stroganov \cite{raz-strog1, raz-strog2}. The result is obtained by exploiting the fact that the ground state of the inhomogeneous XXZ spin chain at $Δ=-1/2$ satisfies a set of qKZ equations associated to $\displaystyle{U_q(\hat{sl_2})}$.

math-ph↗
arXiv2011

Doubly-refined enumeration of Alternating Sign Matrices and determinants of 2-staircase Schur functions

We prove a determinantal identity concerning Schur functions for 2-staircase diagrams lambda=(ln+l',ln,l(n-1)+l',l(n-1),...,l+l',l,l',0). When l=1 and l'=0 these functions are related to the partition function of the 6-vertex model at the combinatorial point and hence to enumerations of Alternating Sign Matrices. A consequence of our result is an identity concerning the doubly-refined enumerations of Alternating Sign Matrices.

math.CO↗
arXiv2010

Proof of the Razumov-Stroganov conjecture

The Razumov-Stroganov conjecture relates the ground-state coefficients in the even-length dense O(1) loop model to the enumeration of fully-packed loop configuration on the square, with alternating boundary conditions, refined according to the link pattern for the boundary points. Here we prove this conjecture, by mean of purely combinatorial methods. The main ingredient is a generalization of the Wieland proof technique for the dihedral symmetry of these classes, based on the `gyration' operation, whose full strength we will investigate in a companion paper.

math.CO↗
arXiv2009

Off-Critical SLE(2) and SLE(4): a Field Theory Approach

Using their relationship with the free boson and the free symplectic fermion, we study the off-critical perturbation of SLE(4) and SLE(2) obtained by adding a mass term to the action. We compute the off-critical statistics of the source in the Loewner equation describing the two dimensional interfaces. In these two cases we show that ratios of massive by massless partition functions, expressible as ratios of regularised determinants of massive and massless Laplacians, are (local) martingales for the massless interfaces. The off-critical drifts in the stochastic source of the Loewner equation are proportional to the logarithmic derivative of these ratios. We also show that massive correlation functions are (local) martingales for the massive interfaces. In the case of massive SLE(4), we use this property to prove a factorisation of the free boson measure.

math-ph↗
arXiv2009

qKZ equations and ground state of the O(1) loop model with open boundary conditions

We consider the qKZ equations based on the two boundaries Temperley Lieb algebra. We construct their solution in the case $s=q^{-3/2}$ using a recursion relation. At the combinatorial point $q^{1/2}= e^{-2πi/3}$ the solution reduces to the ground state of the dense O(1) loop model on a strip with open boundary conditions. We present an alternative construction of such ground state based on the knowledge of the ground state of the same model with mixed boundary conditions and prove that the sum rule as of its components is given by the product of four symplectic characters.

math-ph↗
arXiv2007

Algebraic Bethe Ansatz for the two species ASEP with different hopping rates

An ASEP with two species of particles and different hopping rates is considered on a ring. Its integrability is proved and the Nested Algebraic Bethe Ansatz is used to derive the Bethe Equations for states with arbitrary numbers of particles of each type, generalizing the results of Derrida and Evans. We present also formulas for the total velocity of particles of a given type and their limit for large size of the system and finite densities of the particles.

cond-mat.stat-mech↗