Tiago J. Oliveira

We consider the number of ways all the sites of a kagome lattice can be covered by nonoverlapping linear rigid rods where each rod covers three sites. We establish a two-to-one correspondence between the configurations of trimers on the kagome lattice to the covering by dimers of a related hexagonal lattice to show that entropy of coverings per trimer ${s}_{\text{tri,kag}}$ equals the entropy per dimer ${s}_{\text{dim,hex}}$, and is given by ${s}_{\text{tri,kag}}={s}_{\text{dim,hex}}=\frac{1}{2\ensuremath{\pi}}{\ensuremath{\int}}_{0}^{2\ensuremath{\pi}/3}ln(2+2\phantom{\rule{0.16em}{0ex}}cosk)dk\ensuremath{\approx}0.323\phantom{\rule{0.16em}{0ex}}065\phantom{\rule{0.16em}{0ex}}947...$.

We consider the number of ways all the sites of a kagome lattice can be covered by non-overlapping linear rigid rods where each rod covers 3 sites. We establish a 2-to-1 correspondence between the configurations of trimers on the kagome lattice to the covering by dimers of a related hexagonal lattice to show that entropy of coverings per trimer $s_{\text{tri,kag}}$ equals the entropy per dimer $ s_{\text{dim,hex}} $, and is given by $ s_{\text{tri,kag}} = s_{\text{dim,hex}} = \frac{1}{2 π} \int_0^{ 2 π/3} \log( 2 + 2 \cos k) dk \approx 0.323065947\ldots$.

Two-dimensional (2D) Kardar-Parisi-Zhang (KPZ) growth is usually investigated on substrates of lateral sizes $L_x=L_y$, so that $L_x$ and the correlation length ($\xi$) are the only relevant lengths determining the scaling behavior. However, in cylindrical geometry, as well as in flat rectangular substrates $L_x\ne L_y$ and, thus, the surfaces can become correlated in a single direction, when $\xi \sim L_x\ll L_y$. From extensive simulations of several KPZ models, we demonstrate that this yields a dimensional crossover in their dynamics, with the roughness scaling as $W\sim t^{{\beta }_{\text{2D}}}$ for $t\ll t_c$ and $W\sim t^{{\beta }_{\text{1D}}}$ for $t\gg t_c$, where $t_c\sim L_x^{1/z_{2\text{D}}}$. The height distributions (HDs) also cross over from the 2D flat (cylindrical) HD to the asymptotic Tracy-Widom Gaussian orthogonal ensemble (Gaussian unitary ensemble) distribution. Moreover, 2D to one-dimensional (1D) crossovers are found also in the asymptotic growth velocity and in the steady-state regime of flat systems, where a family of universal HDs exists, interpolating between the 2D and 1D ones as $L_y/L_x$ increases. Importantly, the crossover scalings are fully determined and indicate a possible way to solve 2D KPZ models.

Two-dimensional (2D) KPZ growth is usually investigated on substrates of lateral sizes $L_x=L_y$, so that $L_x$ and the correlation length ($ξ$) are the only relevant lengths determining the scaling behavior. However, in cylindrical geometry, as well as in flat rectangular substrates $L_x \neq L_y$ and, thus, the surfaces can become correlated in a single direction, when $ξ\sim L_x \ll L_y$. From extensive simulations of several KPZ models, we demonstrate that this yields a dimensional crossover in their dynamics, with the roughness scaling as $W \sim t^{β_{\text{2D}}}$ for $t \ll t_c$ and $W \sim t^{β_{\text{1D}}}$ for $t \gg t_c$, where $t_c \sim L_x^{1/z_{2\text{D}}}$. The height distributions (HDs) also cross over from the 2D flat [cylindrical] HD to the asymptotic Tracy-Widom GOE [GUE] distribution. Moreover, 2D-to-1D crossovers are found also in the asymptotic growth velocity and in the steady state regime of flat systems, where a family of universal HDs exists, interpolating between the 2D and 1D ones as $L_y/L_x$ increases. Importantly, the crossover scalings are fully determined and indicate a possible way to solve 2D KPZ models.

In recent work [Rodrigues et al., Phys. Rev. E 100, 022121 (2019)], evidence was found that the surface adsorption transition of interacting self-avoiding trails (ISATs) placed on the square lattice displays a nonuniversal behavior at the special adsorption point (SAP) where the collapsing polymers adsorb. In fact, different surface exponents ${\varphi }^{\left(s\right)}$ and $1/{\delta }^{\left(s\right)}$ were found at the SAP depending on whether the surface orientation is horizontal (HS) or diagonal (DS). Here, we revisit these systems and study other ones, through extensive Monte Carlo simulations, considering much longer trails than previous works. Importantly, we demonstrate that the different exponents observed in the reference above are due to the presence of a surface-attached-globule (SAG) phase in the DS system, which changes the multicritical nature of the SAP and is absent in the HS case. By considering a modified horizontal surface (mHS), on which the trails are forbidden from having two consecutive steps, resembling the DS situation, a stable SAG phase is found in the phase diagram, and both DS and mHS systems present similar $1/{\delta }^{\left(s\right)}$ exponents at the SAP, namely, $1/{\delta }^{\left(s\right)}\approx 0.44$, whereas $1/{\delta }^{\left(s\right)}\approx 0.34$ in the HS case. Intriguingly, while ${\varphi }^{\left(s\right)}\approx 1/{\delta }^{\left(s\right)}$ is found for the DS and HS scenarios, as expected, in the mHS case ${\varphi }^{\left(s\right)}$ is about 10% smaller than $1/{\delta }^{\left(s\right)}$. These results strongly indicate that at least two universality classes exist for the SAPs of adsorbing ISATs on the square lattice.

While the one-point height distributions (HDs) and two-point covariances of $(2+1)$ Kardar-Parisi-Zhang (KPZ) systems have been investigated in several recent works for flat and spherical geometries, for the cylindrical one the HD was analyzed for few models and nothing is known about the spatial and temporal covariances. Here, we report results for these quantities, obtained from extensive numerical simulations of discrete KPZ models, for three different setups yielding cylindrical growth. Beyond demonstrating the universality of the HD and covariances, our results reveal other interesting features of this geometry. For example, the spatial covariances measured along the longitudinal and azimuthal directions are different, with the former being quite similar to the curve for flat $(2+1)$ KPZ systems, while the latter resembles the ${\mathrm{Airy}}_2$ covariance of circular $(1+1)$ KPZ interfaces. We also argue (and present numerical evidence) that, in general, the rescaled temporal covariance $\mathcal{A}(t/t_0)$ decays asymptotically as $\mathcal{A}\left(x\right)\sim x^{-\overline{\lambda }}$ with an exponent $\overline{\lambda }=\beta +d^{*}/z$, where $d^{*}$ is the number of interface sides kept fixed during the growth (being $d^{*}=1$ for the systems analyzed here). Overall, these results complete the picture of the main statistics for the $(2+1)$ KPZ class.

In recent work [PRE 100, 022121 (2019)] evidence was found that the surface adsorption transition of interacting self-avoiding trails (ISATs) placed on the square lattice displays a non-universal behavior at the special adsorption point (SAP) where the collapsing polymers adsorb. In fact, different surface exponents $φ^{(s)}$ and $1/δ^{(s)}$ were found at the SAP depending on whether the surface orientation is horizontal (HS) or diagonal (DS). Here, we revisit these systems and study other ones, through extensive Monte Carlo simulations, considering much longer trails than previous works. Importantly, we demonstrate that the different exponents observed in the reference above are due to the presence of a previously unseen surface-attached-globule (SAG) phase in the DS system, which changes the multicritical nature of the SAP and is absent in the HS case. By considering a modified horizontal surface (mHS) where the trails are forbidden of having two consecutive steps along it, resembling the DS situation, a stable SAG phase is found in the phase diagram, and both DS and mHS systems present similar $1/δ^{(s)}$ exponents at the SAP, being $1/δ^{(s)} \approx 0.44$, whilst $1/δ^{(s)} \approx 0.34$ in the HS case. Intriguingly, while $φ^{(s)} \approx 1/δ^{(s)}$ is found for the DS and HS scenarios, as expected, in the mHS case $φ^{(s)}$ is about $10$\% smaller than $1/δ^{(s)}$. These results strongly indicate that at least two universality classes exist for the SAPs of adsorbing ISATs on the square lattice.

While the 1-point height distributions (HDs) and 2-point covariances of $(2+1)$ KPZ systems have been investigated in several recent works for flat and spherical geometries, for the cylindrical one the HD was analyzed for few models and nothing is known about the spatial and temporal covariances. Here, we report results for these quantities, obtained from extensive numerical simulations of discrete KPZ models, for three different setups yielding cylindrical growth. Beyond demonstrating the universality of the HD and covariances, our results reveal other interesting features of this geometry. For example, the spatial covariances measured along the longitudinal and azimuthal directions are different, with the former being quite similar to the curve for flat $(2+1)$ KPZ systems, while the latter resembles the Airy$_2$ covariance of circular $(1+1)$ KPZ interfaces. We also argue (and present numerical evidence) that, in general, the rescaled temporal covariance $\mathcal{A}(t/t_0)$ decays asymptotically as $\mathcal{A}(x) \sim x^{-\barλ}$ with an exponent $\barλ = β+ d^*/z$, where $d^*$ is the number of interface sides kept fixed during the growth (being $d^* = 1$ for the systems analyzed here). Overall, these results complete the picture of the main statistics for the $(2+1)$ KPZ class.

The determination of the exact exponents of the KPZ class in any substrate dimension $d$ is one of the most important open issues in Statistical Physics. Based on the behavior of the dimensional variation of some exact exponent differences for other growth equations, I find here that the KPZ growth exponents (related to the temporal scaling of the fluctuations) are given by $\beta_d = \frac{7}{8d+13}$. These exponents present an excellent agreement with the most accurate estimates for them in the literature. Moreover, they are confirmed here through extensive Monte Carlo simulations of discrete growth models and real space renormalization group (RG) calculations for directed polymers in random media (DPRM), up to $d=15$. The left-tail exponents of the probability density functions for the DPRM energy provide another striking verification of the analytical result above.