Exact extreme-value statistics at mixed-order transitions

We study extreme-value statistics for spatially extended models exhibiting mixed-order phase transitions (MOT). These are phase transitions that exhibit features common to both first-order (discontinuity of the order parameter) and second-order (diverging correlation length) transitions. We consider here the truncated inverse distance squared Ising model, which is a prototypical model exhibiting MOT, and study analytically the extreme-value statistics of the domain lengths The lengths of the domains are identically distributed random variables except for the global constraint that their sum equals the total system size $L$. In addition, the number of such domains is also a fluctuating variable, and not fixed. In the paramagnetic phase, we show that the distribution of the largest domain length $l_{max}$ converges, in the large $L$ limit, to a Gumbel distribution. However, at the critical point (for a certain range of parameters) and in the ferromagnetic phase, we show that the fluctuations of $l_{max}$ are governed by novel distributions, which we compute exactly. Our main analytical results are verified by numerical simulations.

Figure 1

Domain representation of the TIDSI model of size $L$. In this configuration, the number of domains is $N=6$. We recall that the interaction is restricted to spins within the same domain. The Boltzmann-Gibbs weight of such a configuration is given by Eq. (7). In this paper we study the statistics of the largest domain length $l_{max}={max}_{1\le i\le N}{l}_i$.

Figure 2

Phase diagram of the model (6) in the $\left(c,T\right)$ plane, with $\mathrm{\Delta }=1$. The different regions of the critical line (I–III) are explained in the text.

Figure 3

Log-linear plot of $p_1\left(x\right|L)$ for $c=3/2$ and $e^{-\beta \mathrm{\Delta }}=0.5$ (hence in the paramagnetic phase). The square symbols correspond to a numerical evaluation of $p_1\left(x\right|L)$ for $L=8000$ (see Appendix pp3 for details) while the solid line corresponds to the exact formula given in Eq. (38).

Figure 4

Plot of $p_1\left(x\right|L)$ for $c=3$ and $e^{-\beta \mathrm{\Delta }}=0.2$ (hence in the ferromagnetic phase). The square symbols correspond to a numerical evaluation of $p_1\left(x\right|L)$ for $L=2000$ (see Appendix pp3 for details) while the circular symbols correspond to the exact limiting distribution $f_{\mathrm{ferro}}\left(y\right)$ obtained by expanding the right hand side of Eq. (23) in powers of $z$. The slight discrepancy between the numerical and the exact asymptotic results is a finite $L$ effect. The solid line is a guide to the eyes indicating the expected algebraic behavior $\propto x^{-c}$, see Eq. (48).

Figure 5

Left: Scaled plot of the PDF of $l_{max}, L{p}_1\left(x\right|L)$ as a function of $x/L$ for $L=100$ and $L=1000$ at criticality and for $c=1.3$. The good collapse of the data [obtained from numerical simulations, (see Appendix pp3 for details)], for these two different values of $L$, corroborate the scaling form given in Eq. (63). The singularity for $x/L=1/2$ is clearly visible on this plot, while other singularities (of higher order and hence not visible on this plot) also exist for $x/L=1/k$, with $k=3,4,...$. Right: Scaled plot of the PDF of $l_{max}, L{p}_1\left(x\right|L)$ as a function of $x/L$ for $L=4000$ at criticality and for $c=1.3$. The diamond symbols correspond to a numerical evaluation of $p_1\left(x\right|L)$ while the blue solid line, for $x/L\ge 1/2$ corresponds to the exact result given in Eqs. (63) and (60). The dotted line for $y<1/2$ corresponds to the asymptotic behavior for $y\to 0$, given in Eq. (64) (and in this case ${\alpha }_0\approx 1.582$).

Figure 6

Scaled plot of $p_1\left(x\right|L)L^{1/(1-c)}$ as a function of $x/L^{1/(c-1)}$, for $L=2000$, according to Eq. (15)—we recall that $p_1\left(x\right|L)=P_1\left(x\right|L)-P_1(x-1|L)$—for $c=3$ at the critical temperature [see Eq. (9)]. The solid line corresponds to the Fréchet distribution with parameters specified in Eq. (15).