Opportunity for Regulating the Collective Effect of Random Expansion with Manifestations of Finite Size Effects in a Moderate Number of Finite Systems

One reports computational study revealing a set of general requirements, fulfilling of which would allow employing changes in ambient conditions to regulate accomplishing the collective outcome of emerging active network patterns in an ensemble of a moderate number of finite discrete systems. The patterns within all these component systems emerge out of random expansion process governed by certain local rule. The systems modeled are of the same type but different in details, finite discrete spatial domains of the expansion within the systems are equivalent regular hexagonal arrays. The way in which elements of a component system function in the local information transmission allows dividing them into two classes. One class is represented by zero-dimensional entities coupled into pairs identified at the array sites being nearest neighbors. The pairs preserve their orientation in the space while experiencing conditional hopping to positions close by and transferring certain information portions. Messenger particles hopping to signal the pairs for the conditional jumping constitute the other class. Contribution from the hopping pairs results in finite size effects being specific feature of accomplishing the mean expected network pattern representing the collective outcome. It is shown how manifestations of the finite size effects allow using changes in parameters of the model ambient conditions of the ensemble evolution to regulate accomplishing the collective outcome representation.Comment: 22 pages, 10 eps figures, corrected URL address placing in text, minor editorial correction in sec.2, author e-mail change

On condensation properties of Bethe roots associated with the XXZ chain

I prove that the Bethe roots describing either the ground state or a certain class of "particle-hole" excited states of the XXZ spin-$1/2$ chain in any sector with magnetisation $\mathfrak{m} \in [0;1/2]$ exist and form, in the infinite volume limit, a dense distribution on a subinterval of $\mathbb{R}$. The results holds for any value of the anisotropy $\Delta \geq -1$. In fact, I establish an even stronger result, namely the existence of an all order asymptotic expansion of the counting function associated with such roots. As a corollary, these results allow one to prove the existence and form of the infinite volume limit of various observables attached to the model -the excitation energy, momentum, the zero temperature correlation functions, so as to name a few- that were argued earlier in the literature.Comment: 54 pages, 2 figures. Some details in proof adde

On lacunary Toeplitz determinants

By using Riemann--Hilbert problem based techniques, we obtain the asymptotic expansion of lacunary Toeplitz determinants $\det_N\big[ c_{\ell_a-m_b}[f] \big]$ generated by holomorhpic symbols, where $\ell_a=a$ (resp. $m_b=b$) except for a finite subset of indices $a=h_1,\dots, h_n$ (resp. $b=t_1,\dots, t_r$). In addition to the usual Szeg\"{o} asymptotics, our answer involves a determinant of size $n+r$.Comment: 11 page