We prove a generalization of Kirchhoff's matrix-tree theorem in which large class combinatorial objects are represented by non-Gaussian Grassmann integrals. As special case, we show that unrooted spanning forests, arise as q-->0 limit the Potts model, can be theory involving Gaussian term and particular bilocal four-fermion term. this latter model mapped, to all orders perturbation theory, onto N-vector at N=-1 or, equivalently, sigma taking values unit supersphere R(1|2). It follows that,...

We study typical case properties of the 1-in-3 satisfiability problem, Boolean satisfaction where a clause is satisfied by exactly one literal, in an enlarged random ensemble parametrized average connectivity and probability negation variable clause. Random exact 3-cover are special cases this ensemble. interpolate between these from region can be typically decided for all connectivities polynomial time to deciding hard, some interval connectivities. derive several rigorous results first...

Recent work on percolation in d=2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> [J. Phys. A: Math. Theor. 55, 204002 (2015)] introduced an operator that gives a weight k^{\ell} display="inline"><mml:msup><mml:mi>k</mml:mi><mml:mi>ℓ</mml:mi></mml:msup></mml:math> to configurations with \ell display="inline"><mml:mi>ℓ</mml:mi></mml:math> ‘nested paths’ (NP), i.e. disjoint cycles...

We prove that, for X , Y A and B matrices with entries in a non-commutative ring such that \hbox{$[X_{ij},Y_{k\ell}]=-A_{i\ell} B_{kj}$}, satisfying suitable commutation relations (in particular, is Manin matrix), row-pseudo-commutative matrix (a the following identity holds: \mathrm {col-det } \ { col-det = \langle 0\mid (aA + (I-a^{\dagger} B)^{-1} Y)\mid 0\rangle Furthermore, if also matrix, [Y_{ij},Y_{kl}]=0 i\neq k j\neq l =\int \mathcal{D}(\psi, \bar{\psi}) \exp \big(\sum_{k \geq...

The Abelian Sandpile generates complex and beautiful patterns seems to display allometry. On the plane, beyond patches, periodic in both dimensions, we remark presence of structures one dimension, that call strings. We classify completely their constituents terms principal vector k, momentum. derive a simple relation between momentum string its density particles, E, which is reminiscent dispersion relation, E=k^2. Strings interact: they can merge split within these processes conserved....

Since the work of Creutz, identifying group identities for Abelian sandpile model (ASM) on a given lattice is puzzling issue: rectangular portions complex quasi-self-similar structures arise. We study ASM square lattice, in different geometries, and variant with directed edges. Cylinders, through their extra symmetry, allow an easy determination identity, which homogeneous function. The geometry shows remarkable exact structure, asymptotically self-similar.

We introduce several infinite families of critical exponents for the random-cluster model and present scaling arguments relating them to $k$-arm exponents. then Monte Carlo simulations confirming these predictions. These provide a convenient way determine from simulations. An understanding also leads radically improved implementation Sweeny algorithm. In addition, our data allow us conjecture an exact expression shortest-path fractal dimension ${d}_{\text{min}}$ in two dimensions:...

The height probabilities for the recurrent configurations in Abelian Sandpile Model on square lattice have analytic expressions, terms of multidimensional quadratures. At first, these quantities been evaluated numerically with high accuracy, and conjectured to be certain cubic rational-coefficient polynomials 1/pi. Later their values determined by different methods. We revert direct derivation probabilities, computing analytically corresponding integrals. Yet another time, we confirm...

We prove, by simple manipulation of commutators, two noncommutative generalizations the Cauchy–Binet formula for determinant a product. As special cases we obtain elementary proofs Capelli identity from classical invariant theory and Turnbull's Capelli-type identities symmetric antisymmetric matrices.

We compute the two-point correlation function for spin configurations which are obtained by solving Euclidean matching problem, one family of points on a grid, and second chosen uniformly at random, when cost depends power p distance.We provide analytic solution in thermodynamic limit, number cases (p > 1 open b.c. = 2 periodic b.c., both criticality), analyse numerically other parts phase diagram.

The present paper is part of our ongoing work on supersymmetric σ-models, their relation with the Potts model at q = 0 and spanning forests, rigorous analytic continuation partition function as an entire N − 2M, a feature first predicted by Parisi Sourlas in 1970s. Here we accomplish two main steps. First, analyze detail role Ising variables that arise when constraint solved, point out situations which forest decouple. Second, establish for some special cases: underlying graph forest,...

Given a hypergraph G, we introduce Grassmann algebra over the vertex set and show that class of integrals permits an expansion in terms spanning hyperforests. Special cases provide generating functions for rooted unrooted (hyper)forests (hyper)trees. All these results are generalizations Kirchhoff's matrix-tree theorem. Furthermore, describing is induced by theory with underlying OSP(1|2) supersymmetry.

In the six-vertex model with domain wall boundary conditions, emptiness formation probability is that a rectangular region in top left corner of lattice frozen. We generalize this notion to case where frozen has shape generic Young diagram. derive here multiple integral representation for correlation function.

We study the generating function of rooted and unrooted hyperforests in a general complete hypergraph with n vertices by using novel Grassmann representation their functions. show that this new approach encodes known results about exponential functions for different number vertices. consider also some applications as counting k-uniform one hyperedges all dimensions. Some feature asymptotic regimes large connected components is discussed.