By Vladimir Kazakov (auth.), Vadim Malyshev, Anatoly Vershik (eds.)

New and outstanding effects acquired lately from a thorough research of asymptotic combinatorics have ended in a brand new, larger point of knowing of comparable difficulties: the idea of integrable platforms, the Riemann-Hilbert challenge, asymptotic illustration conception, spectra of random matrices, combinatorics of younger diagrams and variations, or even a few features of quantum box theory.

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1991) String Theory in Two Dimensions, Lectures delivered at the ICTP Spring School on String Theory and Quantum Gravity, Trieste, April 1991; hep-th/9108019. (Review sur la corde a une dimension - surfaces aleatoires en 1 dimension, les approches matriciel et conforme) Boulatov, D. and Kazakov, (1993) One dimensional string theory with vortices as an upside down matrix oscillator, J. Mod. Phys. A, 8, 809; hep-th/0012228. Kazakov, V. and Migdal, A. (1988) Recent Progress in the Theory of Non-critical Strings, Nucl.

The ground state wave function is a unitary singlet, and thus a function of the N eigenvalues Ai alone. N (Ai - Aj). the scalar product (119) takes the form (\]I 11'1/'2) = J dNA Ll 2 (A) \]I 1 (A) \]12 (A) (120) and therefore if we introduce the wave function X(A) = Ll(A)\]I(A) (121) the scalar product becomes (122) and the Schr6dinger equation for the ground state reads (123) 48 The ground state wave function w(A) is also invariant under permutation of the A's (permutations are particular unitary transformations) and therefore X(A) is totally antisymmetric.

J dAe- NV (A)Pm(A)A 8Pn(A) = [Q8]n,m, and dAe-NV(A)Pm(A)V'(A)Pn(A) = [V'(Q)]n,m). The relation (60), plus the antisymmetry of P, fixes this matrix completely. e. M is of type [a, b] if Mij vanishes for i > j + a or i < j - b. The matrix Q is [1,1], thus QS of type [s, s], and then from (60) and the antisymmetry, P is of type [2k - 1, 2k -1], if 2k is the degree of the potential. 37 Finally from the definitions (49) and (57) follows the commutation relation [Q,P] = 1 (61) which is central to the construction (it follows immediately from the differentiation with respect to A of Acpn(A) = L:m Qn,mCPm(A) ).