In recent years, random lasing materials (e.g. powders, porous media, precipitates in solution, or photonic crystals with impurities) have been extensively studied experimentally. Pumping energy into these systems causes them to re-emit multi-mode coherent light, with a spectrum displaying randomly arranged peaks in frequency. Starting from the structure and geometry of the atoms and molecules that scatter the light waves, one would eventually want a theory that predicts the onset, the nature and the features of the light modes and answering the following questions.

- What shape and size do light modes display in space ?
- In which dimension and under which conditions do they localize because of disorder?
- On which frequencies do light modes emit in cavity-less media?
- Can there be a random laser pulse in time?
- Do competing random laser modes phase-lock as in multimode standard lasers?
- How strong is the coupling magnitude and how is it related to the coupling modes spatial overlap and the etherogeneous optical susceptibility?

The latter two questions are connected to the coupling property of depending on the spatial overlap of the electromagnetic fields of the interacting modes. This feature ascribes to the problem of assessing the structure of an interacting network of light-modes in a statistical mechanics representation. Indeed, a set of modes can interact only if their electromagnetic fields overlap in space and, in the lasing regime, non-linear amplification occurs only if the frequencies of the modes satisfy some kind of mode-locking condition. These rules strongly influence the set of feasible interactions in which each mode is viewed as a network node. A key challenge that we address is the characterization of the structure of this network of wave-modes, including the strengths and signs of the relevant random interactions, as is required, e.g., in order to distinguish apart physical regimes of laser stationary behaviour. To this aim a Hamiltonian theory has been derived and investigated in systems with different kinds of bond-disorder, ranging from standard ordered multimode mode-locking lasers to recently introduced glassy random lasers.

**Glassy Random Laser and Experimental Measurement of Replica Symmetry Breaking **

The investigation of the glassy behaviour of light in the framework of our theory is made possible by means of a newly introduced overlap parameter, the *Intensity Fluctuation Overlap* (IFO) measuring the correlation between intensity fluctuations of waves in random media. This order parameter allows to identify the laser transition in arbitrary physical regimes, with varying amount of disorder and non-linearity. In particular, in random media it allows for the identification of the glassy nature of some kind of random laser, in terms of emission spectra data, the only data so far accessible in random laser measurements. The model devised from first principles in whose framework the parameter is defined is the nonlinear phasor statistical mechanical model. This is a generalised complex spherical spin-glass model solvable in the mean-field approximation by *Replica Symmetry Breaking* theory. IFO measurements are possible in real experiments, recently leading to a validation of the RSB theory and a new characterisation of lasers in terms of spectral intensity fluctuations.

**Interference of Coupling of Waves in Random Media **

The light modes interaction network has to be inferred starting from data acquired in measurements, of spectra and correlations of phases and amplitudes of the light modes, and this inference problem is closely analogous to those in our other areas of application of *statistical inference*. Starting with the analysis of the *inverse problem* in statistical mechanical systems with continuous variables, like XY and complex phasors, our inference project is concerned with the bottom-up approach for studying statistical models for application to wave and optics. The parameters describing a given model system, like active links in the network system and external field affecting the system, are inferred using the data set which is made available by experimental or numerical measurements.

We adopt various inference techniques to reconstruct the interaction networks and to estimate the coupling values: mean-field approach, Pseudo Likelihood Maximization (PLM) with L1 and L2 regularizations and PLM with decimation. Such inverse problems for network reconstruction are considered on graphs of different kinds, from 2D and 3D nearest-neighbour lattices, Bethe and Erdos-Renyi sparse random graph to dense random graphs.

- F Antenucci,
**Statistical Physics of Wave Interactions**, Springer (2016).
- P Tyagi, A Marruzzo, A Pagnani, F Antenucci, L Leuzzi,
**Regularization and decimation pseudolikelihood approaches to statistical inference in XY-spin models**, Physical Review B 94, 024203 (2016) Doi: 10.1103/PhysRevB.94.024203.
- F Antenucci, A Crisanti, M Ibáñez-Berganza, A Marruzzo, L Leuzzi,
**Statistical mechanics models for multimode lasers and random lasers.** Philosophical Magazine 96, 704-731 (2016) Doi: 10.1080/14786435.2016.1145359.
- F Antenucci, MI Berganza, L Leuzzi,
**Statistical physics of nonlinear wave interaction,** Physical Review B 92, 014204 (2015) Doi: 10.1103/PhysRevB.92.014204 .
- P Tyagi, A Pagnani, F Antenucci, M Ibanez Berganza, L Leuzzi,
**Inference for interacting linear waves in ordered and random media**, Journal of Statistical Mechanics: Theory and Experiment 2015 (5), Doi: 10.1088/1742-5468/2015/05/P05031
- F Antenucci, A Crisanti, L Leuzzi,
**Complex spherical 2+ 4 spin glass: A model for nonlinear optics in random media**, Physical Review A 91, 053816 (2015) Doi: 10.1103/PhysRevA.91.053816.
- F Antenucci, MI Berganza, L Leuzzi,
**Statistical physical theory of mode-locking laser generation with a frequency comb**. Physical Review A 91, 043811 (2015) Doi: 10.1103/PhysRevA.91.043811 .
- A Marruzzo, L Leuzzi,
**Nonlinear XY and p-clock models on sparse random graphs: Mode-locking transition of localized waves,** Physical Review B 91, 054201 (2015) Doi:10.1103/PhysRevB.91.054201 .
- F Antenucci, C Conti, A Crisanti, L Leuzzi,
**General phase diagram of multimodal ordered and disordered lasers in closed and open cavities.** Physical Review Letters 114, 043901 (2015) Doi: 10.1103/PhysRevLett.114.043901 .
- N Ghofraniha, I Viola, F Di Maria, G Barbarella, G Gigli, L Leuzzi, C Conti,
**Experimental evidence of replica symmetry breaking in random lasers,** Nature communications 6, 5 (2015) Doi:10.1038/ncomms7058 .
- F Antenucci, A Crisanti, L Leuzzi,
**The glassy random laser: replica symmetry breaking in the intensity fluctuations of emission spectra**, Scientific reports 5, 16792 (2015) Doi:10.1038/srep16792 .
- F Antenucci, M Ibanez Berganza, L Leuzzi,
**Statistical mechanical theory of mode-locked multimode lasers in closed cavity: determination of thresholds, spectra, pulse phase delays and pulse correlations.** Phys. Rev. A 91, 043811 (2014) Doi: 10.1103/PhysRevA.91.043811.

**Other Selected Publications**

- V Folli, A Puglisi, L Leuzzi, C Conti,
**Shaken Granular Lasers,** Physical Review Letters 108, 248002 (2012) Doi: 10.1103/PhysRevLett.108.248002.
- L Leuzzi, C Conti, V Folli, L Angelani, G Ruocco,
**Phase Diagram and Complexity of Mode-Locked Lasers: From Order to Disorder,** Physical Review Letters 102, 083901 (2009) Doi:10.1103/PhysRevLett.102.083901 .

**Statistical mechanics of disordered granular laser systems: theory and experiment,**” funded by the Italian Ministry of Research (MIUR) program *futuro in ricerca*. (2010-2015),

**NETADIS: **Networks across disciplines, FP7-PEOPLE-2011-ITN Project (2011-2015).